Abstract
We present a new generalization of the nonlinear variational wave equation. We prove existence of local, smooth solutions for this system. As a limiting case, we recover the nonlinear variational wave equation.
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1 Introduction
The nonlinear variational wave equation (NVW) is given by
where \(u=u(t,x)\), \(t\ge 0\) and \(x\in {\mathbb {R}}\), with initial data
In this paper we modify (1.1) by adding two transport equations and coupling terms. The resulting system is given by
with initial data
It is clear that when \(\rho =\sigma =0\) we recover (1.1). We are interested in studying conservative solutions of the initial value problem (1.3), (1.4) for initial data \(u_{0}\), \(u_{0,x}\), \(u_{1}\), \(\rho _{0}\), \(\sigma _{0}\in L^{2}({\mathbb {R}})\).
We assume that \(c\in C^{2}({\mathbb {R}})\) and satisfies
for some \(\kappa \ge 1\). In addition, we assume that
for positive constants \(k_{1}\) and \(k_{2}\).
The NVW equation was introduced by Saxton in [16], where it is derived by applying the variational principle to the functional
The equation appears in the study of liquid crystals, where it describes the director field of a nematic liquid crystal, and where the function c is given by
where \(\alpha \) and \(\beta \) are positive physical constants. We refer to [13, 16] for information about liquid crystals, and the derivation of the equation.
A key property of (1.1), and hence also (1.3), is that solutions can loose regularity in finite time, even for smooth initial data, see [10]. The loss of regularity is due to the formation of singularities in the derivatives of u. A singularity means that either \(u_{x}\) or \(u_{t}\) becomes unbounded pointwise while u remains continuous. Therefore, one has to consider weak solutions of (1.1). For smooth solutions of the NVW equation, the energy
is independent of time. In the general case, the singularities in the derivatives are characterized by the fact that \(u_{x}(t,\cdot )\) and \(u_{t}(t,\cdot )\) remain in \(L^{2}({\mathbb {R}})\) after they become pointwise unbounded. Moreover, we have concentration of energy at points where the derivative blows up. Thus, it is reasonable to look for weak solutions with bounded energy. This naturally leads to the two following notions of solutions: dissipative and conservative solutions. The difference between these solutions comes from the continuation after the formation of a singularity. For dissipative solutions the energy decreases at the blow-up time, see [6, 17,18,19], while for conservative solutions the energy remains unchanged. In the latter case a semigroup of solutions has been constructed in [8, 12].
Uniqueness of weak solutions to the NVW equation is a delicate subject, as the characteristics in general are not unique. The uniqueness of conservative solutions has been studied in [2, 5], where uniqueness is established for the solutions constructed in [8] given that certain conditions hold, which yield unique characteristics.
A result on the regularity of conservative solutions to (1.1) is established in [3]. For initial data satisfying certain smoothness conditions, it is shown that the solution u is piecewise smooth and that the derivative \(u_{x}\) can become pointwise unbounded at finitely many characteristics. An asymptotic description of these solutions in a neighborhood of the singularities is shown in [7]. Moreover, in this setting a Lipschitz metric for conservative solutions has been constructed in [4].
We will use the approach from [12] for (1.1) to obtain conservative solutions of (1.3).
For the moment, we assume that u, \(\rho \), and \(\sigma \) are smooth solutions of (1.3). Then the energy is given by
and is independent of time. Next, we introduce the functions R and S defined as
Note that R and S are smooth by assumption. Using (1.8) we can express the energy in (1.7) as
From (1.3) we obtain
which yields
Combining (1.10) and (1.11), we finally obtain
In the view of (1.10), we interpret \(R^{2}+c(u)\rho ^{2}\) and \(S^{2}+c(u)\sigma ^{2}\) as the left and right traveling part of the energy density, respectively. Moreover, the right-hand sides of the two equations in (1.10) are equal up to the sign, which means that the right and the left part can interact with each other. That is, energy can swap back and forth between the two parts, while the total energy remains unchanged because of (1.12).
Following [12], we add to the initial data, in order to allow for energy concentration, two positive Radon measures \(\mu _{0}\) and \(\nu _{0}\), such that the absolutely continuous parts equal the classical energy densities, i.e., \((\mu _{0})_{\text {ac}}=\frac{1}{4}(R_{0}^{2}+c(u_{0})\rho _{0}^{2})\,dx\) and \((\nu _{0})_{\text {ac}}=\frac{1}{4}(S_{0}^{2}+c(u_{0})\sigma _{0}^{2})\,dx\). The singular parts of the measures on the other hand contain information about the concentration of energy. We denote the set of initial data by \({\mathcal {D}}\), which consists of tuples \((u,R,S,\rho ,\sigma ,\mu ,\nu )\).
The construction of a semigroup of weak, global, conservative solutions of (1.3) follows to a large extent the procedure for the NVW equation in [12] and is presented in Sects. 1–6. We state the results in Sects. 1–6 without proof, since the proofs are slight modifications of the ones in [12]. The details can be found in [15].
Our main results, which are contained in Sect. 7, are the following. We consider smooth initial data \(u_{0}\), \(R_{0}\), \(S_{0}\), \(\rho _{0}\), and \(\sigma _{0}\), and absolutely continuous measures \(\mu _{0}\) and \(\nu _{0}\) on a finite interval \([x_{l},x_{r}]\). If \(\rho _{0}\) and \(\sigma _{0}\) are strictly positive on this interval, then for every time \(t\in \big [0,\frac{1}{2\kappa }(x_{r}-x_{l})\big ]\), the solutions \(\rho (t,x)\) and \(\sigma (t,x)\) will also be strictly positive for all \(x\in [x_{l}+\kappa t,x_{r}-\kappa t]\). That is, the strict positivity of \(\rho _{0}\) and \(\sigma _{0}\) is preserved. This has a regularizing effect on the solution at time \(t\in \big [0,\frac{1}{2\kappa }(x_{r}-x_{l})\big ]\) in the sense that u(t, x), R(t, x), S(t, x), \(\rho (t,x)\), and \(\sigma (t,x)\) are smooth, and the measures \(\mu (t)\) and \(\nu (t)\) are absolutely continuous for all \(x\in [x_{l}+\kappa t,x_{r}-\kappa t]\), see Theorem 7.1. The region where regularity holds is bounded by the backward and forward characteristics as indicated in Fig. 1.
Characteristics of the NVW equation. The forward characteristic \(x_{1}(t)\) starting from \(x_{l}\) is given by \(x_{1,t}(t)=c(u(t,x_{1}(t)))\), \(x_{1}(0)=x_{l}\), and the backward characteristic \(x_{2}(t)\) starting from \(x_{r}\) is given by \(x_{2,t}(t)=-c(u(t,x_{2}(t)))\), \(x_{2}(0)=x_{r}\). Because of (1.5), they intersect at a time t such that \(\frac{1}{2\kappa }(x_{r}-x_{l})\le t\le \frac{\kappa }{2}(x_{r}-x_{l})\)
In Theorem 7.9, we consider a sequence of smooth solutions \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})\) with initial data satisfying \(u_{0}^{n}\rightarrow u_{0}\) in \(L^{\infty }([x_{l},x_{r}])\), \(R_{0}^{n}\rightarrow R_{0}\), \(S_{0}^{n}\rightarrow S_{0}\), \(\rho _{0}^{n}\rightarrow 0\), \(\sigma _{0}^{n}\rightarrow 0\) in \(L^{2}([x_{l},x_{r}])\), where \(u_{0}\), \(R_{0}\) and \(S_{0}\) are smooth, and the associated measures \(\mu _{0}\) and \(\nu _{0}\) are absolutely continuous on \([x_{l},x_{r}]\). Then, \(u^{n}(t,\cdot )\rightarrow u(t,\cdot )\) in \({L^\infty }([x_{l}+\kappa t,x_{r}-\kappa t])\) for all \(t\in \big [0,\frac{1}{2\kappa }(x_{r}-x_{l})\big ]\), where u is a solution of the NVW equation with initial data \((u_{0},R_{0},S_{0},\mu _{0},\nu _{0})\). A central ingredient in the proof is a Gronwall inequality in two variables, see [9].
We point out that these are local results. The main reason for this is that we require the initial data \(\rho _{0}\) and \(\sigma _{0}\) corresponding to the Eqs. (1.3b) and (1.3c) to be bounded from below by a strictly positive constant and to belong to \(L^{2}({\mathbb {R}})\), which is not possible globally.
We hope that further studies of the smooth approximations will be helpful in the understanding of singularities to (1.1).
The motivation for studying (1.3) comes from the two-component Camassa–Holm (2CH) system
where \(\kappa \in {\mathbb {R}}\) and \(\eta \in (0,\infty )\) are given numbers. In [11], global, weak, conservative solutions of (1.13) are constructed. It is shown that the solution of (1.13) is regular if initially \(\rho _{0}\) is strictly positive. Moreover, a sequence of regular solutions, with \(\rho _{0}^{n}\rightarrow 0\), converges in \({L^\infty }({\mathbb {R}})\) to the global, weak conservative solution of the CH equation, which corresponds to \(\rho =0\).
Loosely speaking, since the CH equation has one family of characteristics, see Fig. 2, an extra variable \(\rho \) is needed to preserve the positivity of \(\rho _{0}\) in the characteristic direction. Since the NVW equation has both forward and backward characteristics, we need two extra variables \(\rho \) and \(\sigma \) to preserve the positivity of \(\rho _{0}\) and \(\sigma _{0}\) in each characteristic direction. We emphasize that the system (1.3) is constructed in order to have similar properties as (1.13), and is not derived from physical considerations.
We mention that a regularizing system for the Hunter–Saxton equation has been studied in [14].
Next we give a brief outline of the used method.
As for the classical wave equation, (1.3) has two families of characteristics: forward and backward characteristics. The backward characteristics transport the energy described by the measure \(\mu \), while the forward characteristics transport the energy described by the measure \(\nu \). We interpret the characteristics as particles. At points where the measures are nonsingular there is a finite amount of energy, and there is exactly one forward and one backward characteristic starting from that point. This particle is mapped to one point in the new coordinates (X, Y).
Characteristics of the CH equation. The characteristic \(x_{1}(t)\) starting from \(x_{l}\) is given by \(x_{1,t}(t)=u(t,x_{1}(t))\), \(x_{1}(0)=x_{l}\), and the characteristic \(x_{2}(t)\) starting from \(x_{r}\) is given by \(x_{2,t}(t)=u(t,x_{2}(t))\), \(x_{2}(0)=x_{r}\)
At a point where one of the measures is singular and the other is not, there is an infinite amount of energy. Hence, there are infinitely many characteristics corresponding to the singular measure starting from that point, while the nonsingular measure yields one characteristic. This single point is mapped to a horizontal or vertical line in the new coordinates, depending on which measure is singular.
The situation where both measures are singular at a point, means that there is an infinite amount of both backward and forward energy at that point. Infinitely many characteristics of both types start out from that point, and all these particles correspond to a box in the (X, Y)-plane.
The derivation of the system of equations corresponding to (1.3) in the new coordinates is illustrated by assuming that u is smooth, and \(\mu \) and \(\nu \) are absolutely continuous. Then, the method of characteristics yields solutions X and Y of the equations
The operators acting on X and Y are the two factors \(\frac{\partial }{\partial t}-c(u)\frac{\partial }{\partial x}\) and \(\frac{\partial }{\partial t}+c(u)\frac{\partial }{\partial x}\) corresponding to the highest order derivatives in (1.3a). This defines new coordinates (X, Y). The characteristics for the equations in (1.14) are given by
respectively, for some starting value \(x(0)=x_{0}\). Note that X and Y are constant along characteristics, meaning that particle paths are mapped to straight lines.
Considering the original variables t and x as functions of X and Y, we define \(U(X,Y)= u(t(X,Y),x(X,Y))\), \(P(X,Y)=\rho (t(X,Y),x(X,Y))\), \(p=Px_{X}\), \(Q(X,Y)=\sigma (t(X,Y),x(X,Y))\), and \(q=Qx_{Y}\). We introduce functions J and K, where J corresponds to the energy distribution in the new coordinates. Denoting \(Z=(t,x,U,J,K)\), we end up with the identities
and a semilinear system of equations
where F(Z) is a bilinear and symmetric tensor from \({\mathbb {R}}^5\times {\mathbb {R}}^5\) to \({\mathbb {R}}^5\).
Moreover, we get two additional equations
which correspond to (1.3b) and (1.3c). From (1.15c) we see that solutions of (1.16) corresponding to (1.1) and (1.3) are not identical. In particular, from (1.15c) we see that the solutions t, x, U, J, K of (1.16) are not independent of the solutions p, q of (1.17). From (1.15) it is clear that the vector Z consists of dependent and independent elements. A fixed point argument is used to prove existence of solutions to the system. This requires a curve \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\) parametrized by \(s\in {\mathbb {R}}\) in the (X, Y)-plane that corresponds to the initial time \(t=0\). In the smooth case, the set of points \((X,Y)\in {\mathbb {R}}^{2}\) such that \(t(X,Y)=0\) defines this curve, which is strictly monotone. For general initial data this set is the union of strictly monotone curves, horizontal and vertical lines, and boxes. In the case of a box there are in principle infinitely many possible ways of choosing the curve. One has to take this into account when defining initial data in the Lagrangian coordinates.
The initial data in \({\mathcal {D}}\) is mapped to the Lagrangian variables in \({\mathcal {G}}_{0}\) in two steps, see Sect. 3. First we define a map \({\textbf{L}}\) from \({\mathcal {D}}\) to the set \({\mathcal {F}}\), consisting of elements \(\psi =(\psi _{1},\psi _{2})\) where \(\psi _{1}(X)\) and \(\psi _{2}(Y)\) are six dimensional vectors.
Loosely speaking, the map \({\textbf{L}}\) yields the value of Z and its derivatives, and p and q in each characteristic direction, i.e., in the X and the Y direction. Linking the values of \(\psi _{1}\) and \(\psi _{2}\) yields the set of points in the (X, Y)-plane where time equals zero. For instance, in the case of initial data where both measures are singular at the same point, this set is a box.
The next map picks one curve \(({\mathcal {X}},{\mathcal {Y}})\) from the set where time equals zero, and sets the value of Z and its derivatives, and p and q on the curve. This map is denoted by \({\textbf{C}}\) and maps \({\mathcal {F}}\) to the set \({\mathcal {G}}_{0}\), which is the set of elements \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}}, {\mathfrak {p}},{\mathfrak {q}})\) corresponding to time equals zero. An element \(\Theta \) consists of the initial curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) parametrized by \(s\in {\mathbb {R}}\), and yields the value of Z, \(Z_{X}\), \(Z_{Y}\), p and q on the curve. This means that \({\mathcal {Z}}(s)=Z({\mathcal {X}}(s),{\mathcal {Y}}(s))\), \({\mathcal {V}}(X)=Z_{X}(X,{\mathcal {Y}}({\mathcal {X}}^{-1}(X)))\), \({\mathcal {W}}(Y)=Z_{Y}({\mathcal {X}}({\mathcal {Y}}^{-1}(Y)),Y)\), \({\mathfrak {p}}(X)=p(X,{\mathcal {Y}}({\mathcal {X}}^{-1}(X)))\) and \({\mathfrak {q}}(Y)=q({\mathcal {X}}({\mathcal {Y}}^{-1}(Y)),Y)\), and we write \(\Theta =(Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\). The functions \({\mathcal {Z}}\), \({\mathcal {V}}\), \({\mathcal {W}}\), \({\mathfrak {p}}\) and \({\mathfrak {q}}\) belong, with some modifications, to \({L^\infty }({\mathbb {R}})\). The set \({\mathcal {C}}\), see Definition 3.2, consists of all curves \(({\mathcal {X}},{\mathcal {Y}})\) such that the functions \({\mathcal {X}}\) and \({\mathcal {Y}}\) are continuous, nondecreasing, and have finite distance to the identity. Moreover, the functions satisfy \({\mathcal {X}}+{\mathcal {Y}}=2{{\,\textrm{Id}\,}}\). In the case of a box, the map \({\textbf{C}}\) picks the curve consisting of the left vertical side and the upper horizontal side of the box.
Existence and uniqueness of solutions to (1.16) with initial data \(\Theta \) follow from a fixed point argument. The solution is first constructed on small rectangular domains \(\Omega \) in the (X, Y)-plane, where the initial curve \(({\mathcal {X}},{\mathcal {Y}})\) connects the lower left corner with the upper right corner of the rectangle, see Theorem 4.8. Here, a solution basically means that Z, \(Z_{X}\), \(Z_{Y}\), p and q are pointwise bounded in the box, and that (1.16) and (1.17) are satisfied almost everywhere in \(\Omega \), see Definition 4.5. We consider solutions satisfying some additional properties, i.e., they satisfy the identities in (1.15) and some monotonicity conditions. These properties are contained in Definition 4.7. The set of such solutions is denoted by \({\mathcal {H}}(\Omega )\). For initial data \(\Theta \in {\mathcal {G}}_{0}\) we have \({\mathcal {V}}_{2}+{\mathcal {V}}_{4}>0\) and \({\mathcal {W}}_{2}+{\mathcal {W}}_{4}>0\) almost everywhere. This property is preserved in the solution and is important in proving that the solution operator from \({\mathcal {D}}\) to \({\mathcal {D}}\) is a semigroup.
A pointwise uniform bound on the functions Z, \(Z_{X}\), \(Z_{Y}\), p and q in strip like domains containing small rectangles allows us to prove, by an induction argument, existence and uniqueness of solutions in \({\mathcal {H}}(\Omega )\) on arbitrarily large rectangular domains \(\Omega \), see Sect. 4.2.
If a function (Z, p, q) in \({\mathcal {H}}(\Omega )\) is a solution on any rectangular domain \(\Omega \), and there exists a curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) such that \((Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {G}}\), we say that (Z, p, q) is a global solution to (1.16) and (1.17), see Definition 4.13. Here, \({\mathcal {G}}\) is the analogue of \({\mathcal {G}}_{0}\), corresponding to time t different from zero. The set of global solutions is denoted by \({\mathcal {H}}\). The functions Z, \(Z_{X}\), \(Z_{Y}\), p and q are, with some modifications, bounded globally. In particular, the Lagrangian counterpart to the energy is bounded.
In Theorem 4.15 a global solution is constructed by using local solutions in boxes. The procedure is as follows. First we construct solutions on rectangles with diagonal points that lie on the initial curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\). These solutions are then used to construct initial data for adjacent rectangles, and we obtain solutions there as well. Continuing like this one obtains a global solution. We denote the solution map that to any initial data \(\Theta \in {\mathcal {G}}\) yields a unique solution \((Z,p,q)\in {\mathcal {H}}\) by \({\textbf{S}}\).
Having constructed a global solution \((Z,p,q)\in {\mathcal {H}}\), the goal is to map it back to Eulerian coordinates \({\mathcal {D}}\) for any time \(T>0\). As addressed before, the points \((X,Y)\in {\mathbb {R}}^{2}\) such that \(t(X,Y)=T\) may contain boxes. In order to use the sets previously defined for time equal to zero, we shift time to zero, i.e., for \((Z,p,q)\in {\mathcal {H}}\) we define \(({\bar{Z}},{\bar{p}},{\bar{q}})\in {\mathcal {H}}\) where \({\bar{t}}(X,Y)=t(X,Y)-T\). The other elements of \(({\bar{Z}},{\bar{p}},{\bar{q}})\) are identical to the ones in (Z, p, q). We call this map \({\textbf{t}}_{T}\). In the case of a box, the curve \(({\mathcal {X}},{\mathcal {Y}})\) corresponding to time T is defined by picking the left vertical side and the upper horizontal side of the box. The element \(\Theta \in {\mathcal {G}}_{0}\) is then defined as \(\Theta =({\bar{Z}},{\bar{p}},{\bar{q}})\bullet ({\mathcal {X}},{\mathcal {Y}})\), and we denote the map by \({\textbf{E}}:{\mathcal {H}}\rightarrow {\mathcal {G}}_{0}\). Because of the monotonicity of the function t, the curve corresponding to time T lies below the initial curve. For any \(\Theta \in {\mathcal {G}}_{0}\) we define a map \({\textbf{D}}\) that associates an element \(\psi \in {\mathcal {F}}\). The definition of these maps are contained in Sect. 5.1. The operator \(S_{T}={\textbf{D}}\circ {\textbf{E}}\circ {\textbf{t}}_{T}\circ {\textbf{S}}\circ {\textbf{C}}\) that for any initial data in \({\mathcal {F}}\) yields a solution in \({\mathcal {F}}\) corresponding to time \(T>0\), is a semigroup, see Theorem 5.6.
