Abstract
The linear transport equation allows to advect levelset functions to represent moving sharp interfaces in multiphase flows as zero levelsets. A recent development in computational fluid dynamics is to modify the linear transport equation by introducing a nonlinear term to preserve certain geometrical features of the levelset function, where the zero levelset must stay invariant under the modification. The present work establishes mathematical justification for a specific class of modified levelset equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton–Jacobi equations. The first main result is the existence of smooth solutions defined in a timeglobal tubular neighborhood of the zero levelset, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; the smooth solution is shown to possess the desired geometrical feature. The second main result is the existence of timeglobal viscosity solutions defined in the whole domain, where standard Perron’s method and the comparison principle are exploited. In the first and second main results, the zero levelset is shown to be identical with the original one. The third main result is that the viscosity solution coincides with the localinspace smooth solution in a timeglobal tubular neighborhood of the zero levelset, where a new aspect of localized doubling the number of variables is utilized.
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1 Introduction
The linear transport equation \(\frac{\partial f}{\partial t}+v\cdot \nabla f=0\) describes the passive advection of a scalar quantity f by a velocity field v. We start with a brief overview of the fundamental role of the linear transport equation in fluid dynamics. Suppose that a domain \(\Omega \subset {{\mathbb {R}}}^3\) is occupied by a fluid. The Lagrangian specification of a fluid flow is to look at the position of each fluid element, i.e., for each time \(t\in [0,\infty )\) the position of the fluid element being at \(\xi \in \Omega \) at time \(\tau \in [0,\infty )\), which is called the “fluid element \((\tau ,\xi )\)”, is denoted by
Then, the velocity of each fluid element \((\tau ,\xi )\) is defined for each \(t\ge 0\) as
Assuming that \(X(t,\tau ,\xi )=x\) is equivalent to \(\xi =X(\tau ,t,x)\) for all \(t,\tau \in [0,\infty )\) and \(x,\xi \in \Omega \), one obtains the Eulerian specification of the fluid flow, i.e., the velocity field v defined as
which leads to
Hence, X can be seen as the flow of the kinematic ordinary differential equation (ODE)
Let \(F(t,\xi )\) be a scalar quantity at time t that is associated with the fluid element \((0,\xi )\). The Eulerian description f of F is defined as
Suppose that \(F(t,\xi )\equiv \phi ^0(\xi )\), i.e., each fluid element \((0,\xi )\) preserves the quantity \(F(0,\xi )=\phi ^0(\xi )\). Then, noting that \(f(t,X(t,0,\xi ))\equiv F(t,\xi )\), we have the following identity
With (1.1), we find that f satisfies the linear transport equation
where \(\nabla =(\partial _{x_1},\partial _{x_2},\partial _{x_3})\). It is intuitively clear (and mathematically true as well) that the solution of (1.3) is given as
This observation leads to the method of characteristics for more general first order partial differential equations (PDEs). Note that if v and \(\phi ^0\) are not \(C^1\)smooth, the meaning of solution must be generalized. A typical example of F being preserved by each fluid element is the density in an incompressible fluid, where the velocity v comes from the incompressible Navier–Stokes equations. We refer to [24] and [13] for recent development of mathematical analysis for the system of the linear transport equation and the incompressible Navier–Stokes equations. We refer also to [17] and [4] for generalization of ODEbased classical theory of the linear transport equations to the case with velocity fields being less regular.
We now discuss the transport equation in the context of the levelset method in twophase flow problems. Suppose that \(\Omega \) is occupied by two immiscible fluids (distinguished by the superscript ±) in such a way that at \(t=0\) the domain \(\Omega \) is divided into two disjoint connected open sets and their interface: \(\Omega ^+(0)\subsetneq \Omega \) with \(\partial \Omega ^+(0)\cap \partial \Omega =\emptyset \) is a connected open set filled by fluid\({}^+\), \(\Omega ^(0):=\Omega {\setminus } \overline{\Omega ^+(0)}\) is filled by fluid\({}^\) and \(\Sigma (0):=\partial \Omega ^+(0)\cap \partial \Omega ^(0)=\partial \Omega ^+(0)\) is the interface. Note that, for now, we discuss the case where \(\Sigma (0)\) does not touch \(\partial \Omega \), while in Sect. 2.2 we will consider the other case. We suppose that for each \(t>0\) the open set