The remaining step back to Eulerian coordinates is the map \({\textbf{M}}:{\mathcal {F}}\rightarrow {\mathcal {D}}\), which yields an element \((u,R,S,\rho ,\sigma ,\mu ,\nu )(T)\in {\mathcal {D}}\) at time \(T>0\), see Definition 5.7. Thus, the map \({\bar{S}}_{T}={\textbf{M}}\circ S_{T}\circ {\textbf{L}}\) yields an element in \({\mathcal {D}}\) given any initial data in \({\mathcal {D}}\). If \({\textbf{L}}\circ {\textbf{M}}={{\,\textrm{Id}\,}}\), it follows from the semigroup property of \(S_{T}\) that also \({\bar{S}}_{T}\) is a semigroup. However, this identity does not hold in general. This is because for any element \(\psi \in {\mathcal {F}}\) we have \(x_{i}+J_{i}\in G\), where the group G is given by all invertible functions f such that \(f-{{\,\textrm{Id}\,}}\), \(f^{-1}-{{\,\textrm{Id}\,}}\in {W^{1,\infty }}({\mathbb {R}})\) and \((f-{{\,\textrm{Id}\,}})'\in L^{2}({\mathbb {R}})\), while the element \({\bar{\psi }}\in {\mathcal {F}}\) given by the map \({\textbf{L}}\) satisfies \({\bar{x}}_{i}+{\bar{J}}_{i}={{\,\textrm{Id}\,}}\). Therefore, in general one has \(\psi \ne {\bar{\psi }}\). To overcome this problem, one considers the following approach. Assume that \(x_{1}+J_{1}=f\) and \(x_{2}+J_{2}=g\) where \(f,g\in G\), and consider \({\tilde{\psi }}\) defined by relabeling such that \({\tilde{x}}_{1}=x_{1}\circ f^{-1}\), \({\tilde{J}}_{1}=J_{1}\circ f^{-1}\), \({\tilde{x}}_{2}=x_{2}\circ g^{-1}\) and \({\tilde{J}}_{2}=J_{2}\circ g^{-1}\). It then follows that \({\tilde{x}}_{i}+{\tilde{J}}_{i}={{\,\textrm{Id}\,}}\), \(i=1,2\). Introduce \(\phi =(f^{-1},g^{-1})\) and \({\tilde{\psi }}=\psi \cdot \phi \), which defines an action of \(G^{2}\) on the set \({\mathcal {F}}\). In particular, the transformation of \(\psi \) to \({\tilde{\psi }}\) defines a projection \(\Pi \) from \({\mathcal {F}}\) on the set
which contains exactly one element of each equivalence class of \({\mathcal {F}}\) with respect to \(G^{2}\), see Sect. 5.4. Thus, we have \({\tilde{\psi }}=\Pi (\psi )\). It turns out that the map \(S_{T}:{\mathcal {F}}\rightarrow {\mathcal {F}}\) is invariant under the group acting on \({\mathcal {F}}\), i.e., \(S_{T}(\psi \cdot \phi )=S_{T}(\psi )\cdot \phi \), where \(\phi \in G^{2}\), since all the maps which \(S_{T}\) is composed of, are invariant under the group action. The action of \(G^{2}\) on \({\mathcal {G}}\) and the set of curves \({\mathcal {C}}\) naturally follows from the definition of the action on \({\mathcal {F}}\). On the set of curves \({\mathcal {C}}\), the action corresponds to stretching the curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) in the X and Y direction. For the set \({\mathcal {H}}\), the action is defined such that it commutes with the \(\bullet \) operation.
A key result is that the map \({\textbf{M}}\) satisfies \({\textbf{M}}={\textbf{M}}\circ \Pi \). This implies that to each element in \({\mathcal {D}}\) there correspond infinitely many elements in \({\mathcal {F}}\), all belonging to the same equivalence class. The mapping \({\textbf{L}}:{\mathcal {D}}\rightarrow {\mathcal {F}}_{0}\) on the other hand picks one member of each equivalence class, but we could also pick a different one. Applying the solution operator to all elements belonging to the same equivalence class yields infinitely many solutions in \({\mathcal {F}}\), which form an equivalence class. Using the mapping \({\textbf{M}}:{\mathcal {F}}\rightarrow {\mathcal {D}}\) on all of these solutions yields the same element in \({\mathcal {D}}\). Since we get the same solution in the end, we can think of each member of the equivalence class as a different "parametrization" of the initial data in \({\mathcal {F}}\), which are connected through relabeling. Hence, the map \({\bar{S}}_{T}\) is a semigroup. Moreover, the solution produced by the map is a global weak solution of (1.3), see Sect. 6. It is conservative in the sense that for all \(T\ge 0\), \(\mu (T)({\mathbb {R}})+\nu (T)({\mathbb {R}})=\mu _{0}({\mathbb {R}})+\nu _{0}({\mathbb {R}})\), where \(\mu (T)\) and \(\nu (T)\) are the measures at time T, and \(\mu _{0}\) and \(\nu _{0}\) are the initial measures. This is a consequence of the fact that the energy function J in Lagrangian coordinates is such that the limit \(\lim _{s\rightarrow \pm \infty }J({\mathcal {X}}(s),{\mathcal {Y}}(s))\) is independent of the curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\), see Lemma 4.14. Thus, the same limiting values of J are obtained for curves corresponding to different times.
2 Equivalent system
In this section we formally derive a set of equations corresponding to (1.3) in new variables. We assume that u, \(\rho \), and \(\sigma \) are sufficiently smooth and bounded.
For the first equation in (1.14) we find the characteristics
where we assume that \(t(0,\xi )=0\) and \(x(0,\xi )=\xi \) for all \(\xi \in {\mathbb {R}}\). From the last equation in (2.1) we obtain
We compute the determinant of the Jacobian corresponding to the map \((s,\xi )\rightarrow (t,x)\) and get
Since \(0<x_{\xi }(s,\xi )<\infty \) we have from the inverse function theorem that the Jacobian corresponding to the map \((t,x)\rightarrow (s,\xi )\) satisfies
From (2.1) we get
so that \(s(t,x)=t\) and \(\xi _{t}(t,x)=-x_{s}(t,\xi (t,x))\xi _{x}(t,x)=c(u(t,\xi (t,x)))\xi _{x}(t,x)\).
Furthermore, (1.14) and (2.1) imply that \(X(t(s,\xi ),x(s,\xi ))=X(0,\xi )=g(\xi )\), for some strictly increasing function \(g\in C^1({\mathbb {R}})\). Differentiation, combined with (2.1) and (2.3) yields
which implies \(0<X_{t}<\infty \) and \(0<X_{x}<\infty \).
Next, we study Y(t, x) with the method of characteristics. We obtain, from (1.14), \(\frac{d}{ds}Y(t(s,\xi ),x(s,\xi ))=0\) with the characteristics given by
where we assume that \(t(0,\xi )=0\) and \(x(0,\xi )=\xi \) for all \(\xi \in {\mathbb {R}}\). If \(Y(0,\xi )=h(\xi )\) for some strictly increasing function \(h\in C^1({\mathbb {R}})\), then \(Y(s,x(s,\xi ))=h(\xi )\). As in the computations above we find
so that \(-\infty<Y_{t}<0\) and \(0<Y_{x}<\infty \), where
Now we consider the mapping from the (t, x)-plane to the (X, Y)-plane. The determinant of the Jacobian of this map reads
The inverse function theorem then implies that the Jacobian corresponding to the map \((X,Y)\rightarrow (t,x)\) satisfies
From the above equality many identities can be read off, and we only mention some of them. By using (1.14) and (2.8), we obtain
which imply
We observe from (2.9) that \(t_{X}\), \(t_{Y}\), \(x_{X}\), and \(x_{Y}\) are nonzero and finite.
Let \(U(X,Y)=u(t(X,Y),x(X,Y))\). We insert the derivatives of \(u(t,x)=U(X(t,x),Y(t,x))\) in (1.3a) and get
Due to (1.14) all second order derivatives of U drop out except for the term containing the mixed derivative \(U_{XY}\). We compute the remaining terms. From (1.14) and (2.9) we have
By differentiating (1.14) and using (1.8) and (2.9) we obtain
Thus, (2.11) is equivalent to
We introduce
which we think of as the left and right traveling part of the energy density in the new variables, respectively. Now (2.13) yields
We find it convenient to introduce the function K defined by
which satisfies
In view of (1.11) and (1.12) we can think of \(K_{X}\) and \(K_{Y}\) as the left and right traveling part of the second conserved quantity \(\frac{1}{c(u)}(R^{2}+c(u)\rho ^{2}-S^{2}-c(u)\sigma ^{2})\) in the new coordinates, respectively.
Next, let us derive the equations for \(t_{XY}\), \(x_{XY}\), \(J_{XY}\), and \(K_{XY}\). We have \(x_{XY}=x_{YX}\), which by using (2.10) is the same as \((c(U)t_{X})_{Y}=(-c(U)t_{Y})_{X}\). This leads to
We find the equation for \(x_{XY}\) by using (2.10) in \(t_{XY}=t_{YX}\), which yields \((\frac{x_{X}}{c(U)})_{Y}=(-\frac{x_{Y}}{c(U)})_{X}\) and finally
Using \(J_{XY}=J_{YX}\), \(K_{XY}=K_{YX}\), and (2.16) we get
and
Let \(\rho (t,x)=P(X(t,x),Y(t,x))\). By (1.3b) we get
and from (2.17) we see that this is the same as
We define \(p=Px_{X}\), so that
Let \(\sigma (t,x)=Q(X(t,x),Y(t,x))\). From (1.3c) we have
and by (2.17) we find
We define \(q=Qx_{Y}\), so that
Finally we end up with the following system of differential equations
We introduce the vector \(Z=(t,x,U,J,K)\). The system (2.19a)–(2.19e) then rewrites as
where F(Z) is a bilinear and symmetric tensor from \({\mathbb {R}}^5\times {\mathbb {R}}^5\) to \({\mathbb {R}}^5\). Due to the relations (2.10), either one of the equations in (2.19a) and (2.19b) is redundant: one could remove one of them, and the system would remain well-posed, and one retrieves t or x by using (2.10). Similarly, either one of the equations (2.19d) and (2.19e) becomes redundant by (2.16). However, we find it convenient to work with the complete set of variables, i.e., \(Z=(t,x,U,J,K)\).
To prove existence of solutions to (2.19), we need a curve \(({\mathcal {X}},{\mathcal {Y}})\) in the (X, Y)-plane that corresponds to the initial time, i.e., it consists of all points \((X,Y)\in {\mathbb {R}}^{2}\) such that \(t(X,Y)=0\). The aim of the next section is to define this curve for general initial data, and to assign values of Z, \(Z_{X}\), \(Z_{Y}\), p and q on the curve. Moreover, to solve (2.19), we will require that (2.10), (2.16), (2.18) together with some monotonicity properties hold on the curve, see Definition 4.7.
3 From Eulerian to Lagrangian coordinates
We first define the set \({\mathcal {D}}\), which consists of possible initial data corresponding to (1.3) in Eulerian coordinates.
Definition 3.1
The set \({\mathcal {D}}\) consists of the tuples \((u,R,S,\rho ,\sigma ,\mu ,\nu )\) such that \(u,R,S,\rho ,\sigma \in L^{2}({\mathbb {R}})\),
and \(\mu \) and \(\nu \) are finite positive Radon measures with
Note that this definition allows for initial data with concentrated energy.
In Lagrangian coordinates we will admit curves of the following type.
Definition 3.2
We denote by \({\mathcal {C}}\) the set of curves in the plane \({\mathbb {R}}^2\) parametrized by \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\) with \(s \in {\mathbb {R}}\), such that
with the normalization
We set
An essential role throughout the paper is played by G, the group of so-called relabeling functions.
Definition 3.3
The group G is given by all invertible functions f such that
and
Note that if f, \(g\in G\), then also \(f^{-1}\), \(g^{-1}\) and \(f\circ g\) belong to G. The next step is to map elements from \({\mathcal {D}}\) to a set \({\mathcal {F}}\), which is defined as follows.
Definition 3.4
The set \({\mathcal {F}}\) consists of all functions \(\psi =(\psi _{1},\psi _{2})\) such that
and
satisfy the following regularity and decay conditions
and the additional conditions
Moreover, for any curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) such that
we have
for all \(s \in {\mathbb {R}}\) and
for almost all \(s\in {\mathbb {R}}\).
For all our analysis it is vital that \(x_{i}+J_{i}\) for \(i=1,2\) are relabeling functions. Next, we define the map from \({\mathcal {D}}\) to \({\mathcal {F}}\).
Definition 3.5
Given \((u,R,S,\rho ,\sigma ,\mu ,\nu )\in {\mathcal {D}}\), we define \(\psi _{1}=(x_{1},U_{1},J_{1},K_{1},V_{1},H_{1})\) and \(\psi _{2}=(x_{2},U_{2},J_{2},K_{2},V_{2},H_{2})\) as
and
We denote by \(\textbf{L}: {\mathcal {D}}\rightarrow {\mathcal {F}}\) the mapping which to any \((u,R,S,\rho ,\sigma ,\mu ,\nu )\in {\mathcal {D}}\) associates the element \(\psi =(\psi _{1},\psi _{2})\in {\mathcal {F}}\) as defined above.
As mentioned before, solutions can develop singularities in finite time and energy can concentrate on sets of measure zero. If this is the case one has to put some extra effort into understanding (3.9e) and (3.9g) since they might be of the form \(0\cdot \infty \), when \(x_1'(X)=0\). One has, in the smooth case for \(X_1<X_2\) that
and
If we now consider the nonsmooth case, (3.10) still holds and the above calculations imply that \(V_1(X)\) exists and is bounded. Furthermore, if \(x_1'(X)=0\), we must have that \(V_1(X)=0\).
Given an element in \({\mathcal {F}}\) we want to define a curve \(({\mathcal {X}},{\mathcal {Y}})\) and the values of \(\psi \) on that curve. We define the set \({\mathcal {G}}\) which consists of curves \(({\mathcal {X}},{\mathcal {Y}})\) and five functions \({\mathcal {Z}}\), \({\mathcal {V}}\), \({\mathcal {W}}\), \({\mathfrak {p}}\), and \({\mathfrak {q}}\), next. The idea is that these functions in the smooth case are given through
and
Let us explain how the curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) may be defined for regular initial data, and how we assign the value of Z, \(Z_{X}\), \(Z_{Y}\), p, and q on the curve. In the derivation in Sect. 2, where we assumed that \(u_{0}\), \(R_{0}\), \(S_{0}\), \(\rho _{0}\), and \(\sigma _{0}\) are smooth and bounded, and the measures \(\mu _{0}\) and \(\nu _{0}\) are absolutely continuous, we found that \(0<t_{X}<\infty \) and \(-\infty<t_{Y}<0\). This implies that both \({\mathcal {X}}(s)\) and \({\mathcal {Y}}(s)\), which are implicitly given through \(t({\mathcal {X}}(s),{\mathcal {Y}}(s))=0\), are strictly increasing functions. Indeed, by differentiating \(t({\mathcal {X}}(s),{\mathcal {Y}}(s))=0\) and using (3.3c) we get \({\dot{{\mathcal {X}}}}=-\frac{2t_{Y}}{t_{X}-t_{Y}}\) and \({\dot{{\mathcal {Y}}}}=\frac{2t_{X}}{t_{X}-t_{Y}}\), which implies \({\dot{{\mathcal {X}}}}>0\) and \({\dot{{\mathcal {Y}}}}>0\). Thus, in this case \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\) is a strictly monotone curve. Next, we use the identities derived in Sect. 2 to assign the value of Z, \(Z_{X}\), \(Z_{Y}\), p, and q on the curve \(({\mathcal {X}},{\mathcal {Y}})\). In particular, we set
Moreover, to get a set of closed equations, we let
Following closely the procedure in [12, Section 2] and using (2.14), we define \(x({\mathcal {X}}(s),{\mathcal {Y}}(s))\) implicitly as
Note that the left-hand side is a strictly increasing function with respect to x, so that (3.12) uniquely defines \(x({\mathcal {X}}(s),{\mathcal {Y}}(s))\). Furthermore, we have
and
Finally using (2.12), (2.14), (2.18), and (3.11), we get the two additional equations
As a motivation for the regularity conditions that are imposed in the definition of the set \({\mathcal {G}}\), we first note from (3.12) that x(X, Y) is increasing with respect to both its arguments and is therefore unbounded. However, from (3.12) we get
which belongs to \(L^{\infty }({\mathbb {R}})\). Therefore, we require that \({\mathcal {Z}}_{2}-{{\,\textrm{Id}\,}}\) belongs to \(L^{\infty }({\mathbb {R}})\).
It is convenient to introduce the following notation: to any triplet \(({\mathcal {Z}},{\mathcal {V}},{\mathcal {W}})\) of five dimensional vector functions we associate a triplet \(({\mathcal {Z}}^{a},{\mathcal {V}}^{a},{\mathcal {W}}^{a})\) given by
for \(i \in \{3,4,5 \}\).
Definition 3.6
The set \({\mathcal {G}}\) is the set of all elements \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\), which consist of a curve \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\in {\mathcal {C}}\), three vector-valued functions
and two functions \({\mathfrak {p}}(X)\) and \({\mathfrak {q}}(Y)\). We set
and
The element \(\Theta \) belongs to \({\mathcal {G}}\) if
-
(i)
$$\begin{aligned} \left\| \Theta \right\| _{{\mathcal {G}}}<\infty \quad \text {and} \quad \vert \hspace{-1pt}\vert \hspace{-1pt}\vert \Theta \vert \hspace{-1pt}\vert \hspace{-1pt}\vert _{{\mathcal {G}}}<\infty ; \end{aligned}$$(3.16)
-
(ii)
$$\begin{aligned} {\mathcal {V}}_{2},{\mathcal {W}}_{2},{\mathcal {V}}_{4},{\mathcal {W}}_{4}\ge 0; \end{aligned}$$(3.17)
-
(iii)
for almost every \(s\in {\mathbb {R}}\), we have
$$\begin{aligned} {\dot{{\mathcal {Z}}}}(s)={\mathcal {V}}({\mathcal {X}}(s)){\dot{{\mathcal {X}}}}(s)+{\mathcal {W}}({\mathcal {Y}}(s)){\dot{{\mathcal {Y}}}}(s) \end{aligned}$$(3.18) -
(iv)
$$\begin{aligned}&{\mathcal {V}}_{2}({\mathcal {X}}(s))=c({\mathcal {Z}}_{3}(s)){\mathcal {V}}_{1}({\mathcal {X}}(s)),{} & {} {\mathcal {W}}_{2}({\mathcal {Y}}(s))=-c({\mathcal {Z}}_{3}(s)){\mathcal {W}}_{1}({\mathcal {Y}}(s)), \end{aligned}$$(3.19a)$$\begin{aligned}&{\mathcal {V}}_{4}({\mathcal {X}}(s))=c({\mathcal {Z}}_{3}(s)){\mathcal {V}}_{5}({\mathcal {X}}(s)),{} & {} {\mathcal {W}}_{4}({\mathcal {Y}}(s))=-c({\mathcal {Z}}_{3}(s)){\mathcal {W}}_{5}({\mathcal {Y}}(s)), \end{aligned}$$(3.19b)
and
$$\begin{aligned} 2{\mathcal {V}}_{4}({\mathcal {X}}(s)){\mathcal {V}}_{2}({\mathcal {X}}(s))&=(c({\mathcal {Z}}_{3}(s)){\mathcal {V}}_{3}({\mathcal {X}}(s)))^{2}+c({\mathcal {Z}}_{3}(s)){\mathfrak {p}}^{2}({\mathcal {X}}(s)), \end{aligned}$$(3.19c)$$\begin{aligned} 2{\mathcal {W}}_{4}({\mathcal {Y}}(s)){\mathcal {W}}_{2}({\mathcal {Y}}(s))&=(c({\mathcal {Z}}_{3}(s)){\mathcal {W}}_{3}({\mathcal {Y}}(s)))^{2}+c({\mathcal {Z}}_{3}(s)){\mathfrak {q}}^{2}({\mathcal {Y}}(s)), \end{aligned}$$(3.19d) -
(v)
$$\begin{aligned} \displaystyle \lim _{s \rightarrow -\infty } {\mathcal {Z}}_{4}(s)=0. \end{aligned}$$(3.20)
We denote by \({\mathcal {G}}_{0}\) the subset of \({\mathcal {G}}\) which parametrize the data at time \(t=0\), that is,
For \(\Theta \in {\mathcal {G}}_{0}\), we get by using (3.18) and (3.19a), that
This implies that
Note that for an element \(\Theta \in {\mathcal {G}}\) we have \({\mathcal {V}}_{2}+{\mathcal {V}}_{4}>0\) and \({\mathcal {W}}_{2}+{\mathcal {W}}_{4}>0\) almost everywhere. As we shall see, this property is preserved in the solution and is important in proving that the solution operator from \({\mathcal {D}}\) to \({\mathcal {D}}\) is a semigroup.
Definition 3.7
For any \(\psi =(\psi _{1},\psi _{2})\in {\mathcal {F}}\), we define \(({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\) as
and set \({\mathcal {Y}}(s)=2s-{\mathcal {X}}(s)\). We have
We define
and
Denote by \(\textbf{C}:{\mathcal {F}}\rightarrow {\mathcal {G}}_{0}\) the mapping which to any \(\psi \in {\mathcal {F}}\) associates the element \(({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}_{0}\) as defined above.