is occupied by fluid\({^+}\) (resp. fluid\(^{}\)) and the common interface of fluid\(^{\pm }\) is given as
where the continuity of the flow X implies that (1.6) is welldefined even though no unique fluid elements are associated to the points on \(\Sigma (0)\). In other words, the velocity field v coming from the twophase Navier–Stokes equations is assumed to be such that (1.2) generates a proper flow from \({\bar{\Omega }}\) to itself with \(\partial \Omega \) being flow invariant; see Sect. 2 below for more details. The interface given as (1.6) is called a material interface, as opposed to a nonmaterial interface formed by a twophase flow with phase change (see [10] for investigations of (1.2) in the case of nonmaterial interfaces). Let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be a smooth function such that \(\phi ^0>0\) on \(\Omega ^+(0)\) and \(\phi ^0<0\) on \(\Omega ^(0)\), which implies that \(\phi ^0=0\) only on \(\Sigma (0)\). We assign to each fluid element \((0,\xi )\) the number \(\phi ^0(\xi )\). Let \(F(t,\xi )\) be the label of the fluid element \((0,\xi )\) at time t, which must be equal to \(\phi ^0(\xi )\) for any \(t\ge 0\). Then, the Eulerian description f of F, i.e., \(f(t,x):=F(t,\xi )_{\xi =X(0,t,x)}=\phi ^0(X(0,t,x))\), satisfies the transport equation (1.3) with the representation (1.4). In particular, we have
We call f a levelset function and the linear transport equation for levelset functions the levelset equation. Throughout the paper, the levelset means the zero level of a levelset function. Suppose that \(\Sigma (0)\) is equal to the levelset of a \(C^2\)function \(\phi ^0\) such that \(\nabla \phi ^0\ne 0\) on \(\Sigma (0)\), where \(\Sigma (0)\) is a \(C^2\)smooth closed surface (compact manifold without boundary). If \(X(t,\tau ,\cdot ):\Omega ^+(\tau )\cup \Sigma (\tau )\cup \Omega ^(\tau )\rightarrow \Omega ^+(t)\cup \Sigma (t)\cup \Omega ^(t)\) is a \(C^2\)diffeomorphism for each \(t,\tau \ge 0\), we see that
and \(\Sigma (t)\) keeps being a \(C^2\)smooth closed surface for all \(t>0\). In particular, the unit normal vector \( \nu (t,x)\) and the total (twice the mean) curvature \(\kappa (t,x)\) of \(\Sigma (t)\) at each point x are welldefined and represented as
where \(\cdot \) denotes the Euclidean norm. In a twophase flow problem, the Navier–Stokes equations for the velocity field are coupled with the levelset equation on the interface through \(\nu \) and \(\kappa \). We refer to [1] and [25] for recent developments of mathematical analysis of multiphase flow problems and to [20] for mathematical analysis of levelset methods beyond fluid dynamics.
In computational fluid dynamics, the levelset equation is often used to represent a moving interface. In this context, the levelset approach has several advantages, such as a very accurate approximation of the mean curvature and a straightforward handling of topological changes of the interface (e.g., breakup and coalescence of droplets). In a numerical simulation, it is common to choose an initial levelset function \(\phi ^0\) that coincides locally with the signed distance function of a given closed surface \(\Sigma (0)\), where \(\phi ^0\) is characterized by \(\nabla \phi ^0 \equiv 1\) in a neighborhood of \(\Sigma (0)\). However, it is known that the local signed distance property is not preserved by (1.3), i.e., \(f(t,\cdot )\) does not coincide even locally with the signed distance function of \(\Sigma (t)\) for \(t>0\) in general. In fact, a short calculation [19] shows that, along each curve \(x(\cdot )\) determined by (1.2) (it is called a characteristic curve) such that \(x(t)\in \Sigma (t)\),