Observe from (3.9a), (3.9b), (3.23), and (3.25b), that for initial data in \({\mathcal {D}}\) such that \(u_{0}\), \(R_{0}\), \(S_{0}\), \(\rho _{0}\), and \(\sigma _{0}\) are smooth, and \(\mu _{0}\) and \(\nu _{0}\) are absolutely continuous, we recover the identity (3.12) with \(x({\mathcal {X}}(s),{\mathcal {Y}}(s))\) replaced by \({\mathcal {Z}}_2(s)\).
4 Existence of solutions for the equivalent system
4.1 Existence of short-range solutions
In the following we denote rectangular domains by
and we set \(s_{l}=\frac{1}{2}(X_{l}+Y_{l})\) and \(s_{r}=\frac{1}{2}(X_{r}+Y_{r})\). We define curves in rectangular domains as follows.
Definition 4.1
Given \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\), we denote by \({\mathcal {C}}(\Omega )\) the set of curves in \(\Omega \) parametrized by \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\) with \(s \in [s_{l},s_{r}]\) such that \(({\mathcal {X}}(s_{l}),{\mathcal {Y}}(s_{l}))=(X_{l},Y_{l})\), \(({\mathcal {X}}(s_{r}),{\mathcal {Y}}(s_{r}))=(X_{r}, Y_{r})\), and
We set
We introduce the counterpart of \({\mathcal {G}}\) on bounded domains, which we denote by \({\mathcal {G}}(\Omega )\).
Definition 4.2
Given \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\), we denote by \({\mathcal {G}}(\Omega )\) the set of all elements which consist of a curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}(\Omega )\), three vector-valued functions \({\mathcal {Z}}(s)\), \({\mathcal {V}}(X)\) and \({\mathcal {W}}(Y)\), and two functions \({\mathfrak {p}}(X)\) and \({\mathfrak {q}}(Y)\). We denote \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\) and set
and
The element \(\Theta \) belongsFootnote 1 to \({\mathcal {G}}(\Omega )\), if
-
(i)
$$\begin{aligned} \vert \hspace{-1pt}\vert \hspace{-1pt}\vert \Theta \vert \hspace{-1pt}\vert \hspace{-1pt}\vert _{{\mathcal {G}}(\Omega )}<\infty , \end{aligned}$$
-
(ii)
$$\begin{aligned} {\mathcal {V}}_{2},{\mathcal {W}}_{2},{\mathcal {Z}}_{4},{\mathcal {V}}_{4},{\mathcal {W}}_{4}\ge 0, \end{aligned}$$
-
(iii)
for almost every \(s\in {\mathbb {R}}\), we have
$$\begin{aligned} {\dot{{\mathcal {Z}}}}(s)={\mathcal {V}}({\mathcal {X}}(s)){\dot{{\mathcal {X}}}}(s)+{\mathcal {W}}({\mathcal {Y}}(s)){\dot{{\mathcal {Y}}}}(s), \end{aligned}$$(4.2) -
(iv)
$$\begin{aligned} {\mathcal {V}}_{2}({\mathcal {X}}(s))&=c({\mathcal {Z}}_{3}(s)){\mathcal {V}}_{1}({\mathcal {X}}(s)), \quad {\mathcal {W}}_{2}({\mathcal {Y}}(s))=-c({\mathcal {Z}}_{3}(s)){\mathcal {W}}_{1}({\mathcal {Y}}(s)), \end{aligned}$$(4.3a)$$\begin{aligned} {\mathcal {V}}_{4}({\mathcal {X}}(s))&=c({\mathcal {Z}}_{3}(s)){\mathcal {V}}_{5}({\mathcal {X}}(s)), \quad {\mathcal {W}}_{4}({\mathcal {Y}}(s))=-c({\mathcal {Z}}_{3}(s)){\mathcal {W}}_{5}({\mathcal {Y}}(s)) \end{aligned}$$(4.3b)
and
$$\begin{aligned} 2{\mathcal {V}}_{4}({\mathcal {X}}(s)){\mathcal {V}}_{2}({\mathcal {X}}(s))&=(c({\mathcal {Z}}_{3}(s)){\mathcal {V}}_{3}({\mathcal {X}}(s)))^{2} +c({\mathcal {Z}}_{3}(s)){\mathfrak {p}}^{2}({\mathcal {X}}(s)), \end{aligned}$$(4.3c)$$\begin{aligned} 2{\mathcal {W}}_{4}({\mathcal {Y}}(s)){\mathcal {W}}_{2}({\mathcal {Y}}(s))&=(c({\mathcal {Z}}_{3}(s)){\mathcal {W}}_{3}({\mathcal {Y}}(s)))^{2} +c({\mathcal {Z}}_{3}(s)){\mathfrak {q}}^{2}({\mathcal {Y}}(s)). \end{aligned}$$(4.3d)
By definition we have for any \(({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}(\Omega )\) that the functions \({\mathcal {X}}\) and \({\mathcal {Y}}\) are nondecreasing. To any nondecreasing function one can associate its generalized inverse, a concept which is presented in, e.g., [1].
Definition 4.3
Given \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\) and \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}(\Omega )\), we define the generalized inverse of \({\mathcal {X}}\) and \({\mathcal {Y}}\) as
respectively. We denote \({\mathcal {X}}^{-1}=\alpha \) and \({\mathcal {Y}}^{-1}=\beta \).
The generalized inverse functions \({\mathcal {X}}^{-1}\) and \({\mathcal {Y}}^{-1}\) satisfy the following properties.
Lemma 4.4
The functions \({\mathcal {X}}^{-1}\) and \({\mathcal {Y}}^{-1}\) are lower semicontinuous and nondecreasing. We have
and
Now we define solutions of (2.19) on rectangular domains. Consider the Banach spaces
The corresponding norms for \(f:\Omega \mapsto {\mathbb {R}}\) are defined as
We introduce the function \(Z^{a}\), defined as
in order to conveniently express the decay of Z at infinity in the diagonal direction. Although we are not yet concerned with the behavior at infinity, the notation will be useful when introducing global solutions.
Definition 4.5
We say that (Z, p, q) is a solution of (2.19) in \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\) if
-
(i)
$$\begin{aligned}&Z^{a}\in [{W^{1,\infty }}(\Omega )]^{5}, \quad Z^{a}_{X}\in [W^{1,\infty }_{Y}(\Omega )]^{5}, \quad Z^{a}_{Y}\in [W^{1,\infty }_{X}(\Omega )]^{5}, \nonumber \\&\quad p\in W^{1,\infty }_{Y}(\Omega ), \quad q\in W^{1,\infty }_{X}(\Omega ), \end{aligned}$$(4.6)
-
(ii)
for almost every \(X\in [X_{l},X_{r}]\),
$$\begin{aligned} (Z_{X}(X,Y))_{Y}=F(Z)(Z_{X},Z_{Y})(X,Y), \end{aligned}$$(4.7) -
(iii)
for almost every \(Y\in [Y_{l},Y_{r}]\),
$$\begin{aligned} (Z_{Y}(X,Y))_{X}=F(Z)(Z_{X},Z_{Y})(X,Y), \end{aligned}$$(4.8) -
(iv)
for almost every \(X\in [X_{l},X_{r}]\),
$$\begin{aligned} p_{Y}(X,Y)=0, \end{aligned}$$(4.9) -
(v)
for almost every \(Y\in [Y_{l},Y_{r}]\),
$$\begin{aligned} q_{X}(X,Y)=0. \end{aligned}$$(4.10)We say that (Z, p, q) is a global solution of (2.19), if these conditions hold for any rectangular domain \(\Omega \).
The following lemma shows that the imposed regularity in Definition 4.5 is necessary to extract relevant data from a curve. Slightly abusing the notation, we denote
Lemma 4.6
Let \(\Omega \) be a rectangular domain in \({\mathbb {R}}^{2}\) and assume that
Given a curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}(\Omega )\), let \(({\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\) be defined as
and
or equivalently
Then \({\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}}\in L^{\infty }_{loc }({\mathbb {R}})\) and we denote \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\) by
We now introduce the set \({\mathcal {H}}(\Omega )\) of all solutions of (2.19) on rectangular domains, which satisfy (2.10), (2.16), (2.18), and some additional constraints.
Definition 4.7
Given \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\), let \({\mathcal {H}}(\Omega )\) be the set of all solutions (Z, p, q) to (2.19) in the sense of Definition 4.5, which satisfy the following properties
We have the following short-range existence theorem.
Theorem 4.8
Given \(\Omega =[X_l,X_r]\times [Y_l,Y_r]\), then for any \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}(\Omega )\), there exists a unique solution \((Z,p,q)\in {\mathcal {H}}(\Omega )\) such that
if \(s_{r}-s_{l}\le 1/C(|||\Theta |||_{{\mathcal {G}}(\Omega )})\). Here C denotes an increasing function dependent on \(\Omega \), \(\kappa \), \(k_1\), and \(k_2\).
4.2 Existence of local solutions
We begin with some a priori estimates.
Given a positive constant L, we call domains of the type
strip domains, which correspond to domains where time is bounded. We have the following a priori estimates for the solution of (2.19).
Lemma 4.9
Given \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\) and \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}(\Omega )\), let \((Z,p,q)\in {\mathcal {H}}(\Omega )\) be a solution of (2.19) such that \(\Theta =(Z,p,q)\,\bullet \, ({\mathcal {X}},{\mathcal {Y}})\). Let \({\mathcal {E}}_{0}=||{\mathcal {Z}}_{4}||_{L^{\infty }([s_{l},s_{r}])}+||{\mathcal {Z}}_{5}||_{L^{\infty }([s_{l},s_{r}])}\). Then the following statements hold:
-
(i)
Boundedness of the energy, that is,
$$\begin{aligned} 0\le J(X,Y)\le {\mathcal {E}}_{0} \quad \text {for all } (X,Y)\in \Omega \end{aligned}$$(4.14a)and
$$\begin{aligned} ||K||_{L^{\infty }(\Omega )}\le (1+\kappa ){\mathcal {E}}_{0}. \end{aligned}$$(4.14b) -
(ii)
The functions Z, \(Z_{X}\), \(Z_{Y}\), p, and q remain uniformly bounded in strip domains which contain \(\Omega \), that is, there exists a nondecreasing function \(C_{1}=C_{1}(\vert \hspace{-1pt}\vert \hspace{-1pt}\vert \Theta \vert \hspace{-1pt}\vert \hspace{-1pt}\vert _{{\mathcal {G}}(\Omega )},L)\) such that for any \(L>0\) and any \((X,Y)\in \Omega \) such that \(|X-Y|\le 2L\), we have
$$\begin{aligned} |Z^{a}(X,Y)|\le & {} C_{1}, \quad |Z_{X}(X,Y)|\le C_{1}, \quad |Z_{Y}(X,Y)|\le C_{1}, \end{aligned}$$(4.15a)$$\begin{aligned} |p(X,Y)|\le & {} C_{1}, \quad |q(X,Y)|\le C_{1} \end{aligned}$$(4.15b)and
$$\begin{aligned}{} & {} \frac{1}{x_{X}+J_{X}}(X,Y)\le C_{1}, \quad \frac{1}{x_{Y}+J_{Y}}(X,Y)\le C_{1}. \end{aligned}$$(4.15c)Condition (ii) is equivalent to the following:
-
(iii)
For any curve \(({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})\in {\mathcal {C}}(\Omega )\), we have
$$\begin{aligned} \vert \hspace{-1pt}\vert \hspace{-1pt}\vert (Z,p,q) \bullet ({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})\vert \hspace{-1pt}\vert \hspace{-1pt}\vert _{{\mathcal {G}}(\Omega )}\le C_{1}, \end{aligned}$$(4.16)where \(C_{1}=C_{1}(||({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})||_{{\mathcal {C}}(\Omega )}, |||\Theta |||_{{\mathcal {G}}(\Omega )})\) is an increasing function with respect to both its arguments.
We have the following existence and uniqueness result.
Lemma 4.10
(Existence and uniqueness on arbitrarily large rectangles) Given a rectangular domain \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\) and \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}(\Omega )\), there exists a unique solution \((Z,p,q)\in {\mathcal {H}}(\Omega )\) such that
Lemma 4.11
(A Gronwall lemma for curves) Let \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\) and assume that \((Z,p,q)\in {\mathcal {H}}(\Omega )\) and \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}(\Omega )\). Then, for any \(({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})\in {\mathcal {C}}(\Omega )\), we have
where \(C=C(||({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})||_{{\mathcal {C}}(\Omega )},|||(Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})|||_{{\mathcal {G}}(\Omega )})\) is an increasing function with respect to both its arguments.
Lemma 4.12
(Stability in \(L^{2}\)) Let \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\) and assume that (Z, p, q), \(({\tilde{Z}},{\tilde{p}},{\tilde{q}})\in {\mathcal {H}}(\Omega )\) and \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}(\Omega )\). Then, for any \(({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})\in {\mathcal {C}}(\Omega )\), we have
where \(D=D(||({\bar{{\mathcal {X}}}},{\bar{{\mathcal {Y}}}})||_{{\mathcal {C}}(\Omega )},|||(Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})|||_{{\mathcal {G}}(\Omega )},|||({\tilde{Z}},{\tilde{p}},{\tilde{q}})\bullet ({\mathcal {X}},{\mathcal {Y}})|||_{{\mathcal {G}}(\Omega )})\) is an increasing function with respect to all its arguments.
4.3 Existence of global solutions in \({\mathcal {H}}\)
Definition 4.13
(Global solutions) Let \({\mathcal {H}}\) be the set of all functions (Z, p, q) such that
-
(i)
\((Z,p,q)\in {\mathcal {H}}(\Omega )\) for all rectangular domains \(\Omega \);
-
(ii)
there exists a curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) such that \((Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {G}}\).
The following lemma shows that condition (ii) does not depend on the particular curve for which it holds. In particular, we can replace this condition by the requirement that \((Z,p,q)\bullet ({\mathcal {X}}_{d},{\mathcal {Y}}_{d})\in {\mathcal {G}}\) for the diagonal, which is given by \({\mathcal {X}}_{d}(s)={\mathcal {Y}}_{d}(s)=s\).
Lemma 4.14
Given \((Z,p,q)\in {\mathcal {H}}\), we have \((Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {G}}\) for any curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\). Moreover, \(\lim _{s\rightarrow \infty }J({\mathcal {X}}(s),{\mathcal {Y}}(s))\) is independent of the curve \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\).
We have the following global existence theorem.
Theorem 4.15
(Existence and uniqueness of global solutions) For any initial data \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}\), there exists a unique, global solution \((Z,p,q)\in {\mathcal {H}}\) such that \(\Theta =(Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\). We denote this solution mapping by
5 From Lagrangian to Eulerian coordinates
5.1 Mapping from \({\mathcal {H}}\) to \({\mathcal {F}}\)
Given an element (Z, p, q) in \({\mathcal {H}}\) we now want to map it to an element in the set \({\mathcal {G}}\) and then further to one in \({\mathcal {F}}\). For a solution in \({\mathcal {H}}\) corresponding to time \(T>0\), i.e., \(t(X,Y)=T\), we find it convenient to first shift the time to zero so that we can map the solution to an element in \({\mathcal {G}}_{0}\) in the next step.
Definition 5.1
Given \(T\ge 0\) and \((Z,p,q)\in {\mathcal {H}}\), we define
and
We denote by \({\textbf{t}}_{T}:{\mathcal {H}}\rightarrow {\mathcal {H}}\) the mapping which associates to any \((Z,p,q)\in {\mathcal {H}}\) the element \(({\bar{Z}},{\bar{p}},{\bar{q}})\in {\mathcal {H}}\). We have
Definition 5.2
Given \((Z,p,q)\in {\mathcal {H}}\), we define
and \({\mathcal {Y}}(s)=2s-{\mathcal {X}}(s)\). Then, we have \(({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {C}}\) and \((Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {G}}_{0}\). We denote by \(\textbf{E}:{\mathcal {H}}\rightarrow {\mathcal {G}}_{0}\) the mapping which associates to any \((Z,p,q)\in {\mathcal {H}}\) the element \((Z,p,q)\bullet ({\mathcal {X}},{\mathcal {Y}})\in {\mathcal {G}}_{0}\).
Definition 5.3
Given \(({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}}) \in {\mathcal {G}}_{0}\), let \(\psi _{1}=(x_{1},U_{1},J_{1},K_{1},V_{1},H_{1})\) and \(\psi _{2}=(x_{2},U_{2},J_{2},K_{2},V_{2},H_{2})\) be defined as
and
We denote by \(\textbf{D}:{\mathcal {G}}_{0}\rightarrow {\mathcal {F}}\) the mapping which to any \(({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}}) \in {\mathcal {G}}_{0}\) associates the element \(\psi \in {\mathcal {F}}\) as defined above.
5.2 Semigroup of solutions in \({\mathcal {F}}\)
We define the solution operator on the set \({\mathcal {F}}\).
Definition 5.4
For any \(T\ge 0\), we define the mapping \(S_{T}:{\mathcal {F}}\rightarrow {\mathcal {F}}\) by
In order to show that \(S_{T}\) is a semigroup we need the following result.
Lemma 5.5
We have
It follows that \({\textbf{S}}\circ {\textbf{C}}=({\textbf{D}}\circ {\textbf{E}})^{-1}\) and the sets \({\mathcal {F}}\) and \({\mathcal {H}}\) are in bijection.
Theorem 5.6
The mapping \(S_{T}\) is a semigroup.
Proof
We have
\(\square \)
5.3 Mapping from \({\mathcal {F}}\) to \({\mathcal {D}}\)
Definition 5.7
Given \(\psi =(\psi _{1},\psi _{2})\in {\mathcal {F}}\), we define \((u,R,S,\rho ,\sigma ,\mu ,\nu )\) asFootnote 2
or, equivalently,
The relations (5.5c)–(5.5f) are equivalent to
respectively, for almost every X and Y. We denote by \(\textbf{M}:{\mathcal {F}}\rightarrow {\mathcal {D}}\) the mapping which to any \(\psi \in {\mathcal {F}}\) associates the element \((u,R,S,\rho ,\sigma ,\mu ,\nu )\in {\mathcal {D}}\) as defined above.
Lemma 5.8
Given \(\psi =(\psi _{1},\psi _{2})\in {\mathcal {F}}\), let \((u,R,S,\rho ,\sigma ,\mu ,\nu )={\textbf{M}}(\psi _{1},\psi _{2})\). Then, for any \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\in {\mathcal {G}}_{0}\) such that \((\psi _{1},\psi _{2})={\textbf{D}}({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\), we have
The relations (5.7b) and (5.7d) are equivalent to
for any s such that \({\dot{{\mathcal {X}}}}(s)>0\). The relations (5.7c) and (5.7e) are equivalent to
for any s such that \({\dot{{\mathcal {Y}}}}(s)>0\).
By using the semigroup \(S_{T}\) we can, together with the mappings from \({\mathcal {D}}\) to \({\mathcal {F}}\) and vica versa, study the solution in the original set of variables, for given initial data in \({\mathcal {D}}\).
Lemma 5.9
Given \((u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\in {\mathcal {D}}\), let
and
Then, we have
for all \((X,Y)\in {\mathbb {R}}^{2}\),
for almost every \((X,Y)\in {\mathbb {R}}^{2}\) such that \(x_{X}(X,Y)>0\), and
for almost every \((X,Y)\in {\mathbb {R}}^{2}\) such that \(x_{Y}(X,Y)>0\). Furthermore, we have
in the sense of distributions.
5.4 Semigroup of solutions in \({\mathcal {D}}\)
Now we can define a mapping on \({\mathcal {D}}\), the original set of variables.
Definition 5.10
For any \(T>0\), let \({\bar{S}}_{T}:{\mathcal {D}}\rightarrow {\mathcal {D}}\) be defined as
Since
it would immediately follow from the semigroup property of \(S_{T}\) that \({\bar{S}}_{T}\) is also a semigroup if we had \({\textbf{L}}\circ {\textbf{M}}={{\,\textrm{Id}\,}}\), but this identity does not hold in general, unless one introduces an equivalence relation based on relabeling. An essential role is played by G, cf. Definition 3.3.
First we define the action of \(G^{2}\) on the set \({\mathcal {F}}\).
Definition 5.11
For any \(\psi =(\psi _{1},\psi _{2})\in {\mathcal {F}}\) and \(f,g\in G\), we define \({\bar{\psi }}=({\bar{\psi }}_{1},{\bar{\psi }}_{2})\) as
The mapping from \({\mathcal {F}}\times G^{2}\) to \({\mathcal {F}}\) given by \(\psi \times (f,g)\mapsto {\bar{\psi }}\) defines an action of the group \(G^{2}\) on \({\mathcal {F}}\) and we denote \({\bar{\psi }}=\psi \cdot (f,g)\).
We refer to [15] for the definition of the action of \(G^2\) on the sets \({\mathcal {C}}\), \({\mathcal {G}}\), and \({\mathcal {H}}\). A key result is the following.
Lemma 5.12
The mapping \(S_{T}\) is \(G^{2}\)-equivariant, that is, for all \(\phi =(f,g)\in G^{2}\), we have
for all \(\psi \in {\mathcal {F}}\).
Definition 5.13
We denote by \({\mathcal {F}}/G^{2}\) the quotient of \({\mathcal {F}}\) with respect to the action of the group \(G^{2}\) on \({\mathcal {F}}\). More specifically, we define the equivalence relation, \(\sim \), on \({\mathcal {F}}\) as
For an element \(\psi \in {\mathcal {F}}\), we denote the equivalence class by
We define the quotient space as
We now define the set \({\mathcal {F}}_{0}\), which plays a key role in the following lemma.
Definition 5.14
Let
and \(\Pi :{\mathcal {F}}\rightarrow {\mathcal {F}}_{0}\) be the projection on \({\mathcal {F}}_{0}\) given by \({\bar{\psi }}=({\bar{\psi }}_{1},{\bar{\psi }}_{2})=\Pi (\psi )\) where \({\bar{\psi }}\in {\mathcal {F}}_{0}\) is defined as follows. Let
and denote \(\phi =(f,g)\in G^{2}\). We set
Lemma 5.15
The following statements hold:
-
(i)
For any \(\psi \) and \({\bar{\psi }}\) in \({\mathcal {F}}\), we have
$$\begin{aligned} \psi \sim {\bar{\psi }}\quad \text {if and only if}\quad \Pi (\psi )=\Pi ({\bar{\psi }}), \end{aligned}$$(5.16a)so that the sets \({\mathcal {F}}/G^{2}\) and \({\mathcal {F}}_{0}\) are in bijection.
-
(ii)
We have
$$\begin{aligned} {\textbf{M}}\circ \Pi ={\textbf{M}} \end{aligned}$$(5.16b)and
$$\begin{aligned} {\textbf{L}}\circ {\textbf{M}}|_{{\mathcal {F}}_{0}}={{\,\textrm{Id}\,}}|_{{\mathcal {F}}_{0}} \quad \text {and} \quad {\textbf{M}}\circ {\textbf{L}}={{\,\textrm{Id}\,}}, \end{aligned}$$(5.16c)so that the sets \({\mathcal {D}}\), \({\mathcal {F}}_{0}\), and \({\mathcal {F}}/G^{2}\) are in bijection.
-
(iii)
We have
$$\begin{aligned} \Pi \circ S_{T}\circ \Pi =\Pi \circ S_{T}. \end{aligned}$$(5.16d)
Note that the first identity in (5.16c) is equivalent to
Now we are finally in position to prove that \({\bar{S}}_{T}\) is a semigroup.
Theorem 5.16
The mapping \({\bar{S}}_{T}\) is a semigroup.
Proof
The proof relies on Lemma 5.15 and Theorem 5.6. From Definition 5.10 we have
\(\square \)
6 Existence of weak global conservative solutions
It remains to prove that the solution obtained by using the operator \({\bar{S}}_{T}\) is a weak solution of (1.3).
Theorem 6.1
Let \(t>0\) and \(\xi _{0}=(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\in {\mathcal {D}}\). Then \((u,R,S,\rho ,\sigma ,\mu ,\nu )(t) = {\bar{S}}_{t}(\xi _{0})\) is a weak solution of (1.3), meaning that
for all \(\phi =\phi (t,x)\) in \(C^{\infty }_{0}((0,\infty )\times {\mathbb {R}})\), where
in the sense of distributions.
Moreover, the measures \(\mu \) and \(\nu \) satisfy the equations
in the sense of distributions.
Note that if the two measures \(\mu \) and \(\nu \) are absolutely continuous, (6.2) coincides with (1.12) in the sense of distributions. Moreover, the difference of the sign in front of \(\mu \) and \(\nu \) indicates the two opposite traveling directions.
The semigroup of solutions, \({\bar{S}}_{t}\), is conservative in the following sense.
Theorem 6.2
Given \(\xi _{0}=(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\in {\mathcal {D}}\), let \((u,R,S,\rho ,\sigma ,\mu ,\nu )(t)={\bar{S}}_{t}(\xi _{0})\). We have:
-
(i)
For all \(t\ge 0\),
$$\begin{aligned} \mu (t)({\mathbb {R}})+\nu (t)({\mathbb {R}})=\mu _{0}({\mathbb {R}}) +\nu _{0}({\mathbb {R}}). \end{aligned}$$ -
(ii)
For almost every \(t\ge 0\), the singular parts of \(\mu (t)\) and \(\nu (t)\) are concentrated on the set where \(c'(u)=0\).
Theorem 6.3
(Finite speed of propagation) For initial data \(\xi _{0}=(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\) and \({\bar{\xi }}_{0}=({\bar{u}}_{0},{\bar{R}}_{0},{\bar{S}}_{0},{\bar{\rho }}_{0},{\bar{\sigma }}_{0},{\bar{\mu }}_{0},{\bar{\nu }}_{0})\) in \({\mathcal {D}}\), we consider the solutions \((u,R,S,\rho ,\sigma ,\mu ,\nu )(t)={\bar{S}}_{t}(\xi _{0})\) and \(({\bar{u}},{\bar{R}},{\bar{S}},{\bar{\rho }},{\bar{\sigma }},{\bar{\mu }},{\bar{\nu }})(t)={\bar{S}}_{t}({\bar{\xi }}_{0})\). Given \({\textbf{t}}>0\) and \({\textbf{x}}\in {\mathbb {R}}\), if \(\xi _{0}(x)={\bar{\xi }}_{0}(x)\) for \(x\in [{\textbf{x}}-\kappa {\textbf{t}},{\textbf{x}}+\kappa {\textbf{t}}]\), then \(u({\textbf{t}},{\textbf{x}})={\bar{u}}({\textbf{t}},{\textbf{x}})\).
7 Regularity of solutions
7.1 Existence of local smooth solutions
Theorem 7.1
Let \(-\infty<x_{l}<x_{r}<\infty \) and consider \((u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\in {\mathcal {D}}\). Let \(m\in {\mathbb {N}}\) and assume that,
-
(A1)
\(u_{0}\in {L^\infty }([x_{l},x_{r}])\),
-
(A2)
\(R_{0},S_{0},\rho _{0},\sigma _{0}\in W^{m-1,\infty }([x_{l},x_{r}])\),
-
(A3)
there are constants \(d>0\) and \(e>0\) such that \(\rho _{0}(x)\ge d\) and \(\sigma _{0}(x)\ge e\) for all \(x\in [x_{l},x_{r}]\),
-
(A4)
\(\mu _{0}\) and \(\nu _{0}\) are absolutely continuous on \([x_{l},x_{r}]\),
-
(A5)
\(c\in C^{m-1}({\mathbb {R}})\) and \({\max _{u\in {\mathbb {R}}}}\bigg |\frac{d^{i}}{du^{i}}c(u)\bigg |\le k_{i}\) for constants \(k_{i}, i=3,4,5,\dots ,m-1\).
For any \(\tau \in \big [0,\frac{1}{2\kappa }(x_{r}-x_{l})\big ]\) consider \((u,R,S,\rho ,\sigma ,\mu ,\nu )(\tau )={\bar{S}}_{\tau }(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\). Then
-
(P1)
\(u(\tau ,\cdot )\in W^{m,\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\),
-
(P2)
\(R(\tau ,\cdot ),S(\tau ,\cdot ),\rho (\tau ,\cdot ),\sigma (\tau ,\cdot )\in W^{m-1,\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\),
-
(P3)
there are constants \({\bar{d}}>0\) and \({\bar{e}}>0\) such that \(\rho (\tau ,x)\ge {\bar{d}}\) and \(\sigma (\tau ,x)\ge {\bar{e}}\) for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\),
-
(P4)
\(\mu (\tau ,\cdot )\) and \(\nu (\tau ,\cdot )\) are absolutely continuous on \([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\).
For \(\tau \in \big [-\frac{1}{2\kappa }(x_{r}-x_{l}),0\big ]\), the solution satisfies the same properties on the interval \(\big [x_{l}-\kappa \tau ,x_{r}+\kappa \tau \big ]\).
Note that since \((u_{0})_{x}=\frac{1}{2c(u_{0})}(R_{0}-S_{0})\), it follows from assumptions (A1), (A2) and (A5) that \(u_{0}\in W^{m,\infty }([x_{l},x_{r}])\).
Specifically, (A4) means that \(\mu _{0}((-\infty ,x_{l}))=\mu _{0}((-\infty ,x_{l}])\), \(\mu _{0}((-\infty ,x_{r}))=\mu _{0}((-\infty ,x_{r}])\), \(\nu _{0}((-\infty ,x_{l}))=\nu _{0}((-\infty ,x_{l}])\), and \(\nu _{0}((-\infty ,x_{r}))=\nu _{0}((-\infty ,x_{r}])\).
By (1.5) and (1.6), (A5) holds for \(i=0,1,2\).
Proof
In the following, we will consider the case \(0<\tau \le \frac{1}{2\kappa }(x_{r}-x_{l})\). The case \(-\frac{1}{2\kappa }(x_{r}-x_{l})\le \tau <0\) can be treated in the same way.
We decompose the proof into three steps.
Step 1. We first consider the case \(m=1\).
(i) Consider \((\psi _{1},\psi _{2})={\textbf{L}}(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\). Since \(\mu _{0}\) is absolutely continuous on \([x_{l},x_{r}]\) we have from (3.9a), \(x_{1}(X)+\mu _{0}((-\infty ,x_{1}(X)))=X\) for all \(x_{1}(X)\in [x_{l},x_{r}]\). For \(X_{l}\) and \(X_{r}\) satisfying \(x_{1}(X_{l})=x_{l}\) and \(x_{1}(X_{r})=x_{r}\) we have \(X_{l}=x_{l}+\mu _{0}((-\infty ,x_{l}))\) and \(X_{r}=x_{r}+\mu _{0}((-\infty ,x_{r}))\). Therefore, since \(x_{1}\) is nondecreasing, we get
for all \(X\in [X_{l},X_{r}]\). Similarly we find by using (3.9b), \(x_{2}(Y)+\nu _{0}((-\infty ,x_{2}(Y)))=Y\) for all \(Y\in [Y_{l},Y_{r}]\), where \(x_{2}(Y_{l})=x_{l}\), \(x_{2}(Y_{r})=x_{r}\), \(Y_{l}=x_{l}+\nu _{0}((-\infty ,x_{l}))\), and \(Y_{r}=x_{r}+\nu _{0}((-\infty ,x_{r}))\). We define \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\). From now on we only consider \((X,Y)\in \Omega \). Rewriting (7.1) yields
Differentiating implies that
Since \(R_{0},\rho _{0}\in {L^\infty }([x_{l},x_{r}])\), we get the lower bound
and since \(\rho _{0}(x)\ge d\), we find the upper bound
Similarly, we find
(ii) Let \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})={\textbf{C}}(\psi _{1},\psi _{2})\). By (3.24) we have \(x_{1}({\mathcal {X}}(s))=x_{2}({\mathcal {Y}}(s))\), which after differentiating and using \({\mathcal {X}}(s)+{\mathcal {Y}}(s)=2s\), yields
This implies, by (7.3)–(7.5), that
for all s such that \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\in \Omega \), that is, for all values of s which satisfy \(X_{l}\le {\mathcal {X}}(s)\le X_{r}\) and \(Y_{l}\le {\mathcal {Y}}(s)\le Y_{r}\). Using this together with the identity \({\mathcal {X}}(s)+{\mathcal {Y}}(s)=2s\), we find that (7.7) is valid for all \(s\in [s_{l},s_{r}]\), where \(s_{l}=\frac{1}{2}(X_{l}+Y_{l})\) and \(s_{r}=\frac{1}{2}(X_{r}+Y_{r})\). Hence, \({\mathcal {X}}(s)\) and \({\mathcal {Y}}(s)\) are strictly increasing functions on \([s_l,s_r]\).
(iii) Consider \((Z,p,q)={\textbf{S}}(\Theta )\). First we prove that
for all \((X,Y)\in \Omega \), where C depends on \(\kappa \), \(k_{1}\), \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\), \(X_{r}-X_{l}\), and \(Y_{r}-Y_{l}\). From (4.12c), we have \(2J_{X}x_{X}\ge (c(U)U_{X})^{2}\) and \(2J_{Y}x_{Y}\ge (c(U)U_{Y})^{2}\). By Young’s inequality, we get \(|U_{X}|\le \frac{\kappa }{\sqrt{2}}(x_{X}+J_{X})\) and \(|U_{Y}|\le \frac{\kappa }{\sqrt{2}}(x_{Y}+J_{Y})\). This implies, after using (2.19b) and (2.19d), that
By Gronwall’s inequality,
Since x and J are nondecreasing with respect to both variables, we have
where we used \(s_{l}=\frac{1}{2}(X_{l}+Y_{l})\), \(s_{r}=\frac{1}{2}(X_{r}+Y_{r})\), and the definition of \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\). Defining \(C=\exp \big \{\frac{k_{1}\kappa ^{2}}{\sqrt{2}}(2|||\Theta |||_{{\mathcal {G}}(\Omega )}+\frac{1}{2}(X_{r}-X_{l})+\frac{1}{2}(Y_{r}-Y_{l}))\big \}\) completes the proof of (7.8). Note that C is strictly positive.
We have \({\mathcal {V}}_{2}+{\mathcal {V}}_{4}=\frac{1}{2}x_{1}'+J_{1}'\) and since \(J_{1}'=1-x_{1}'\), \(x_{1}'\ge 0\) and \(J_{1}'\ge 0\), this implies that \(\frac{1}{2}\le {\mathcal {V}}_{2}+{\mathcal {V}}_{4}\le 1\). Thus, we get
By (3.26f), (3.9g), (A3) and (7.3) we obtain
Similarly we obtain
Then, since \(p_{Y}=0\), we get for all \((X,Y)\in \Omega \),
where we used (4.12c) and (7.9). Hence,
Using that \(q_{X}=0\), we find in the same way that
where \({\tilde{C}}\) is strictly positive and depends on \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\), \(X_r-X_l\), and \(Y_r-Y_l\). From (4.12a), we then get
(iv) Let \(0<\tau \le \frac{1}{2\kappa }(x_{r}-x_{l})\) and consider \(\Theta (\tau )={\textbf{E}}\circ {\textbf{t}}_{\tau }(Z,p,q)\).
We claim that \(({\mathcal {X}}(\tau ,s), {\mathcal {Y}}(\tau ,s))\) lies below the curveFootnote 3\(({\mathcal {X}}(s),{\mathcal {Y}}(s))\). For any \({\bar{s}}\) and s such that \({\mathcal {X}}(\tau ,{\bar{s}})={\mathcal {X}}(s)\) we have \(t({\mathcal {X}}(s),{\mathcal {Y}}(s))=0<\tau =t({\mathcal {X}}(\tau ,{\bar{s}}),{\mathcal {Y}}(\tau ,{\bar{s}}))\), so that by (4.12a) and (4.12d),
which proves the claim.
We prove that there exist \({\bar{s}}_{\text {min}}\) and \({\bar{s}}_{\text {max}}\) satisfying \(s_{l}<{\bar{s}}_{\text {min}}\le {\bar{s}}_{\text {max}}<s_{r}\) such that \(({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) belongs to \(\Omega \) for all \(s\in [{\bar{s}}_{\text {min}},{\bar{s}}_{\text {max}}]\). First we need an estimate. Since \(c(u)\le \kappa \), \(t_{X}(X,Y)>0\), and \(t_{Y}(X,Y)<0\), we have
where we used (4.12a). Consider \({\bar{s}}_{\text {max}}\) such that \({\mathcal {X}}(\tau ,{\bar{s}}_{\text {max}})=X_{r}\). Note that since \({\mathcal {X}}(\tau ,{\bar{s}}_{\text {max}})={\mathcal {X}}(s_{r})=X_r\), we get from (7.14), \({\mathcal {Y}}(\tau ,{\bar{s}}_{\text {max}})<{\mathcal {Y}}(s_{r})=Y_{r}\). From (7.15) we get \(t(X_{r},{\mathcal {Y}}(\tau ,{\bar{s}}_{\text {max}}))=\tau \le \frac{1}{2\kappa }(x_{r}-x_{l})\le t(X_{r},Y_{l})\), so that \({\mathcal {Y}}(\tau ,{\bar{s}}_{\text {max}})\ge Y_{l}\). In particular, we have
Since \({\mathcal {X}}(\tau ,\cdot )\) and \({\mathcal {Y}}(\tau ,\cdot )\) are nondecreasing and continuous, there exists \({\bar{s}}_{\text {min}}\) such that \({\bar{s}}_{\text {min}}\le {\bar{s}}_{\text {max}}\), \({\mathcal {Y}}(\tau ,{\bar{s}}_{\text {min}})=Y_{l}\) and \({\mathcal {X}}(\tau ,{\bar{s}}_{\text {min}})\le X_{r}\). Since \(t(X_{l},Y_{l})=0<\tau =t({\mathcal {X}}(\tau ,{\bar{s}}_{\text {min}}),Y_{l})\) we find that \({\mathcal {X}}(\tau ,{\bar{s}}_{\text {min}})>X_{l}\) and we have
For any \(s\in [{\bar{s}}_{\text {min}},{\bar{s}}_{\text {max}}]\) we get from (7.16) and (7.17), since \({\mathcal {X}}(\tau ,\cdot )\) and \({\mathcal {Y}}(\tau ,\cdot )\) are nondecreasing, that
In other words, the curve \(({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) lies in \(\Omega \) for all \(s\in [{\bar{s}}_{\text {min}},{\bar{s}}_{\text {max}}]\), and hence all the estimates obtained in (iii) are valid along this curve. Observe that \(s_{l}=\frac{1}{2}(X_{l}+Y_{l})<\frac{1}{2}\big ({\mathcal {X}}(\tau ,{\bar{s}}_{\text {min}})+{\mathcal {Y}}(\tau ,{\bar{s}}_{\text {min}})\big )={\bar{s}}_{\text {min}}\) and \(s_{r}=\frac{1}{2}(X_{r}+Y_{r})>\frac{1}{2}\big ({\mathcal {X}}(\tau ,{\bar{s}}_{\text {max}})+{\mathcal {Y}}(\tau ,{\bar{s}}_{\text {max}})\big )={\bar{s}}_{\text {max}}\), which implies \(s_{l}<{\bar{s}}_{\text {min}}\le {\bar{s}}_{\text {max}}<s_{r}\).
By differentiating \(t({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))=\tau \) and using that \({\dot{{\mathcal {X}}}}(\tau ,s)+{\dot{{\mathcal {Y}}}}(\tau ,s)=2\), where \(\, \cdot \,\) denotes the derivative with respect to s, we obtain
By (4.5), we have \(|t_{X}(X,Y)|\le ||Z_{X}^{a}||_{W^{1,\infty }_{Y}(\Omega )} +\frac{\kappa }{2}\) and \(|t_{Y}(X,Y)|\le ||Z_{Y}^{a}||_{W^{1,\infty }_{X}(\Omega )}+\frac{\kappa }{2}\) for all \((X,Y)\in \Omega \). Using this and (7.13) in (7.18), we find
for all \(s\in [{\bar{s}}_{\text {min}},{\bar{s}}_{\text {max}}]\). Similarly, one proves that
for some positive constant \(\alpha _{2}\) that depends on \(d_{2}\), C, \(\kappa \), \(||Z_{X}^{a}||_{W^{1,\infty }_{Y}(\Omega )}\), and \(||Z_{Y}^{a}||_{W^{1,\infty }_{X}(\Omega )}\).
Next, we prove that there exist \({\bar{s}}_{1}\) and \({\bar{s}}_{2}\) satisfying \({\bar{s}}_{\text {min}}\le {\bar{s}}_{1}\le {\bar{s}}_{2}\le {\bar{s}}_{\text {max}}\) such that
Observe that since x is increasing with respect to both variables this will imply
for all \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\) and in particular, \(({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) belongs to \(\Omega \) for all \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\), see Figs. 3 and 4.
By (4.12a), (4.12d), and (1.5), we have
Using the lower bound on c, we end up with
The region bounded by the characteristics \(x_{2}(t)\) (forward) and \(x_{5}(t)\) (backward) starting from \(x_{l}\) and \(x_{r}\), respectively, at \(t=0\). Here, \(0<\tau _{1}<\tau _{2}<\frac{1}{2\kappa }(x_{r}-x_{l})\). The remaining functions are given by \(x_{1}(t)=x_{l}+\frac{t}{\kappa }\), \(x_{3}(t)=x_{l}+\kappa t\), \(x_{4}(t)=x_{r}-\kappa t\) and \(x_{6}(t)=x_{r}-\frac{t}{\kappa }\). We have \(x_{3}(\tau _{1})=x_{l}+\kappa \tau _{1}=x({\mathcal {X}}(\tau _{1},{\bar{s}}_{1}),{\mathcal {Y}}(\tau _{1},{\bar{s}}_{1}))\) and \(x_{4}(\tau _{1})=x_{r}-\kappa \tau _{1}=x({\mathcal {X}}(\tau _{1},{\bar{s}}_{2}),{\mathcal {Y}}(\tau _{1},{\bar{s}}_{2}))\)
The region from Fig. 3 in Lagrangian coordinates. The curves \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\), \(({\mathcal {X}}(\tau _{1},s),{\mathcal {Y}}(\tau _{1},s))\) and \(({\mathcal {X}}(\tau _{2},s),{\mathcal {Y}}(\tau _{2},s))\) correspond to \(t=0\), \(t=\tau _{1}\) and \(t=\tau _{2}\), respectively
Following the same lines, one obtains
Since \(0<\tau \le \frac{1}{2\kappa }(x_{r}-x_{l})\) we have \(x_{l}+\kappa \tau =x_{l}+2\kappa \tau -\kappa \tau \le x_{l}+x_{r}-x_{l}-\kappa \tau = x_{r}-\kappa \tau \), which implies, since x is continuous with respect to both variables, that there exist \({\bar{s}}_{1}\) and \({\bar{s}}_{2}\) such that \({\bar{s}}_{1}\le {\bar{s}}_{2}\) and
From (7.22), (7.24), and the fact that x increases along the curve \(({\mathcal {X}}(\tau ,s), {\mathcal {Y}}(\tau ,s))\), we have \({\mathcal {X}}(\tau ,{\bar{s}}_{\text {min}})\le {\mathcal {X}}(\tau ,{\bar{s}}_{1})\) which implies \({\bar{s}}_{\text {min}}\le {\bar{s}}_{1}\). By (7.23) and (7.24) we get \({\mathcal {Y}}(\tau ,{\bar{s}}_{2})\le {\mathcal {Y}}(\tau ,{\bar{s}}_{\text {max}})\) and \({\bar{s}}_{2}\le {\bar{s}}_{\text {max}}\). This concludes the proof of (7.21).
We prove (P1). From (5.7a) we have \(u(\tau ,x)={\mathcal {Z}}_{3}(\tau ,s)\) if \(x={\mathcal {Z}}_{2}(\tau ,s)\). Since the function \({\mathcal {Z}}_{2}(\tau ,s)\) is nondecreasing, the smallest and biggest value it can attain for \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\) are
respectively. The function \({\mathcal {Z}}_{2}(\tau ,\cdot )\) is in fact strictly increasing for \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\), as we now show. We differentiate the relation \({\mathcal {Z}}_{2}(\tau ,s)=x({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) and get
From (7.11), (7.12), (7.19) and (7.20) we have
Hence, the mapping \(s\mapsto {\mathcal {Z}}_{2}(\tau ,s)\) is strictly increasing for \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\) and therefore invertible on \([{\bar{s}}_{1},{\bar{s}}_{2}]\). For any \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\) we get
where \({\mathcal {Z}}_{2}^{-1}(\tau ,x)\) denotes the inverse of the function \(s\mapsto {\mathcal {Z}}_2(\tau ,s)\). Since
we have
Next, we differentiate (7.26) and get
We have
so that \({\dot{{\mathcal {Z}}}}_{3}(\tau ,\cdot )\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\). By (7.25) we end up with
which implies that
From (7.27) and (7.29) we conclude that (P1) holds.
We prove (P2). From (5.8a) and (5.8b), we have
and
for all \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\). We multiply these equations by \(2{\dot{{\mathcal {X}}}}(\tau ,s)\) and use (3.22) to get
and
which yields
and
for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). Since \({\mathcal {V}}_{3}(\tau ,{\mathcal {X}}(\tau ,s))=U_{X}({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\), we have \(|{\mathcal {V}}_{3}(\tau ,{\mathcal {X}}(\tau ,s))|\le ||U_{X}||_{W^{1,\infty }_{Y}(\Omega )}\) or in other words \({\mathcal {V}}_{3}(\tau ,{\mathcal {X}}(\tau ,\cdot ))\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\), and since \({\mathfrak {p}}(\tau ,{\mathcal {X}}(\tau ,s))=p({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) it follows that \({\mathfrak {p}}(\tau ,{\mathcal {X}}(\tau ,\cdot ))\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\). Using (7.25) in (7.30) and (7.31) we obtain
for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). Therefore \(R(\tau ,\cdot ),\rho (\tau ,\cdot )\in {L^\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\). In a similar way one shows that \(S(\tau ,\cdot ),\sigma (\tau ,\cdot )\in {L^\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\) and we have proved (P2).
We prove (P3). By inserting
into (7.31) we get
for all \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\). Since \(p_{Y}=0\) we get from (7.10),
for all \(X\in [X_{l},X_{r}]\). Recalling (7.9) and that \(J_X(X,Y)\ge 0\) we get \(\rho (\tau ,x)\ge d_{2}C^{-1}\) for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). Similarly we find \(\sigma (\tau ,x)\ge e_{2}{\tilde{C}}^{-1}\) for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). This concludes the proof of (P3).
We prove (P4). Let \(M\subset [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\) be a Borel set. By an argument as in [12, Proof of Lemma 11], we have
where \(A=\{s\in {\mathbb {R}} \ | \ {\mathcal {V}}_{2}(\tau ,{\mathcal {X}}(\tau ,s))>0 \}\). We have \(\text {meas}(A^{c})=0\), which implies that \(\mu _{\text {sing}}(\tau ,M)\le 2|||\Theta (\tau )|||_{{\mathcal {G}}(\Omega )}\,\text {meas}(A^{c})=0\). This proves that \(\mu (\tau )\) is absolutely continuous on \([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). Similarly, one shows that \(\nu (\tau )\) is absolutely continuous on \([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\).
Step 2. Assume that \(m=2\). By Step 1, (P3) and (P4) hold. Moreover, (P1) and (P2) hold for \(m=1\). It remains to prove that \(u_{xx}(\tau ,\cdot ),R_{x}(\tau ,\cdot ),S_{x}(\tau ,\cdot ),\rho _{x}(\tau ,\cdot ),\sigma _{x}(\tau ,\cdot )\in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\). In order to do so, we have to show that \(Z_{XX}=(t_{XX},x_{XX},U_{XX},J_{XX},K_{XX})\) and \(Z_{YY}=(t_{YY},x_{YY},U_{YY},J_{YY},K_{YY})\) exist and are bounded. We first consider \(Z_{XX}\), which is the unique solution of the semilinear system \(Z_{XXY}(X,Y)=f(X,Y,Z_{XX})\), where f is Lipschitz continuous with respect to the \(Z_{XX}\) variable. This can be seen by differentiating the governing equations (2.19). For instance, we have
and
if we assume without loss of generality that \(Y\le {\mathcal {Y}}(X)\). The other case is similar.
Let us find a bound on \(x_{XX}\) at time \(\tau =0\). We differentiate (7.2) and get
which, by (7.4), implies that
and we conclude that \(x_{1}''\in {L^\infty }([X_{l},X_{r}])\).
By (3.26b), we have \(x_{X}(X,{\mathcal {Y}}(X))={\mathcal {V}}_{2}(X)=\frac{1}{2}x_{1}'(X)\). We differentiate and get
so that by (7.19),
and \(x_{XX}(\cdot ,{\mathcal {Y}}(\cdot ))\in {L^\infty }([X_{l},X_{r}])\). Here we used the estimate
We estimate \(x_{XXY}\). Since \(|Z_{X}^{a}|\le ||Z_{X}^{a}||_{W^{1,\infty }_{Y}(\Omega )}\) and \(|Z_{Y}^{a}|\le ||Z_{Y}^{a}||_{W^{1,\infty }_{X}(\Omega )}\), we get from (2.19) that
where \(\eta \) depends on \(||Z_{X}^{a}||_{W^{1,\infty }_{Y}(\Omega )}\), \(||Z_{Y}^{a}||_{W^{1,\infty }_{X}(\Omega )}\), \(\kappa \), and \(k_{1}\). We obtain from (7.32),
We insert (7.36) into (7.33) and get
Following the same lines for the other components of \(Z_{XX}\), we obtain
where \(C_{1}\) and \(C_{2}\) depend on \(||Z_{X}^{a}||_{W^{1,\infty }_{Y}(\Omega )}\), \(||Z_{Y}^{a}||_{W^{1,\infty }_{X}(\Omega )}\), \(\kappa \), \(k_{1}\), and \(k_{2}\). By Gronwall’s lemma, we obtain
A similar procedure yields
where \({\tilde{C}}_{1}\) and \({\tilde{C}}_{2}\) depend on \(||Z_{X}^{a}||_{W^{1,\infty }_{Y}(\Omega )}\), \(||Z_{Y}^{a}||_{W^{1,\infty }_{X}(\Omega )}\), \(\kappa \), \(k_{1}\), and \(k_{2}\).
We also need estimates for \({\ddot{{\mathcal {X}}}}(\tau ,s)\) and \({\ddot{{\mathcal {Y}}}}(\tau ,s)\). By (3.21), we have \(x_{X}({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s)){\dot{{\mathcal {X}}}} (\tau ,s) = x_{Y}({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s)){\dot{{\mathcal {Y}}}}(\tau ,s)\). We differentiate and get
Since \({\mathcal {X}}(\tau ,s)+{\mathcal {Y}}(\tau ,s)=2s\), we have \({\ddot{{\mathcal {Y}}}}(\tau ,s)=-{\ddot{{\mathcal {X}}}}(\tau ,s)\), so that
By (7.11), (7.12), (7.37), and (7.38), we find
Thus, we conclude that \({\ddot{{\mathcal {X}}}}(\tau ,\cdot )\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\).
We differentiate \({\mathcal {Z}}_{2}(\tau ,s)=x({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) twice and get, by (7.39), that
Using (7.35) and (7.37), we end up with
and \({\ddot{{\mathcal {Z}}}}_{2}(\tau ,\cdot )\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\).
Following the same lines we prove that \({\ddot{{\mathcal {Z}}}}_{3}(\tau ,\cdot )\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\).
We compute \(u_{xx}\) from (7.28) and get
By (7.25) we obtain
and we conclude that \(u_{xx}(\tau ,\cdot )\in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
By differentiating (7.30) we get
where we denote \({\dot{{\mathcal {V}}}}_{3}(\tau ,{\mathcal {X}}(\tau ,s))=\frac{d}{ds}{\mathcal {V}}_{3}(\tau ,{\mathcal {X}}(\tau ,s))\).
Since \({\mathcal {V}}_{3}(\tau ,{\mathcal {X}}(\tau ,s)) = U_X({\mathcal {X}}(\tau ,s), {\mathcal {Y}}(\tau ,s))\), we have
From (7.35) and (7.37) we obtain
Therefore we have \({\dot{{\mathcal {V}}}}_{3}(\tau ,{\mathcal {X}}(\tau ,\cdot ))\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\). By (7.25) we get
which implies that \(R_{x}(\tau ,\cdot )\in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
We differentiate (7.31) and get
where we denote \({\dot{{\mathfrak {p}}}}(\tau ,{\mathcal {X}}(\tau ,s))=\frac{d}{ds}{\mathfrak {p}}(\tau ,{\mathcal {X}}(\tau ,s))\). Since \({\mathfrak {p}}(\tau ,{\mathcal {X}}(\tau ,s))=p({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\) and \(p_{Y}(X,Y)=0\) we have
Furthermore, by (3.26f) and (3.9g), we have
We differentiate and get, since \(p_{Y}(X,Y)=0\),
Using (7.4), this leads to the estimate
which implies that
Recalling (7.34), it follows that \({\dot{{\mathfrak {p}}}}(\tau ,{\mathcal {X}}(\tau ,\cdot ))\in {L^\infty }([{\bar{s}}_{1},{\bar{s}}_{2}])\). Using (7.25), we end up with
and we conclude that \(\rho _{x}(\tau ,\cdot )\in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
In a similar way one shows that \(S_{x}(\tau ,\cdot ),\sigma _{x}(\tau ,\cdot )\in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
Step 3. Assume that the result holds for \(m=n\), that is, if \(u_{0}\in {L^\infty }([x_{l},x_{r}])\) and \(R_{0},S_{0},\rho _{0},\sigma _{0}\in W^{n-1,\infty }([x_{l},x_{r}])\), then \(u(\tau ,\cdot )\in W^{n,\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\), \(R(\tau ,\cdot )\), \(S(\tau ,\cdot ),\rho (\tau ,\cdot )\), \(\sigma (\tau ,\cdot )\in W^{n-1,\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\) and (P3) and (P4) hold. We show by induction that the result also holds for \(m=n+1\). Note that it is enough to show that if \(R_{0}\), \(S_{0}\), \(\rho _{0}\), \(\sigma _{0}\) belong to \(W^{n,\infty }([x_{l},x_{r}])\), then \(u(\tau ,\cdot )\in W^{n+1,\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\) and \(R(\tau ,\cdot ),S(\tau ,\cdot ),\rho (\tau ,\cdot ),\sigma (\tau ,\cdot )\in W^{n,\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
Since the result holds for \(m=n\) we get, following closely the argument used in Step 2 to derive (7.37) and (7.38), that
Since \(R_{0},S_{0},\rho _{0},\sigma _{0}\in W^{n,\infty }([x_{l},x_{r}])\) we get by Definitions 3.5 and 3.7 that \(\frac{\partial ^{\alpha +\beta }}{\partial X^{\alpha }\partial Y^{\beta }}Z\), \(\alpha ,\beta =0,1,\dots ,n+1\), \(\alpha +\beta =n+1\) is bounded on the curve \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\), \(s\in [s_{l},s_{r}]\).
Since the governing Eq. (2.19) is semilinear, there exists a unique solution \(\frac{\partial ^{n+1}}{\partial X^{\alpha }\partial Y^{\beta }}Z\), \(\alpha ,\beta =0,1,\dots ,n+1\), \(\alpha +\beta =n+1\) in \(\Omega \) of the system
where f and \(g_{\alpha ,\beta }\) depend on derivatives up to order n and \(g_{\alpha ,\beta }\) denotes \(n+1\) five dimensional vectors. By (7.42), the functions f and \(g_{\alpha ,\beta }\) are bounded. To clarify the notation, let us compute (7.43) for \(n=2\). We have
By Gronwall’s lemma, we obtain \(\frac{\partial ^{n+1}}{\partial X^{\alpha }\partial Y^{\beta }}Z\in [{L^\infty }(\Omega )]^{5}\), \(\alpha ,\beta =0,1,\ldots ,n+1\), \(\alpha +\beta =n+1\). This implies, since \(x_{X}({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s)){\dot{{\mathcal {X}}}}(\tau ,s)=x_{Y}({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s)){\dot{{\mathcal {Y}}}}(\tau ,s)\), \({\mathcal {Z}}_{2}(\tau ,s)=x({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\), and \({\mathcal {Z}}_{3}(\tau ,s)=U({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\), that \(\frac{d^{n+1}}{ds^{n+1}}{\mathcal {X}}(\tau ,\cdot )\), \(\frac{d^{n+1}}{ds^{n+1}}{\mathcal {Y}}(\tau ,\cdot )\), \(\frac{d^{n+1}}{ds^{n+1}}{\mathcal {Z}}_{2}(\tau ,\cdot )\), \(\frac{d^{n+1}}{ds^{n+1}}{\mathcal {Z}}_{3}(\tau ,\cdot )\in {L^\infty }([s_{l},s_{r}])\). It then follows from (7.25) and (7.26) that \(\frac{\partial ^{n+1}}{\partial x^{n+1}}u(\tau ,\cdot ) \in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\), and from (7.25) and (7.30) we obtain \(\frac{\partial ^{n}}{\partial x^{n}}R(\tau ,\cdot ) \in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
By (7.41), \(\frac{\partial ^{k}}{\partial X^{k}}p(\cdot ,{\mathcal {Y}}(\cdot ))\), \(k=0,1,\dots ,n\), is bounded on the curve \(({\mathcal {X}}(s),{\mathcal {Y}}(s))\), \(s\in [s_{l},s_{r}]\). Since \(\frac{\partial ^{k}}{\partial X^{k}}p(X,{\mathcal {Y}}(X))=\frac{\partial ^{k}}{\partial X^{k}}p(X,Y)\), we get from (7.25) and (7.31), \(\frac{\partial ^{n}}{\partial x^{n}}\rho (\tau ,\cdot ) \in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
Similarly, one proves that \(\frac{\partial ^{n}}{\partial x^{n}}S(\tau ,\cdot ),\frac{\partial ^{n}}{\partial x^{n}}\sigma (\tau ,\cdot ) \in L^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\). \(\square \)
Remark 7.2
A close inspection of the proof of (7.21) reveals that to any \(0\le \tau \le \frac{1}{2\kappa } (x_r-x_l)\) there exist \({\bar{s}}_{1}\) and \({\bar{s}}_{2}\) such that \(s_{l}<{\bar{s}}_{1}\le {\bar{s}}_{2}<s_{r}\), \(({\mathcal {X}}(\tau ,s),{\mathcal {Y}}(\tau ,s))\in \Omega \) for all \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\) and
In particular, this result also holds in the case \(\rho =0=\sigma \).
Remark 7.3
In [10] it has been highlighted that singularities appear, when either \(R=u_t+c(u)u_x=c(u)\frac{U_X}{x_X}\) or \(S=u_t-c(u)ux=-c(u)\frac{U_Y}{x_Y}\) become unbounded. As a closer look at the above proof reveals, both \(x_X\) and \(x_Y\) remain strictly positive in the studied region, cf. (7.11) and (7.12), while \(U_X\) and \(U_Y\) remain bounded. Thus neither R nor S can become unbounded in the studied region and hence no singularities form.
From Theorem 7.1 we obtain the following result.
Corollary 7.4
Let \(-\infty<x_{l}<x_{r}<\infty \) and consider \((u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\in {\mathcal {D}}\). Assume that
-
(A1’)
\(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0}\in C^{\infty }([x_{l},x_{r}])\),
-
(A2’)
there are constants \(d>0\) and \(e>0\) such that \(\rho _{0}(x)\ge d\) and \(\sigma _{0}(x)\ge e\) for all \(x\in [x_{l},x_{r}]\),
-
(A3’)
\(\mu _{0}\) and \(\nu _{0}\) are absolutely continuous on \([x_{l},x_{r}]\),
-
(A4’)
\(c\in C^{\infty }({\mathbb {R}})\) and \(c^{(m)}\in L^{\infty }({\mathbb {R}})\) for \(m=3,4,5,\ldots \).
For any \(\tau \in \big [0,\frac{1}{2\kappa }(x_{r}-x_{l})\big ]\) consider \((u,R,S,\rho ,\sigma ,\mu ,\nu )(\tau )={\bar{S}}_{\tau }(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0}, \nu _{0})\). Then
-
(P1’)
\(u(\tau ,\cdot ),R(\tau ,\cdot ),S(\tau ,\cdot ),\rho (\tau ,\cdot ),\sigma (\tau ,\cdot )\in C^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\),
-
(P2’)
there are constants \({\bar{d}}>0\) and \({\bar{e}}>0\) such that \(\rho (\tau ,x)\ge {\bar{d}}\) and \(\sigma (\tau ,x)\ge {\bar{e}}\) for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\),
-
(P3’)
\(\mu (\tau ,\cdot )\) and \(\nu (\tau ,\cdot )\) are absolutely continuous on \([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\).
For \(\tau \in \big [-\frac{1}{2\kappa }(x_{r}-x_{l}),0\big ]\), the solution satisfies the same properties on the interval \(\big [x_{l}-\kappa \tau ,x_{r}+\kappa \tau \big ]\).
7.2 Convergence results
In this section we state and prove convergence results needed to prove our main result, which is presented in Sect. 7.3. Our main theorem is a local result, and strictly speaking we only require local convergence of the Lagrangian variables to prove it. However, some of the steps can be done globally, see Lemmas 7.5 and 7.7, and we choose to present them here as it is more challenging than the local case. In the proof of our main result, Theorem 7.9, we will point out whenever the local convergence differs from Lemmas 7.5 and 7.7.
Lemma 7.5
Let \((u,R,S,\rho ,\sigma ,\mu ,\nu )\) and \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})\) belong to \({\mathcal {D}}\), and assume that \(\mu \), \(\nu \), \(\mu ^{n}\), and \(\nu ^{n}\) are absolutely continuous. Consider \(\psi _{1}=(x_{1},U_{1},J_{1},K_{1},V_{1}, H_{1})\), \(\psi _{2}=(x_{2},U_{2},J_{2},K_{2},V_{2},H_{2})\), \(\psi _{1}^{n}=(x_{1}^{n},U_{1}^{n},J_{1}^{n},K_{1}^{n},V_{1}^{n},H_{1}^{n})\), and \(\psi _{2}^{n}=(x_{2}^{n},U_{2}^{n}, \)\( J_{2}^{n},K_{2}^{n},V_{2}^{n},H_{2}^{n})\) defined by \((\psi _{1},\psi _{2})={\textbf{L}}(u,R,S,\rho ,\sigma ,\mu ,\nu )\) and \((\psi _{1}^{n},\psi _{2}^{n})={\textbf{L}}(u^{n},\)\(R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})\). Assume that
Then, \(x_{i}\) and \(x_{i}^{n}\) are strictly increasing, and
for \(i=1,2\).
We mention that the functions which converge in \(L^{1}({\mathbb {R}})\) also converge in \(L^{2}({\mathbb {R}})\), because convergence in \(L^{1}({\mathbb {R}})\) and (uniform) boundedness in \(L^{\infty }({\mathbb {R}})\) imply convergence in \(L^{2}({\mathbb {R}})\).
Proof
We start by proving that \(u^{n}\rightarrow u\) in \({L^\infty }({\mathbb {R}})\). By (3.1) and the Cauchy Schwarz inequality, we have
which implies, since \(R^{n}\rightarrow R\), \(S^{n}\rightarrow S\), and \(u^{n}\rightarrow u\) in \(L^{2}({\mathbb {R}})\), that \(u^{n}\rightarrow u\) in \({L^\infty }({\mathbb {R}})\).
Let \(\psi =(\psi _{1},\psi _{2})={\textbf{L}}(u,R,S,\rho ,\sigma ,\mu ,\nu )\) and \(\psi ^{n}=(\psi _{1}^{n},\psi _{2}^{n})={\textbf{L}}(u^{n},R^{n},S^{n}, \)\( \rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})\). We will only prove the results for the components of \(\psi _{1}\), the proof for \(\psi _{2}\) can be done in a similar way.
Let \(f=\frac{1}{4}(R^{2}+c(u)\rho ^{2})\) and \(f^{n}=\frac{1}{4}((R^{n})^{2}+c(u^{n})(\rho ^{n})^{2})\). From (3.2) we have \(\mu ((-\infty ,x))=\int _{-\infty }^{x}f(z)\,dz\) and \(\mu ^{n}((-\infty ,x))=\int _{-\infty }^{x}f^{n}(z)\,dz\), since \(\mu \) and \(\mu ^{n}\) are absolutely continuous measures. Consider the functions \(F(x)=x+\int _{-\infty }^{x}f(z)\,dz\) and \(F^{n}(x)=x+\int _{-\infty }^{x}f^{n}(z)\,dz\), which are strictly increasing and continuous. By (3.9a), we have that \(F(x_{1}(X))=X\) and \(F^{n}(x_{1}^{n}(X))=X\) for all \(X\in {\mathbb {R}}\), which implies that \(x_{1}\) and \(x_{1}^{n}\) are strictly increasing and continuous. Since \(x_{1}^{-1}=F\) and \((x_{1}^{n})^{-1}=F^{n}\), the inverses \(x_{1}^{-1}\) and \((x_{1}^{n})^{-1}\) exist and are strictly increasing and continuous. We prove that \((x_{1}^{n})^{-1}\rightarrow x_{1}^{-1}\) in \({L^\infty }({\mathbb {R}})\). Since \(|x_{1}^{-1}(x)-(x_{1}^{n})^{-1}(x)|\le ||f-f^{n}||_{L^{1}({\mathbb {R}})}\) we have to show that \(f^{n}\rightarrow f\) in \(L^{1}({\mathbb {R}})\). By using the estimate
and the Cauchy–Schwarz inequality, we find
Since \(R^{n}\rightarrow R\) and \(\rho ^{n}\rightarrow \rho \) in \(L^{2}({\mathbb {R}})\), and \(u^{n}\rightarrow u\) in \({L^\infty }({\mathbb {R}})\), it follows that \(f^{n}\rightarrow f\) in \(L^{1}({\mathbb {R}})\). Therefore \((x_{1}^{n})^{-1}\rightarrow x_{1}^{-1}\) in \({L^\infty }({\mathbb {R}})\).
We prove that \(x_{1}^{n}\rightarrow x_{1}\) in \({L^\infty }({\mathbb {R}})\). By direct calculations, we get for \(x_{1}^{n}(X)\le x_{1}(X)\), that
since \(f^{n}\ge 0\). Treating the case \( x_{1}(X)\le x_{1}^{n}(X)\) in a similar way, we end up with \(|x_{1}(X)-x_{1}^{n}(X)|\le ||f-f^{n}||_{L^{1}({\mathbb {R}})}\), which implies, since \(f^{n}\rightarrow f\) in \(L^{1}({\mathbb {R}})\), that \(x_{1}^{n}\rightarrow x_{1}\) in \({L^\infty }({\mathbb {R}})\). Using (3.9c) we get \(J_{1}^{n}\rightarrow J_{1}\) in \({L^\infty }({\mathbb {R}})\).
We prove that \((x_{1}^{n})'\rightarrow x_{1}'\) in \(L^{1}({\mathbb {R}})\). As in (7.2) we get \((x_{1}^{n})'=\frac{1}{f^{n}\circ x_{1}^{n}+1}\) and \(x_{1}'=\frac{1}{f\circ x_{1}+1}\), so that
For every \(\varepsilon >0\), there exists a function l in \(C_{c}({\mathbb {R}})\) such that \(||f-l||_{L^{1}({\mathbb {R}})}\le \varepsilon \). Here \(C_{c}({\mathbb {R}})\) denotes the space of continuous functions with compact support. Applying the triangle inequality to (7.45) yields
by a change of variables. Furthermore we used that \(0\le x_{1}'\le 1\) and \(0\le (x_{1}^{n})'\le 1\). It remains to show that \(\lim _{n\rightarrow \infty }||l\circ x_{1}-l\circ x_{1}^{n}||_{L^{1}({\mathbb {R}})}=0\). We have \(l\circ x_{1}^{n}\rightarrow l\circ x_{1}\) pointwise almost everywhere. In order to use the dominated convergence theorem we have to show that \(l\circ x_{1}^{n}\) can be uniformly bounded by a function which belongs to \(L^{1}({\mathbb {R}})\). We prove a slightly more general result, which will be used many times throughout the text.
Lemma 7.6
Assume that \(g\in C_{c}({\mathbb {R}})\), and that h and \(h_{n}\) satisfy \(h-{{\,\textrm{Id}\,}},h_{n}-{{\,\textrm{Id}\,}}\in L^{\infty }({\mathbb {R}})\), and \(h_{n}\rightarrow h\) in \(L^{\infty }({\mathbb {R}})\). Then there exists a constant \(0<K<\infty \) that is independent of n such that
where \(\chi _{[-K,K]}\) denotes the indicator function of the interval \([-K, K]\).
Proof
Since g has compact support there is a constant \(0<k<\infty \) such that \(\text {supp}(g)\subset [-k,k]\). Writing \(h_{n}(x)=h_{n}(x)-h(x)+h(x)-x+x\), we get \(\{x\mid h_{n}(x)\in [-k,k]\}\subset [-K_{n},K_{n}]\) where \(K_{n}=k+||h-h_{n}||_{L^{\infty }({\mathbb {R}})}+||h-{{\,\textrm{Id}\,}}||_{L^{\infty }({\mathbb {R}})}\). Since \(h_{n}\rightarrow h\) in \(L^{\infty }({\mathbb {R}})\) we can find a constant M such that \(||h-h_{n}||_{L^{\infty }({\mathbb {R}})}\le M\) for all n. If we set \(K=k+M+||h-{{\,\textrm{Id}\,}}||_{L^{\infty }({\mathbb {R}})}\), we get
which proves (7.47). \(\square \)
From (7.47) we conclude that \(l\circ x_{1}^{n}\) can be uniformly bounded by an \(L^{1}({\mathbb {R}})\) function. By the dominated convergence theorem we obtain \(\lim _{n\rightarrow \infty }||l\circ x_{1}-l\circ x_{1}^{n}||_{L^{1}({\mathbb {R}})}=0\). We conclude that the right-hand side of (7.46) can be made arbitrarily small, so that \((x_{1}^{n})'\rightarrow x_{1}'\) in \(L^{1}({\mathbb {R}})\) and as an immediate consequence \((J_1^n)'\rightarrow J_1'\) in \(L^1({\mathbb {R}})\) by (3.9c).
We show that \(U_{1}^{n}\rightarrow U_{1}\) in \({L^\infty }({\mathbb {R}})\). By (3.9d) and the Cauchy–Schwarz inequality we get
From (3.1) we have \(||u_{x}||_{L^{2}({\mathbb {R}})}\le \frac{\kappa }{2}(||R||_{L^{2}({\mathbb {R}})}+||S||_{L^{2}({\mathbb {R}})})\), and since \(x_{1}^{n}\rightarrow x_{1}\) and \(u^{n}\rightarrow u\) in \(L^{\infty }({\mathbb {R}})\), we conclude that \(U_{1}^{n}\rightarrow U_{1}\) in \({L^\infty }({\mathbb {R}})\).
Let us prove that \(U_{1}^{n}\rightarrow U_{1}\) in \(L^{2}({\mathbb {R}})\). Since \(u\in H^{1}({\mathbb {R}})\) there is for every \(\varepsilon >0\) a function \(\eta \in C_{c}({\mathbb {R}})\) such that \(||u-\eta ||_{L^{2}({\mathbb {R}})}\le \varepsilon \) and \(||u-\eta ||_{L^{\infty }({\mathbb {R}})}\le \varepsilon \). We have
Let \(D_{1}=\big \{X\in {\mathbb {R}} \ | \ x_{1}'(X)<\frac{1}{2}\big \}\) and \(D_{2}=\big \{Y\in {\mathbb {R}} \ | \ x_{2}'(Y)<\frac{1}{2}\big \}\). Observe that \({{\,\textrm{meas}\,}}(D_{1})\le \int _{\mathbb {R}}2J_1'(X)\,dX\) and \({{\,\textrm{meas}\,}}(D_{2})\le \int _{\mathbb {R}}2J_2'(Y)\,dY\), since \(J_i'=1-x_i'\), \(i=1,2\). Since \(\lim _{X\rightarrow -\infty }J_{1}(X)=0=\lim _{Y\rightarrow -\infty }J_{2}(Y)\), we get \(\text {meas}(D_{1})\le 2||J_{1}||_{L^{\infty }({\mathbb {R}})}\) and \(\text {meas}(D_{2})\le 2||J_{2}||_{L^{\infty }({\mathbb {R}})}\). In a similar way we find that the measures of the sets \(D_{1}^{n}=\big \{X\in {\mathbb {R}} \ | \ (x_{1}^{n})'(X)<\frac{1}{2}\big \}\) and \(D_{2}^{n}=\big \{Y\in {\mathbb {R}} \ | \ (x_{2}^{n})'(Y)<\frac{1}{2}\big \}\) have the bounds \(\text {meas}(D_{1}^{n})\le 2||J_{1}^{n}||_{L^{\infty }({\mathbb {R}})}\) and \(\text {meas}(D_{2}^{n})\le 2||J_{2}^{n}||_{L^{\infty }({\mathbb {R}})}\). Since \(J_{i}^{n}\rightarrow J_{i}\) in \({L^\infty }({\mathbb {R}})\), we can find constants \(E_{i}\) that are independent of n such that \(\text {meas}(D_{1}^{n})\le E_{1}\) and \(\text {meas}(D_{2}^{n})\le E_{2}\) for all n. We have
by a change of variables. Similarly,
and
Since \(u_{n}\rightarrow u\) in \(L^{2}({\mathbb {R}})\cap L^{\infty }({\mathbb {R}})\) we get that
For the remaining term in (7.48) we use the dominated convergence theorem. We have \(\eta \circ x_{1}^{n}\rightarrow \eta \circ x_{1}\) pointwise almost everywhere, and by Lemma 7.6 we find that \(\eta \circ x_{1}^{n}\) can be uniformly bounded by an \(L^{2}({\mathbb {R}})\) function. By the dominated convergence theorem we get
From (7.51) and (7.52), and since the right-hand sides of (7.49) and (7.50) can be made arbitrarily small, we conclude that \(U_{1}^{n}\rightarrow U_{1}\) in \(L^{2}({\mathbb {R}})\).
We prove that \(K_{1}^{n}\rightarrow K_{1}\) in \({L^\infty }({\mathbb {R}})\). By (3.9f) we get
For the first term on the right-hand side we use the Cauchy–Schwarz inequality and get
and for the second term we have
which implies that \(K_{1}^{n}\rightarrow K_{1}\) in \({L^\infty }({\mathbb {R}})\).
The above proof in fact also shows that \((K_{1}^{n})'\rightarrow K_{1}'\) in \(L^{1}({\mathbb {R}})\), since
We prove that \(H_{1}^{n}\rightarrow H_{1}\) in \(L^{2}({\mathbb {R}})\). Since \(\rho \) and \(\rho ^{n}\) belong to \(L^{2}({\mathbb {R}})\) there exist for every \(\varepsilon >0\) functions \(\phi \) and \(\phi ^{n}\) in \(C_{c}({\mathbb {R}})\) such that \(||\rho -\phi ||_{L^{2}({\mathbb {R}})}\le \varepsilon \) and \(||\rho ^{n}-\phi ^{n}||_{L^{2}({\mathbb {R}})}\le \varepsilon \). Since \(\rho ^{n}\rightarrow \rho \) in \(L^{2}({\mathbb {R}})\) we can for every \(\varepsilon >0\) choose n so large that \(||\rho -\rho ^{n}||_{L^{2}({\mathbb {R}})}\le \varepsilon \). This implies that for large n we have \(||\phi -\phi ^{n}||_{L^{2}({\mathbb {R}})}\le 3\varepsilon \). From (3.9g) we have
so that
Since \(0\le x_{1}'\le 1\) we get for the first term on the right-hand side by a change of variables,
Similarly we get for the third term,
For the second term we have
where we used that \(0\le x_{1}'\le 1\), \(0\le (x_{1}^{n})'\le 1\), and a change of variables for the last term on the right-hand side. We have \(\phi \circ x_{1}^{n}\rightarrow \phi \circ x_{1}\) pointwise almost everywhere, and by Lemma 7.6 we get that \(\phi \circ x_{1}^{n}\) can be uniformly bounded by an \(L^{2}({\mathbb {R}})\) function. By the dominated convergence theorem we have \(\lim _{n\rightarrow \infty }||\phi \circ x_{1}-\phi \circ x_{1}^{n}||_{L^{2}({\mathbb {R}})}=0\).
Using the estimates (7.55)–(7.57) in (7.54) we observe that all terms on the right-hand side can be made arbitrarily small, which implies that \(H_{1}^{n}\rightarrow H_{1}\) in \(L^{2}({\mathbb {R}})\).
We can prove in more or less the same way that \(V_{1}^{n}\rightarrow V_{1}\) in \(L^{2}({\mathbb {R}})\), where we also have to use that \(U_{1}^{n}\rightarrow U_{1}\) in \(L^{\infty }({\mathbb {R}})\) and the boundedness of c and \(c'\). \(\square \)
Lemma 7.7
Let \((\psi _{1},\psi _{2})\) and \((\psi _{1}^{n},\psi _{2}^{n})\) belong to \({\mathcal {F}}\), where \(\psi _{i}=(x_{i},U_{i},J_{i},K_{i},V_{i},H_{i})\) and \(\psi _{i}^{n}=(x_{i}^{n},U_{i}^{n},J_{i}^{n},K_{i}^{n},V_{i}^{n},H_{i}^{n})\), \(i=1,2\). Assume that \(x_{i}\) and \(x_{i}^{n}\) are strictly increasing, and that \(x_{i}+J_{i}={{\,\textrm{Id}\,}}\), and \(x_i^n+J_i^n={{\,\textrm{Id}\,}}\). Consider \(({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})={\textbf{C}}(\psi _{1},\psi _{2})\) and \(({\mathcal {X}}^{n},{\mathcal {Y}}^{n},{\mathcal {Z}}^{n},{\mathcal {V}}^{n},{\mathcal {W}}^{n},{\mathfrak {p}}^{n},{\mathfrak {q}}^{n})={\textbf{C}}(\psi _{1}^{n},\psi _{2}^{n})\). Assume
for \(i=1,2\). Then \({\mathcal {X}}\), \({\mathcal {Y}}\), \({\mathcal {X}}^{n}\), and \({\mathcal {Y}}^{n}\) are strictly increasing, and
for \(i=1,\dots ,5\).
Proof
Since \(x_{i}\) and \(x_{i}^{n}\) are continuous and strictly increasing for \(i=1,2\), we have by (3.23) that for every \(s\in {\mathbb {R}}\) there exist unique points \({\mathcal {X}}(s)\) and \({\mathcal {X}}^{n}(s)\) such that \(x_{1}({\mathcal {X}}(s))=x_{2}(2s-{\mathcal {X}}(s))\) and \(x_{1}^{n}({\mathcal {X}}^{n}(s))=x_{2}^{n}(2s-{\mathcal {X}}^{n}(s))\). Moreover, \({\mathcal {X}}\) and \({\mathcal {X}}^{n}\) are strictly increasing and continuous. It follows from (3.3c) that \({\mathcal {Y}}\) and \({\mathcal {Y}}^{n}\) are strictly increasing and continuous. Thus, the inverse functions \({\mathcal {X}}^{-1}\), \({\mathcal {Y}}^{-1}\), \(({\mathcal {X}}^{n})^{-1}\) and \(({\mathcal {Y}}^{n})^{-1}\) exist, and they are continuous and strictly increasing.
We prove that \({\mathcal {X}}^{n}\rightarrow {\mathcal {X}}\) in \({L^\infty }({\mathbb {R}})\). To begin with we show that \(J_{i}^{n}\circ (x_{i}^{n})^{-1}\rightarrow J_{i}\circ x_{i}^{-1}\) in \({L^\infty }({\mathbb {R}})\). Write
to get
Here, we used that \(0\le J_{i}'\le 1\), where the upper bound comes from the identity \(x_{i}'+J_{i}'=1\), and \(x_{i}'\ge 0\). Using the convergence assumptions implies that \(J_{i}^{n}\circ (x_{i}^{n})^{-1}\rightarrow J_{i}\circ x_{i}^{-1}\) in \({L^\infty }({\mathbb {R}})\).
We insert \(X={\mathcal {X}}(s)\) in \(x_{1}(X)+J_{1}(X)=X\) and get
Similarly we get
where we used that \(x_{1}({\mathcal {X}}(s))=x_{2}({\mathcal {Y}}(s))\). The expressions for \({\mathcal {X}}^{n}\) and \({\mathcal {Y}}^{n}\) are defined in a similar way. By (3.3c), \({\mathcal {X}}(s)+{\mathcal {Y}}(s)-{\mathcal {X}}^{n}(s)-{\mathcal {Y}}^{n}(s)=0\), which combined with the above expressions yields
Since \(J_{i}^{n}\circ (x_{i}^{n})^{-1}\) is increasing we get
Therefore
and
Thus, we showed that \(x_{1}^{n}\circ {\mathcal {X}}^{n}\rightarrow x_{1}\circ {\mathcal {X}}\) and \(J_{1}^{n}\circ {\mathcal {X}}^{n}\rightarrow J_{1}\circ {\mathcal {X}}\) in \({L^\infty }({\mathbb {R}})\) since \(J_{i}^{n}\circ (x_{i}^{n})^{-1}\rightarrow J_{i}\circ x_{i}^{-1}\) in \({L^\infty }({\mathbb {R}})\). Recallling (3.25b) and (3.25d), we thus have \({\mathcal {Z}}_{2}^{n}\rightarrow {\mathcal {Z}}_{2}\) and \({\mathcal {Z}}_{4}^{n}\rightarrow {\mathcal {Z}}_{4}\) in \(L^{\infty }({\mathbb {R}})\). Furthermore from (7.58), (3.3c), and (7.59) it immediately follows that \({\mathcal {X}}^{n}\rightarrow {\mathcal {X}}\) and \({\mathcal {Y}}^{n}\rightarrow {\mathcal {Y}}\) in \(L^{\infty }({\mathbb {R}})\).
We show that \({\mathcal {Z}}_{5}^{n}\rightarrow {\mathcal {Z}}_{5}\) in \(L^{\infty }({\mathbb {R}})\). By (3.25e) we have
and for the first line we get
A similar estimate for the second line yields \({\mathcal {Z}}_{5}^{n}\rightarrow {\mathcal {Z}}_{5}\) in \(L^{\infty }({\mathbb {R}})\).
By (3.25c) and the Cauchy–Schwarz inequality we have
which shows that \({\mathcal {Z}}_{3}^{n}\rightarrow {\mathcal {Z}}_{3}\) in \(L^{\infty }({\mathbb {R}})\).
We prove that \({\mathcal {Z}}_{3}^{n}\rightarrow {\mathcal {Z}}_{3}\) in \(L^{2}({\mathbb {R}})\). We have
where \(\langle \cdot ,\cdot \rangle \) denotes the inner product on \(L^{2}({\mathbb {R}})\). Since \({\dot{{\mathcal {X}}}}+{\dot{{\mathcal {Y}}}}=2\) we get from (3.25c) and a change of variables,
and similarly
Using that \({\dot{{\mathcal {X}}}}^{n}+{\dot{{\mathcal {Y}}}}^{n}=2\) we have
Since \({\mathcal {Z}}_{3}\in L^{2}({\mathbb {R}})\) there exists for every \(\varepsilon >0\) a function \(\phi \in C^{\infty }_{c}({\mathbb {R}})\) such that \(||{\mathcal {Z}}_{3}-\phi ||_{L^{2}({\mathbb {R}})}\le \varepsilon \). Write
By a change of variables we get
and in a similar way we obtain
where \(T_{2}^{n}\) is equal to \(T_{1}^{n}\) with \({\mathcal {X}}(s)\) and \({\mathcal {X}}^{n}(s)\) replaced by \({\mathcal {Y}}(s)\) and \({\mathcal {Y}}^{n}(s)\), respectively.
Using (7.61)–(7.64) in (7.60) we get
The strong convergence \(U_{i}^{n}\rightarrow U_{i}\) in \(L^{2}({\mathbb {R}})\) implies that \(||U_{i}^{n}||_{L^{2}({\mathbb {R}})}\rightarrow ||U_{i}||_{L^{2}({\mathbb {R}})}\) for \(i=1,2\). Thus, it remains to show that \(T_{i}^{n}\rightarrow 0\) for \(i=1,2\).
Using (3.25c) and \(0\le {\dot{{\mathcal {X}}}}^{n}\le 2\) we get by the Cauchy–Schwarz inequality and a change of variables,
and similarly
Since \(\phi \) has compact support, there exists \(k>0\) such that \(\text {supp}(\phi )\subset [-k,k]\). Integration by parts yields
where the first term in the integration by parts equals zero because \(\phi \) has compact support and the second integral is finite, since
for \(s\in [-k,k]\). We have
and similarly we obtain
By the Cauchy–Schwarz inequality we get
which implies that
Therefore
From (7.66), (7.67) and (7.69) we conclude that \(T_{1}^{n}\rightarrow 0\) as \(n\rightarrow \infty \). In a similar way we can prove that \(T_{2}^{n}\rightarrow 0\). This implies by (7.65) that \({\mathcal {Z}}_{3}^{n}\rightarrow {\mathcal {Z}}_{3}\) in \(L^{2}({\mathbb {R}})\).
Using (3.26a) we get
and recalling (7.44), we see that \({\mathcal {V}}_{1}^{n}\rightarrow {\mathcal {V}}_{1}\) in \(L^{2}({\mathbb {R}})\).
From (3.26b)–(3.26f) and the assumptions we immediately get that \({\mathcal {V}}_{i}^{n}\rightarrow {\mathcal {V}}_{i}\), \(i=2,\dots ,5\) and \({\mathfrak {p}}^{n}\rightarrow {\mathfrak {p}}\) in \(L^{2}({\mathbb {R}})\).
The corresponding results for \({\mathcal {W}}\) and \({\mathfrak {q}}\) can be proved in a similar way. \(\square \)
Theorem 7.9 deals with convergence of the elements u, \(\rho \), and \(\sigma \) for \((u,R,S,\rho ,\sigma ,\mu ,\nu )\) in \({\mathcal {D}}\), see (P1”) and (P2”) therein. The following result indicates the type of convergence we have to assume for elements in \({\mathcal {G}}_{0}\) in order to get weak-star convergence for the remaining elements in \({\mathcal {D}}\).
Lemma 7.8
Let \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})\) and \(\Theta ^{n}=({\mathcal {X}}^{n},{\mathcal {Y}}^{n},{\mathcal {Z}}^{n},{\mathcal {V}}^{n},{\mathcal {W}}^{n},{\mathfrak {p}}^{n},{\mathfrak {q}}^{n})\) belong to \({\mathcal {G}}_{0}\). Consider \((u,R,S,\rho ,\sigma ,\mu ,\nu )={\textbf{M}}\circ {\textbf{D}}(\Theta )\) and \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})={\textbf{M}}\circ {\textbf{D}}(\Theta ^{n})\). Assume that
for \(i=1,\dots ,5\). Then
and
Observe that there are no assumptions on the monotonicity of \(({\mathcal {X}},{\mathcal {Y}})\) and \(({\mathcal {X}}^{n},{\mathcal {Y}}^{n})\). This means that as functions of s they are nondecreasing, but not necessarily strictly increasing.
From these results it follows immediately that \(u^{n}_{x}\overset{*}{\rightharpoonup }u_{x}\) due to (3.1).
Proof
We will use Lemma 5.8. For any x, there exist s and \(s^{n}\), which are not necessarily unique, such that \(x={\mathcal {Z}}_{2}(s)\) and \(x={\mathcal {Z}}_{2}^{n}(s^{n})\). By (5.7a), we have \(u(x)={\mathcal {Z}}_{3}(s)\) and \(u^{n}(x)={\mathcal {Z}}_{3}^{n}(s^{n})\). We have
We estimate the difference \({\mathcal {Z}}_{3}(s)-{\mathcal {Z}}_{3}(s^{n})\). We assume that \(s^{n}\le s\), the other case can be treated similar. We have
by the Cauchy–Schwarz inequality. From (3.19c) and (3.22) we get
where we used that \({\mathcal {Z}}_{2}(s)={\mathcal {Z}}_{2}^{n}(s^{n})\). In a similar way, we obtain
Using \({\mathcal {Z}}_{2}(s)={\mathcal {Z}}_{2}^{n}(s^{n})\) once more we get
Similarly, we get
Combining (7.72)–(7.75) in (7.71) and using that \({\mathcal {Z}}_{2}^{n}\rightarrow {\mathcal {Z}}_{2}\) in \({L^\infty }({\mathbb {R}})\) we find that \({\mathcal {Z}}_{3}(s^{n})\rightarrow {\mathcal {Z}}_{3}(s)\). Using this and that \({\mathcal {Z}}_{3}^{n}\rightarrow {\mathcal {Z}}_{3}\) in \(L^{\infty }({\mathbb {R}})\) in (7.70) we conclude that \(u^{n}\rightarrow u\) in \({L^\infty }({\mathbb {R}})\).
We prove that \(\mu ^{n}\overset{*}{\rightharpoonup }\mu \), that is,
for all \(\phi \in C_{0}({\mathbb {R}})\). Here \(C_{0}({\mathbb {R}})\) is the space of continuous functions that vanish at infinity. Since \(C_{c}^{\infty }({\mathbb {R}})\) is dense in \(C_{0}({\mathbb {R}})\) it suffices to consider test functions \(\phi \) in \(C_{c}^{\infty }({\mathbb {R}})\). By (5.7f) we have
By (3.18) and (3.19b) we have \(2{\mathcal {V}}_{4}({\mathcal {X}}(s)){\dot{{\mathcal {X}}}}(s)={\dot{{\mathcal {Z}}}}_{4}(s)+c({\mathcal {Z}}_{3}(s)){\dot{{\mathcal {Z}}}}_{5}(s)\). Inserting this in (7.76) and using integration by parts yields
Combining this with the corresponding expression for \(\int _{{\mathbb {R}}}\phi (x)\,d\mu ^{n}\) yields
The three integrals on the right-hand side of (7.77) can be treated in more or less the same way, and we only consider the second one. We have
We have
since
We have \(\phi '\circ {\mathcal {Z}}_{2}^{n}\rightarrow \phi '\circ {\mathcal {Z}}_{2}\) pointwise almost everywhere, and by Lemma 7.6 we find that \(\phi '\circ {\mathcal {Z}}_{2}^{n}\) can be uniformly bounded by an \(L^{1}({\mathbb {R}})\) function. By the dominated convergence theorem we get \(\lim _{n\rightarrow \infty }||\phi '\circ {\mathcal {Z}}_{2}-\phi '\circ {\mathcal {Z}}_{2}^{n}||_{L^{1}({\mathbb {R}})}=0\), and (7.79) implies \(\lim _{n\rightarrow \infty }I_{1}^{n}=0\).
Integration by parts yields
which implies
By a change of variables and an estimate as in the proof of Lemma 7.6 we get
for a constant \({\tilde{C}}\) that is independent of n. Since \({\mathcal {Z}}_{2}^{n}\rightarrow {\mathcal {Z}}_{2}\) in \(L^{\infty }({\mathbb {R}})\) we get \(\lim _{n\rightarrow \infty }I_{2}^{n}=0\).
Following the same lines as above we find \(\lim _{n\rightarrow \infty }I_{3}^{n}=0=\lim _{n\rightarrow \infty }I_{4}^{n}\).
We return to (7.78) and obtain
By using (7.80) in (7.77) we get
for all \(\phi \in C_{c}^{\infty }({\mathbb {R}})\), and we conclude that \(\mu ^{n}\overset{*}{\rightharpoonup }\mu \). Similarly we prove that \(\nu ^{n}\overset{*}{\rightharpoonup }\nu \).
Next we show that \(\rho ^{n}\overset{*}{\rightharpoonup }\rho \), that is,
for all \(\phi \in L^{2}({\mathbb {R}})\). Since \(C_{c}^{\infty }({\mathbb {R}})\) is dense in \(L^{2}({\mathbb {R}})\), it suffices to consider test functions \(\phi \) in \(C_{c}^{\infty }({\mathbb {R}})\).
Consider a test function \(\phi \) in \(C_{c}^{\infty }({\mathbb {R}})\). By (5.7d), we get
Since \(\phi \) has compact support, there exists \(k>0\) such that \(\text {supp}(\phi )\subset [-k,k]\). Thus, we only integrate over s such that \(-k\le {\mathcal {Z}}_{2}(s)\le k\) in \(A_1^n\). Since the quantity \({\mathcal {Z}}_{2}-{{\,\textrm{Id}\,}}\) belongs to \(L^{\infty }({\mathbb {R}})\), the region we integrate over is contained in \(-k-||{\mathcal {Z}}_{2}-{{\,\textrm{Id}\,}}||_{L^{\infty }({\mathbb {R}})}\le s \le k+||{\mathcal {Z}}_{2}-{{\,\textrm{Id}\,}}||_{L^{\infty }({\mathbb {R}})}\), or written more compactly, \(-M\le s\le M\) with \(M=k+||{\mathcal {Z}}_{2}-{{\,\textrm{Id}\,}}||_{L^{\infty }({\mathbb {R}})}\). Integration by parts yields
By a change of variables we have
Using an estimate as in (7.68) yields
This implies that the first term on the right-hand side of (7.84) equals zero. Moreover, we get
where we used that
Since \({\mathcal {X}}^{n}\rightarrow {\mathcal {X}}\) in \(L^{\infty }({\mathbb {R}})\) and \({\mathfrak {p}}^{n}\rightarrow {\mathfrak {p}}\) in \(L^{2}({\mathbb {R}})\) we find that \(\lim _{n\rightarrow \infty }A_{1}^{n}=0\).
The second integral in (7.83) is estimated as follows. Using the Cauchy–Schwarz inequality and that \(0\le {\dot{{\mathcal {X}}}}^{n}\le 2\) we obtain
by a change of variables. We have \(\phi \circ {\mathcal {Z}}_{2}^{n}\rightarrow \phi \circ {\mathcal {Z}}_{2}\) pointwise almost everywhere, and by Lemma 7.6 we get that \(\phi \circ {\mathcal {Z}}^{n}_{2}\) can be uniformly bounded by an \(L^{2}({\mathbb {R}})\) function. Thus, the dominated convergence theorem implies that \(\lim _{n\rightarrow \infty } ||\phi \circ {\mathcal {Z}}_{2}-\phi \circ {\mathcal {Z}}^{n}_{2}||_{L^{2}({\mathbb {R}})}=0\). Since also \({\mathfrak {p}}^{n}\rightarrow {\mathfrak {p}}\) in \(L^{2}({\mathbb {R}})\) we find that \(\lim _{n\rightarrow \infty }A_{2}^{n}=0\).
We return to (7.83), and conclude that (7.82) holds. In a similar way one shows that \(\sigma ^{n}\overset{*}{\rightharpoonup }\sigma \).
To show that \(R^{n}\overset{*}{\rightharpoonup }R\), we consider a test function \(\phi \) in \(C_{c}^{\infty }({\mathbb {R}})\), use (5.7b), and write
where the terms on the right-hand side can be treated more or less as demonstrated earlier. In a similar way one shows that \(S^{n}\overset{*}{\rightharpoonup }S\). \(\square \)
7.3 Approximation by local smooth solutions
Theorem 7.9
Let \(-\infty<x_{l}<x_{r}<\infty \). Consider \(\xi _{0}=(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\) and \(\xi _{0}^{n}=(u_{0}^{n},R_{0}^{n},S_{0}^{n},\rho _{0}^{n},\sigma _{0}^{n},\mu _{0}^{n},\nu _{0}^{n})\) in \({\mathcal {D}}\). Assume that for all \(n\in {\mathbb {N}}\),
-
(A1”)
\(u_{0}, R_{0}, S_{0}, \rho _{0}, \sigma _{0}, u_{0}^{n}, R_{0}^{n}, S_{0}^{n}, \rho _{0}^{n}, \sigma _{0}^{n}\in C^{\infty }([x_{l},x_{r}])\),
-
(A2”)
\(\rho _{0}(x)=0\) and \(\sigma _{0}(x)=0\) for all \(x\in [x_{l},x_{r}]\),
-
(A3”)
there are constants \(d_{n}>0\) and \(e_{n}>0\) such that \(\rho _{0}^{n}(x)\ge d_{n}\) and \(\sigma _{0}^{n}(x)\ge e_{n}\) for all \(x\in [x_{l},x_{r}]\),
-
(A4”)
\(u_{0}^{n}\rightarrow u_{0}\) in \(L^{\infty }([x_{l},x_{r}])\), \(R_{0}^{n}\rightarrow R_{0}, \ S_{0}^{n}\rightarrow S_{0}, \ \rho _{0}^{n}\rightarrow \rho _{0}\), and \(\sigma _{0}^{n}\rightarrow \sigma _{0}\) in \(L^{2}([x_{l},x_{r}])\),
-
(A5”)
\(\mu _{0}, \ \nu _{0}, \ \mu _{0}^{n}, \text { and } \nu _{0}^{n}\) are absolutely continuous on \([x_{l},x_{r}]\),
-
(A6”)
\(\mu _{0}((-\infty ,x_{l}))=\mu _{0}^{n}((-\infty ,x_{l}))\), \(\mu _{0}((-\infty ,x_{r},))=\mu _{0}^{n}((-\infty ,x_{r}))\), \(\nu _{0}((-\infty ,x_{l}))=\nu _{0}^{n}((-\infty ,x_{l}))\), and \(\nu _{0}((-\infty ,x_{r},))=\nu _{0}^{n}((-\infty ,x_{r}))\),
-
(A7”)
\(c\in C^{\infty }({\mathbb {R}})\) and \(c^{(m)}\in L^{\infty }({\mathbb {R}})\) for \(m=3,4,5,\ldots \).
For any \(\tau \in \big [0,\frac{1}{2\kappa }(x_{r}-x_{l})\big ]\) consider \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})(\tau )={\bar{S}}_{\tau }(\xi _{0}^{n})\) and \((u,R,S,\rho ,\sigma ,\mu ,\nu )(\tau )={\bar{S}}_{\tau }(\xi _{0})\). Then we have
-
(P1”)
\(u^{n}(\tau ,\cdot )\rightarrow u(\tau ,\cdot )\) in \({L^\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\),
-
(P2”)
\(\rho ^{n}(\tau ,\cdot )\rightarrow 0\) and \(\sigma ^{n}(\tau ,\cdot )\rightarrow 0\) in \(L^{1}([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
The same conclusion holds on the interval \(\big [x_{l}-\kappa \tau ,x_{r}+\kappa \tau \big ]\) in the case where \(\tau \in \big [-\frac{1}{2\kappa }(x_{r}-x_{l}),0\big ]\).
Observe that we assume in (A4”) \(u_{0}^{n}\rightarrow u_{0}\) in \(L^{\infty }([x_{l},x_{r}])\), in contrast to the \(L^2({\mathbb {R}})\) convergence in the global case, see Lemma 7.5. This is because functions in \(H^{1}({\mathbb {R}})\) tend to 0 as \(x\rightarrow \pm \infty \), while here we have no assumptions on the values of \(u_{0}\) and \(u_{0}^{n}\) at the endpoints of \([x_{l},x_{r}]\). Since \([x_{l},x_{r}]\) is a bounded interval, the convergence in \(L^{\infty }([x_{l},x_{r}])\) implies that \(u_{0}^{n}\rightarrow u_{0}\) in \(L^{2}([x_{l},x_{r}])\).
By the assumptions (A5”) and (A6”) we mean that \(\mu _{0}([x_{l},x_{r}])=\mu _{0}^{n}([x_{l},x_{r}])\) and \(\nu _{0}([x_{l},x_{r}])=\nu _{0}^{n}([x_{l},x_{r}])\) for all n.
Note that the approximating sequence in Theorem 7.9 is locally smooth. Indeed, by Corollary 7.4 we have \(u^{n}(\tau ,\cdot ),R^{n}(\tau ,\cdot ),S^{n}(\tau ,\cdot ),\rho ^{n}(\tau ,\cdot ),\sigma ^{n}(\tau ,\cdot )\in C^{\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\). Furthermore, there are constants \({\bar{d}}_{n}>0\) and \({\bar{e}}_{n}>0\) such that \(\rho ^{n}(\tau ,x)\ge {\bar{d}}_{n}\) and \(\sigma ^{n}(\tau ,x)\ge {\bar{e}}_{n}\) for all \(x\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\), and \(\mu ^{n}(\tau ,\cdot )\) and \(\nu ^{n}(\tau ,\cdot )\) are absolutely continuous on \([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\) for all n. However, the limit solution does not in general satisfy these properties, which is illustrated in an example in [15].
In addition, an inspection of the proof shows, that the above theorem actually states that locally, i.e., on \(\{x\in [x_l+\kappa \tau , x_r-\kappa \tau ] \mid \tau \in [0, \frac{1}{2\kappa }(x_r-x_l)]\}\), the conservative solution \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})(\tau )={\bar{S}}_{\tau }(\xi _{0}^{n})\) to (1.3) converges to the conservative solution \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})(\tau )={\bar{S}}_{\tau }(\xi _{0}^{n})\) of the nonlinear variational wave equation (1.1), since \({\mathfrak {p}}^n\rightarrow 0\) and \({\mathfrak {q}}^n\rightarrow 0\).
Proof
We will only consider the case \(0<\tau \le \frac{1}{2\kappa }(x_{r}-x_{l})\). The case \(-\frac{1}{2\kappa }(x_{r}-x_{l})\le \tau <0\) can be treated in the same way.
We split the proof into four steps.
Step 1. Set \((\psi _{1},\psi _{2})={\textbf{L}}(u_{0},R_{0},S_{0},\rho _{0},\sigma _{0},\mu _{0},\nu _{0})\) and \((\psi _{1}^{n},\psi _{2}^{n}) = {{\textbf {L}}}(u^{n}_{0},R^{n}_{0},S^{n}_{0},\rho ^{n}_{0},\sigma ^{n}_{0},\mu ^{n}_{0},\nu ^{n}_{0})\), where \(\psi _{1}=(x_{1},U_{1},J_{1},K_{1},V_{1},H_{1})\), \(\psi _{2}=(x_{2},U_{2},J_{2},K_{2},V_{2},H_{2})\), \(\psi _{1}^{n}=(x_{1}^{n},U_{1}^{n},J_{1}^{n},K_{1}^{n},V_{1}^{n},H_{1}^{n})\), and \(\psi _{2}^{n}=(x_{2}^{n},U_{2}^{n},J_{2}^{n},K_{2}^{n},V_{2}^{n},H_{2}^{n})\). Let us find out what kind of region the interval \([x_{l},x_{r}]\) corresponds to in Lagrangian coordinates (X, Y). Since the measures are assumed to be absolutely continuous we get from (3.9a), \(x_{1}(X)+\mu _{0}((-\infty ,x_{1}(X)))=X\) for all \(x_{1}(X)\in [x_{l},x_{r}]\). We show which range of the X-variable this corresponds to. If \({\hat{X}}\) is such that \(x_{1}({\hat{X}})=x_{l}\) then \(x_{l}+\mu _{0}((-\infty ,x_{l}))={\hat{X}}\). We also have \(x_{1}^{n}(X)+\mu _{0}^{n}((-\infty ,x_{1}^{n}(X)))=X\) for all \(x_{1}^{n}(X)\in [x_{l},x_{r}]\). If \({\check{X}}\) is such that \(x_{1}^{n}({\check{X}})=x_{l}\) we get \(x_{l}+\mu _{0}^{n}((-\infty ,x_{l}))={\check{X}}\). Using (A6”) we obtain \({\hat{X}}={\check{X}}\) and we denote \(X_{l}={\hat{X}}={\check{X}}\). If \({\bar{X}}\) and \({\tilde{X}}\) are such that \(x_{1}({\bar{X}})=x_{1}^{n}({\tilde{X}})=x_{r}\) then \(x_{r}+\mu _{0}((-\infty ,x_{r}))={\bar{X}}\) and \(x_{r}+\mu _{0}^{n}((-\infty ,x_{r}))={\tilde{X}}\), and by (A6”) we get \({\bar{X}}={\tilde{X}}\) which we denote \(X_{r}={\bar{X}}={\tilde{X}}\). We have \(X_{l}\le X_{r}\) since \(X_{l}=x_{l}+\mu _{0}((-\infty ,x_{l}))\le x_{r}+\mu _{0}((-\infty ,x_{r}))=X_{r}\). In other words, we can define the interval \([X_{l},X_{r}]\) using either the measure \(\mu _{0}\) or \(\mu _{0}^{n}\). Observe that this is a consequence of the assumptions (A5”) and (A6”).
In a similar way we find that \(x_{2}(Y)+\nu _{0}((-\infty ,x_{2}(Y)))=Y\) and \(x_{2}^{n}(Y)+\nu _{0}^{n}((-\infty ,x_{2}^{n}(Y)))=Y\) for all \(Y\in [Y_{l},Y_{r}]\) where \(Y_{l}=x_{l}+\nu _{0}((-\infty ,x_{l}))\) and \(Y_{r}=x_{r}+\nu _{0}((-\infty ,x_{r}))\). Let \(\Omega =[X_{l},X_{r}]\times [Y_{l},Y_{r}]\).
Following closely the proof of Lemma 7.5, we obtain that \(x_{1}\), \(x_{1}^{n}\), \(x_{2}\), and \(x_{2}^{n}\) are strictly increasing for \(X\in [X_l, X_r]\) and \(Y\in [Y_l,Y_r]\), respectively, and
Note that (3.9f) does not imply \(K_{1}^{n}\rightarrow K_{1}\) in \(L^{\infty }([X_{l},X_{r}])\) and \(K_{2}^{n}\rightarrow K_{2}\) in \(L^{\infty }([Y_{l},Y_{r}])\). However, we have \(K_{1}^{n}-K_{1}^{n}(X_{l})\rightarrow K_{1}-K_{1}(X_{l})\) in \(L^{\infty }([X_{l},X_{r}])\) and \(K_{2}^{n}-K_{2}^{n}(Y_{l})\rightarrow K_{2}-K_{2}(Y_{l})\) in \(L^{\infty }([Y_{l},Y_{r}])\). The proof of this closely follows the investigation of (7.53) in Lemma 7.5.
Step 2. Let \(\Theta =({\mathcal {X}},{\mathcal {Y}},{\mathcal {Z}},{\mathcal {V}},{\mathcal {W}},{\mathfrak {p}},{\mathfrak {q}})={\textbf{C}}(\psi _{1},\psi _{2})\) and \(\Theta ^{n}=({\mathcal {X}}^{n},{\mathcal {Y}}^{n},{\mathcal {Z}}^{n},{\mathcal {V}}^{n},{\mathcal {W}}^{n},{\mathfrak {p}}^{n},{\mathfrak {q}}^{n})={\textbf{C}}(\psi _{1}^{n},\psi _{2}^{n})\). From (3.23) and since both \(x_{1}\) and \(x_{2}\) are strictly increasing on \([X_{l},X_{r}]\) and \([Y_{l},Y_{r}]\), respectively, the relevant values of s are those satisfying \({\mathcal {X}}(s)\in [X_{l},X_{r}]\) and \({\mathcal {Y}}(s)\in [Y_{l},Y_{r}]\). In other words, since \(x_1(X_l)=x_2(Y_l)=x_l\) and \(x_1(X_r)=x_2(Y_r)=x_r\), we have \(\frac{1}{2}(X_{l}+Y_{l})\le s\le \frac{1}{2}(X_{r}+Y_{r})\) and we let \(s_{l}=\frac{1}{2}(X_{l}+Y_{l})\) and \(s_{r}=\frac{1}{2}(X_{r}+Y_{r})\). The same conclusion holds for \(({\mathcal {X}}^{n}(s),{\mathcal {Y}}^{n}(s))\). In particular, one has
i.e., the lower left and upper right corner of \(\Omega \).
Following closely the proof of Lemma 7.7, we obtain that \({\mathcal {X}}\), \({\mathcal {Y}}\), \({\mathcal {X}}^{n}\), and \({\mathcal {Y}}^{n}\) are strictly increasing on \([s_l, s_r]\), and
for \(i=1,\dots ,5\), \(j=1,\dots ,4\). Using (A2”), (A3”), and (7.3), we get from (3.26f) and (3.9g), that for all \(X\in [X_{l},X_{r}]\), \({\mathfrak {p}}(X)=0\) and \({\mathfrak {p}}^{n}(X)\ge k_{n}>0\) for some constant \(k_{n}\).
Since \(K_{1}^{n}-K_{1}^{n}(X_{l})\rightarrow K_{1}-K_{1}(X_{l})\) in \(L^{\infty }([X_{l},X_{r}])\) and \(K_{2}^{n}-K_{2}^{n}(Y_{l})\rightarrow K_{2}-K_{2}(Y_{l})\) in \(L^{\infty }([Y_{l},Y_{r}])\) we get from (3.25e) and (7.85) that \({\mathcal {Z}}_{5}^{n}-{\mathcal {Z}}_{5}^{n}(s_{l})\rightarrow {\mathcal {Z}}_{5}-{\mathcal {Z}}_{5}(s_{l})\) in \({L^\infty }([s_{l},s_{r}])\).
Step 3. Consider \((Z,p,q)={\textbf{S}}(\Theta )\) and \((Z^{n},p^{n},q^{n})={\textbf{S}}(\Theta ^{n})\). We prove a Gronwall type estimate. We claim that for all (X, Y) in \(\Omega \),
where K depends on \(\kappa \), \(k_{1}\), \(k_{2}\), \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\) and the size of \(\Omega \).
Let \((X,Y)\in \Omega \). Subtracting the equations
and
yields, using \(Z_{X}(X,{\mathcal {Y}}(X))={\mathcal {V}}(X)\) and \(Z_{X}^n(X,{\mathcal {Y}}^n(X))={\mathcal {V}}^n(X)\),
Using (2.19a) we get
Next we need a pointwise uniform bound on the components of \(Z_{X}^{n}\) and \(Z_{Y}^{n}\). By (2.19a) we get
and by doing the same kind of estimate for the other components we get
for a constant \(B_{1}\) that only depends on \(\kappa \) and \(k_{1}\). By (7.87) and the corresponding expressions for \(x^{n}_{X}\), \(U^{n}_{X}\), \(J^{n}_{X}\) and \(K^{n}_{X}\) we obtain
where we used (7.90), and by Gronwall’s inequality,
By (4.12a), (4.12b), (4.12d), and (4.12e) we get
From (4.12c) we have \(2J_{Y}^{n}x_{Y}^{n}\ge (c(U^{n})U_{Y}^{n})^{2}\), and by Young’s inequality, \(|U_{Y}^{n}|\le \frac{\kappa }{\sqrt{2}}(J_{Y}^{n}+x_{Y}^{n})\), which by (7.92) implies
Using this in (7.91) we obtain
for a new constant \(B_{2}\) that again depends on \(\kappa \) and \(k_{1}\). Since \(x^{n}\) and \(J^{n}\) are nondecreasing with respect to both variables, we have
Since
and
we end up with
The convergence \({\mathcal {Z}}_{i}^{n}\rightarrow {\mathcal {Z}}_{i}\) in \(L^{\infty }([s_{l},s_{r}])\) implies that for every \(\varepsilon >0\) there exists N such that for all \(n\ge N\), we have \(||{\mathcal {Z}}_{i}-{\mathcal {Z}}_{i}^{n}||_{L^{\infty }([s_{l},s_{r}])}\le \varepsilon \), \(i=2,4\). Hence,
We use (7.94) and \(Z_{i,X}^{n}(X,{\mathcal {Y}}^{n}(X))={\mathcal {V}}_{i}^{n}(X)\) in (7.93) and obtain
where \(B_{3}\) only depends on \(\kappa \), \(k_{1}\) and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\). Since \(\mu _{0}^{n}\) is absolutely continuous in \([x_{l},x_{r}]\) we obtain as in (7.2) that \(0\le (x_{1}^{n})'(X)\le 1\) for all \(X\in [X_{l},X_{r}]\). Using (3.26a), (3.26b), (3.26d), (3.26e), (3.9c) and (3.9f) we get \(0\le {\mathcal {V}}_{1}^{n}(X)\le \frac{1}{2}\kappa \), \(0\le {\mathcal {V}}_{2}^{n}(X)\le \frac{1}{2}\), \(0\le {\mathcal {V}}_{4}^{n}(X)\le 1\), \(0\le {\mathcal {V}}_{5}^{n}(X)\le \kappa \) for all \(X\in [X_{l},X_{r}]\). From (3.26c) and (3.7c) we have \(|{\mathcal {V}}_{3}^{n}(X)|\le \frac{1}{c(U_{1}^{n}(X))}\sqrt{(J_{1}^{n})'(X)(x_{1}^{n})'(X)}\le \kappa \). Inserting these estimates in (7.95) we get
for a new constant \(B_{4}\), which only depends on \(\kappa \), \(k_{1}\), and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\). Similarly we can show that there exist constants \(B_{5}\), \(B_{6}\), and \(B_{7}\), which only depend on \(\kappa \), \(k_{1}\), and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\), such that for all \((X,Y)\in \Omega \),
and
From (7.90) we get for all \((X,Y)\in \Omega \) that
for a constant D that depends on \(\kappa \), \(k_{1}\), and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\). From (7.96)–(7.99) we get in (7.89), for all \((X,Y)\in \Omega \) that
where \(C_{1}\) depends on \(\kappa \), \(k_{1}\), \(k_{2}\), and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\).
Recall the notation in (4.11). Using the estimates (7.101) and (7.100) in (7.88) we get
To write the estimates more compactly, we denote \(Z=(Z_{1},Z_{2},Z_{3},Z_{4},Z_{5})=(t,x,U,J,K)\), and similar for \(Z^{n}\). By the same procedure as above, we obtain
for \(j=1,2,3,4,5\), where \(C_{j}\) depends on \(\kappa \), \(k_{1}\), \(k_{2}\) and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\), and D depends on \(\kappa \), \(k_{1}\) and \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\).
Following the same lines as above we find estimates for the partial derivatives with respect to Y. We obtain for \(j=1,2,3,4,5\),
We have
To any X in \([X_l, X_r]\), there exist unique s and \(s^{n}\) in \([s_l,s_r]\) such that \(X={\mathcal {X}}(s)\) and \(X={\mathcal {X}}^{n}(s^{n})\) and we can write
This implies, using the alternative notation,
where we used (7.97).
Applying Hölder’s inequality to (7.102)–(7.104) yields for all \((X,Y)\in \Omega \),
where we introduced \(C=\sum _{j=1}^{5}C_{j}^{2}\).
At this point we need a Gronwall inequality, which can be found in [9], see Theorem 1, case 2, at the beginning of the chapter “Gronwall inequalities in higher dimensions”. For a proof, we refer to [15].
Lemma 7.10
Consider a nonnegative function u(x, y) in the region \(x\ge 0\) and \(y\ge 0\). Assume that
where a, b, and c are nonnegative constants. Then we have
Using Lemma 7.10 we get from (7.105),
Since all the differences appearing in the exponential function are bounded by either \(X_{r}-X_{l}\) or \(Y_{r}-Y_{l}\) we can find a new constant K which depends on \(\kappa \), \(k_{1}\), \(k_{2}\), \(|||\Theta |||_{{\mathcal {G}}(\Omega )}\), and the size of \(\Omega \) such that (7.86) holds. Thus, we have proved the claim.
From (7.86) we obtain an estimate for the difference \(Z_{i}(X,2\,s-X)-Z_{i}^{n}(X,2\,s-X)\) for \(i=1,2,3,4\). We have
which implies by (7.96), (7.97), and the Cauchy–Schwarz inequality, that
From (7.86), we have
Integration and a change of variables leads to
From (3.3c) we have \(X+{\mathcal {Y}}\circ {\mathcal {X}}^{-1}(X)=2{\mathcal {X}}^{-1}(X)\) and \(X+{\mathcal {Y}}^{n}\circ ({\mathcal {X}}^{n})^{-1}(X)=2({\mathcal {X}}^{n})^{-1}(X)\), so that
This leads to
To estimate the above integral, we need to introduce another sequence of curves on \([s_l, s_r]\) given by \(({{\hat{{\mathcal {X}}}}}^n(s), {{\hat{{\mathcal {Y}}}}}^n(s))=(\max \{{\mathcal {X}}(s), {\mathcal {X}}^n(s)\}, \min \{{\mathcal {Y}}(s), {\mathcal {Y}}^n(s)\})\). This sequence satisfies that both \(({\mathcal {X}}, {\mathcal {Y}})\) and \(({\mathcal {X}}^n, {\mathcal {Y}}^n)\) lie above or are equal to \(({{\hat{{\mathcal {X}}}}}^n, {{\hat{{\mathcal {Y}}}}}^n)\). This implies that \({\mathcal {X}}^{-1}(X)\ge ({{\hat{{\mathcal {X}}}}}^{n})^{-1}(X)\) and \(({\mathcal {X}}^n)^{-1}(X)\ge ({{\hat{{\mathcal {X}}}}}^{n})^{-1}(X)\) for all X in \([X_l,X_r]\), and that
By a change of variables and integration by parts we get
and similarly we find
Note that \({\mathcal {X}}^{-1}(X_{l})=s_l=({{\hat{{\mathcal {X}}}}}^{n})^{-1}(X_{l})\) and \({\mathcal {X}}^{-1}(X_{r})=s_r=({\hat{{\mathcal {X}}}}^{n})^{-1}(X_{r})\). Therefore,
We obtain in a similar way,
Combining (7.111) and the above estimates we end up with
which inserted in (7.110) yields
By similar calculations we find
Returning to (7.108) we now get
which we insert in (7.107) and get
for \(i=1,\dots ,4\). A similar inequality is valid for
the only difference from (7.114) being that we get \(||({\mathcal {Z}}_{5}-{\mathcal {Z}}_{5}(s_{l}))-({\mathcal {Z}}_{5}^{n}-{\mathcal {Z}}_{5}^{n}(s_{l}))||_{L^{\infty }([s_{l},s_{r}])}\) on the right-hand side.
Step 4. Let \(0<\tau \le \frac{1}{2\kappa }(x_{r}-x_{l})\) and consider \(\Theta (\tau )={\textbf{E}}\circ {\textbf{t}}_{\tau }(Z,p,q)\), \(\Theta ^{n}(\tau )={\textbf{E}}\circ {\textbf{t}}_{\tau }(Z^{n},p^{n},q^{n})\), \((u,R,S,\rho ,\sigma ,\mu ,\nu )(\tau )={\textbf{M}}\circ {\textbf{D}}(\Theta (\tau ))\) and \((u^{n},R^{n},S^{n},\rho ^{n},\sigma ^{n},\mu ^{n},\nu ^{n})(\tau )={\textbf{M}}\circ {\textbf{D}}(\Theta ^{n}(\tau ))\).
We prove (P1”).
Recall Remark 7.2 and consider \(z\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). Then there exist \(s\in [{\bar{s}}_{1},{\bar{s}}_{2}]\) and \(s^{n}\in [{\bar{s}}_{1}^{n},{\bar{s}}_{2}^{n}]\) such that \(z={\mathcal {Z}}_{2}(\tau ,s)={\mathcal {Z}}_{2}^{n}(\tau ,s^{n})\). Using (5.7a) we obtain
Now we can have several different scenarios depending on the order of the points s and \(s^{n}\), and the intervals \([{\bar{s}}_{1},{\bar{s}}_{2}]\) and \([{\bar{s}}_{1}^{n},{\bar{s}}_{2}^{n}]\). We only show one of the challenging cases, the others can be treated in a similar way. Assume that \(s\le {\bar{s}}_{1}^{n}\le {\bar{s}}_{\text {max}}\). Here \([{\bar{s}}_{\text {min}}, {\bar{s}}_{\text {max}}]\) denotes the maximal interval such that \(({\mathcal {X}}(\tau ,s), {\mathcal {Y}}(\tau ,s))\) belongs to \(\Omega \) for all \(s\in [{\bar{s}}_{\text {min}}, {\bar{s}}_{\text {max}}]\). Observe that in this case the point \(({\mathcal {X}}(\tau ,{\bar{s}}_{1}^{n}),{\mathcal {Y}}(\tau ,{\bar{s}}_{1}^{n}))\) is in \(\Omega \), but the point \(({\mathcal {X}}^{n}(\tau ,s),{\mathcal {Y}}^{n}(\tau ,s))\) may be outside \(\Omega \). So when estimating (7.115) we have to carefully choose points on the curve so that we do not end up outside \(\Omega \), see Fig. 5.
An example of the situation described in Step 4. Note that the point \(({\mathcal {X}}^{n}(\tau ,s),{\mathcal {Y}}^{n}(\tau ,s))\) lies outside \(\Omega \)
Write
By (3.18) and the Cauchy–Schwarz inequality we have
From (3.19c) and (3.22) we get
where we used that \({\mathcal {Z}}_{2}(\tau ,s)={\mathcal {Z}}_{2}^{n}(\tau ,s^{n})\) and \({\mathcal {Z}}_{2}^{n}(\tau ,{\bar{s}}_{1}^{n})\le {\mathcal {Z}}_{2}^{n}(\tau ,s^{n})\) since \({\bar{s}}_{1}^{n}\le s^{n}\) and \({\mathcal {Z}}_{2}^{n}(\tau ,\cdot )\) is nondecreasing. We also used that since \(({\mathcal {X}}(\tau ,r),{\mathcal {Y}}(\tau ,r))\in \Omega \) for \(r\in [s,{\bar{s}}_{1}^{n}]\) we can use the estimate in (7.98) to obtain \(|{\mathcal {V}}_{4}(\tau ,{\mathcal {X}}(\tau ,r))|=|J_{X}({\mathcal {X}}(\tau ,r),{\mathcal {Y}}(\tau ,r))|\le B_{6}\). We have
By (3.3c), (4.12a), and since \(t_{X}\ge 0\) and \(t_{Y}\le 0\) we get
Since \(t({\mathcal {X}}(\tau ,{\bar{s}}_{1}^{n}),{\mathcal {Y}}(\tau ,{\bar{s}}_{1}^{n}))=\tau =t^{n}({\mathcal {X}}^{n}(\tau ,{\bar{s}}_{1}^{n}),{\mathcal {Y}}^{n}(\tau ,{\bar{s}}_{1}^{n}))\) we have
Therefore,
and (7.118) yields
We insert this in (7.117) and get
Similarly we get
We have
Using these estimates in (7.116) gives
for a constant \(v_{1}\) that is independent of n.
By (4.12c) we have the estimates \(|U_{X}|\le \frac{\sqrt{2J_{X}x_{X}}}{c(U)}\) and \(|U_{Y}|\le \frac{\sqrt{2J_{Y}x_{Y}}}{c(U)}\), which leads to
since \(x_{Y}\ge 0\) and \(J_{Y}\ge 0\). From (4.12a) we get
where the last equality follows from (7.119). Using (7.98) and (7.99) yields
Now we get
for a constant \(v_{2}\) which is independent of n.
By estimating as we did for \(A_{1}^{n}\) we get
for a constant \(v_{3}\) that is independent of n.
Using (7.121), (7.122) and (7.123) in (7.115) yields
and we conclude, after using (7.114), that \(u^{n}(\tau ,\cdot )\rightarrow u(\tau ,\cdot )\) in \({L^\infty }([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
Since \(p_{Y}=p_{Y}^{n}=0\) we have \({\mathfrak {p}}(\tau ,X)=p(X,{\mathcal {Y}}(\tau ,{\mathcal {X}}^{-1}(\tau ,X)))=p(X,{\mathcal {Y}}({\mathcal {X}}^{-1}(X)))={\mathfrak {p}}(X)\), and similarly \({\mathfrak {p}}^{n}(\tau ,X)={\mathfrak {p}}^{n}(X)\). From Step 2 we get \({\mathfrak {p}}(\tau ,X)=0\) and \({\mathfrak {p}}^{n}(\tau ,X)\ge k_{n}>0\) for all \(X\in [X_{l},X_{r}]\). Then by (5.7d) we obtain
so that by a change of variables,
Since \({\mathfrak {p}}^{n}\rightarrow 0\) in \(L^{2}([X_{l},X_{r}])\) this implies that
As mentioned in the comment before the proof, there is for any n a constant \({\bar{d}}_{n}>0\) such that \(\rho ^{n}(\tau ,z)\ge {\bar{d}}_{n}\) for all \(z\in [x_{l}+\kappa \tau ,x_{r}-\kappa \tau ]\). Since \(\rho ^{n}(\tau ,\cdot )\) is positive, (7.124) is the same as \(\rho ^{n}(\tau ,\cdot )\rightarrow 0\) in \(L^{1}([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\).
Similarly we obtain \(\sigma ^{n}(\tau ,\cdot )\rightarrow 0\) in \(L^{1}([x_{l}+\kappa \tau ,x_{r}-\kappa \tau ])\). This concludes the proof of (P2”). \(\square \)
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
Note that condition (i) implies \(\left\| \Theta \right\| _{{\mathcal {G}}(\Omega )}<\infty \) because \(\Omega \) is bounded.
The push-forward of a measure \(\lambda \) by a function f is the measure \(f_{\#}\lambda \) defined by \(f_{\#}\lambda (B)=\lambda (f^{-1}(B))\) for Borel sets B.
Note that this is also true outside \(\Omega \).
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Acknowledgements
The authors are very grateful to Xavier Raynaud for proposing the regularized system, and to Helge Holden for many discussions. The authors would like to thank Institut Mittag–Leffler for support and hospitality during the program Interactions between Partial Differential Equations & Functional Inequalities.
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Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital - Trondheim University Hospital)
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Research supported by the grants Waves and Nonlinear Phenomena (WaNP) and Wave Phenomena and Stability—a Shocking Combination (WaPheS) from the Research Council of Norway.
This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.
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Grunert, K., Reigstad, A. A regularized system for the nonlinear variational wave equation. Partial Differ. Equ. Appl. 4, 35 (2023). https://doi.org/10.1007/s42985-023-00252-0
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DOI: https://doi.org/10.1007/s42985-023-00252-0