holds for a classical solution f of the standard levelset equation (1.3). Here, \(\langle \cdot , \cdot \rangle \) stands for the inner product of \({{\mathbb {R}}}^3\). Unfortunately, problems with the numerical accuracy emerge if \(\nabla f\) becomes too small or too large, which is the case in general, even though the nondegeneracy condition (1.7) is mathematically guaranteed. This is an important point in practice: on the one hand, it must be possible to resolve \(\nabla f\) by the computational mesh, which implies an upper limit for \(\nabla f\) related to the mesh size; on the other hand, too small values of \(\nabla f\) lead to an inaccurate positioning of the interface, the normal field, the mean curvature field, etc. in the numerical algorithm. In order to keep the norm of the gradient approximately constant, socalled “reinitialization” methods [29, 30] have been developed. Typically, an additional PDE is solved that computes a new function \({\tilde{f}}\) with the same zero contour but with a predefined norm of the gradient (e.g., \(\nabla {\tilde{f}}=1\) on the levelset). We refer to Section 11.6 in [28] for a critical assessment of this approach and to [21] for rigorous mathematical analysis of the reinitialization process. In [27], the authors developed an alternative numerical method to control the size of the gradient based on the levelset equation with a suitable source term that is determined by an extra equation, where the reinitialization procedure was no longer necessary. These methods might be computationally expensive. Moreover, it is known that many reinitialization methods struggle with extra difficulties if the interface touches the domain boundary \(\partial \Omega \) [16] (i.e., if a socalled “contact line” is formed; see Sect. 2.2).
In order to control the norm of the gradient within a single PDE, the following nonlinear modification of the levelset equation has been introduced in the literature of computational fluid dynamics (see [19, 22] for details):
Note that, from here on, we rather use \(\phi \) instead of f to stress the fact that we deal with a modified levelset equation. Since the source term on the righthand side is chosen proportional to the levelset function \(\phi \), the modification term vanishes on the zero interface and, as seen in Sect. 2, one can show that the evolution of the zero levelset is unaffected by the modification (in fact, the configuration component of the characteristic ODEs for (1.9) becomes (1.2) on the levelset). Moreover, a suitable choice of the nonlinear function R allows to control the evolution of \(\nabla \phi \) (at least locally at the levelset). A formal calculation [19] shows that, by choosing,^{Footnote 1}
we indeed obtain
along each characteristic curve x(t) of (1.9) such that \(x(t)\in \Sigma (t)\) for all \(t\ge 0\) or, equivalently,
We will prove this statement rigorously using the method of characteristics (see problem (2.3) in Sect. 2). Notice that, in general, the property (1.11) only holds locally at the levelset. The signed distance function of \(\Sigma (t)\) itself does not solve (1.3) nor (1.9) in general, but another nonlinear PDE, cf. Lemma 3.1 in [22]. From the numerical perspective, it is of interest to study a formulation like (1.9) because the advection of the interface and the preservation of the norm of the gradient are combined into one single PDE, i.e., into a monolithic approach. In addition to the choice (1.10), we will also study a variant in which a cutoff function is applied such that the nonlinear source term is only active in a neighborhood of the levelset (see problem (3.2) below) and another simpler modified levelset equation, which only keeps the norm of the gradient within given bounds; see the initial value problem (2.24) below. We refer to [19] for a numerical investigation of (1.9).
Now, we move to the mathematical analysis of (1.9). It is important to note that, due to the nonlinear source term in (1.9), the ODE (1.2) is no longer the characteristic ODE of (1.9); instead, the system of ODEs (2.5)–(2.7) defines the characteristic curves of (1.9). See Appendix 1 for more details on the method of characteristics as applied to Hamilton–Jacobi equations. Furthermore, since (1.9) is a first order fully nonlinear PDE, the mathematical analysis of (1.9) is not at all as simple as that of (1.3), even if v and R are smooth enough. Existence of a classical solution on the whole domain within an arbitrary time interval is no longer possible in general, i.e., the notion of viscosity solutions is necessary. Then, it is expected that the following statements hold true for R given by (1.10) or its variants:

(i)
(1.9) provides a levelset that is identical to the original one provided by (1.3) for all \(t\ge 0\);

(ii)
(1.9) admits a unique classical solution \(\phi \) at least in a tglobal tubular neighborhood of the levelset (see its definition in Sect. 2) so that the normal field and mean curvature field are welldefined by \(\phi \) and the property (1.11) (or, less restrictively, an a priori bound of \(\nabla \phi \)) holds on the levelset for all \(t\ge 0\);

(iii)
(1.9) admits a unique globalintime viscosity solution defined on \([0,\infty )\times {\bar{\Omega }}\);

(iv)
If initial data is \(C^2\)smooth, the viscosity solution \({\tilde{\phi }}\) coincides with the localinspace classical solution \(\phi \) in a tglobal tubular neighborhood of the levelset, i.e., partial \(C^2\)regularity of \({\tilde{\phi }}\).
The purpose of the current paper is to provide full proofs of (i)–(iv) for the problem (1.9) with a given smooth velocity field v and the abovementioned R, where mathematical analysis on the system of (1.9) and Navier–Stokes type equations for v is an interesting future work. We will exploit the method of characteristics to show (ii) and (i) for the smooth solution; usually, the method of characteristics works only within a short time interval; however, since the nonlinearity of (1.9) becomes arbitrarily small near the levelset, on which \(\nabla \phi \) is appropriately controlled as well, we may iterate the method of characteristics countably many times with a shrinking neighborhood of the levelset to construct a time global solution defined in a tglobal tubular neighborhood of the levelset. To show (iii) and (i) for the viscosity solution, we will apply the standard theory of viscosity solutions to (1.9) with a boundary condition arising formally from the classical solutions. To prove (iv), we adapt the idea of localized doubling the number of variables for the comparison principle of viscosity solutions within a cone of dependence; the difficulty is that we cannot have a cone of dependence that contains a tglobal tubular neighborhood of the levelset; we will demonstrate a reasoning similar to localized doubling the number of variables with an unusual choice of a penalty function in a tglobal tubular neighborhood of the levelset. We emphasize that the result (iv) is particularly important from application points of view in the sense that, once a continuous viscosity solution is obtained, it provides the levelset, its normal field and mean curvature field with the necessary regularity being guaranteed; numerical construction of a viscosity solution on the whole domain would be easier than that of a localinspace smooth solution; there is huge literature pioneered by [12] on rigorous numerical methods of viscosity solutions.
Finally, we compare our results on (i)–(iv) with the work [22]. In [22], the author formulated a modification of the initial value problem of a general Hamilton–Jacobi equation with an autonomous Hamiltonian (including the linear transport equation with \(v=v(x)\)) on the whole space and proved the existence of a unique viscosity solution, where the modification is essentially the same as (1.10); owing to the modification, he showed that the (continuous) viscosity solution of the modified equation stays close to the signed distance function of its own levelset with good upper/lower estimates, from which he obtained differentiability of the viscosity solution on the levelset with the norm of the derivative to be one. Additional regularity of the viscosity solution away from the levelset remained open. Our current paper provides a stronger partial regularity property of viscosity solutions in the same context as [22].
2 \(C^2\)solutions on tubular neighborhood of levelset
Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded connected open set and \(v=v(t,x)\) be a given smooth function defined in \([0,\infty )\times {\bar{\Omega }}\). Our investigation relies on certain properties of the flow generated by v that satisfies certain conditions; in particular, we require flow invariance of \({\bar{\Omega }}\) and \(\partial \Omega \), i.e., the solution \(x(s)=x(s;s_0,\xi )\) of (1.2) with initial condition \(x(s_0)=\xi \), \((s_0,\xi )\in [0,\infty )\times {\bar{\Omega }}\) (resp. \((s_0,\xi )\in [0,\infty )\times \partial \Omega \)) uniquely exists and stays in \({\bar{\Omega }}\) (resp. \(\partial \Omega \)) for all \(s\in [0,\infty )\). Note that since \({\bar{\Omega }}\) is compact, localintime invariance implies globalintime invariance. Typical examples of v fulfilling our requirements are

(i)
\(\partial \Omega \) is smooth and \(v(t,x)\cdot \nu (x)=0\) for all \(x\in \partial \Omega \) and \(t\ge 0\), where \(\nu \) is the unit normal of \(\partial \Omega \) (cf., nonpenetration condition in fluid dynamics),

(ii)
\(v(t,x)=0\) for all \(x\in \partial \Omega \) and \(t\ge 0\) (cf., nonslip condition in fluid dynamics).
In the current paper, we consider a more general situation based on the theory of ODEs on closed sets and flow invariance. For this purpose, we introduce the socalled Bouligand contingent cone \(T_K(x)\) of an arbitrary closed set \(K\subset {{\mathbb {R}}}^d\) at \(x\in K\) as
We say that \(y\in {{\mathbb {R}}}^d\) is subtangential to K at a point \(x\in K\), if \(y\in T_K(x)\). Note that \(T_K(x)={{\mathbb {R}}}^d\) for x in the interior of K. We shall employ the following result on flow invariance.
Lemma 2.1
Let \(J=(a,b)\subset {{\mathbb {R}}}\), \(K \subset {{\mathbb {R}}}^d\) compact and \(g: J \times K \rightarrow {{\mathbb {R}}}^d\) (jointly) continuous^{Footnote 2} and locally Lipschitz in \(x\in K\). Then, the following holds true:

(a)
Suppose that \(\pm g\) are subtangential to K, i.e.,
$$\begin{aligned} \pm g(s,x) \in T_K(x), \quad \forall \, s \in J, \,\,\,\forall \, x \in K. \end{aligned}$$Then, given any \(s_0 \in J\) and \(x_0 \in K\), the initial value problem
$$\begin{aligned} x'(s)=g(s,x(s)),\quad x(s_0)=x_0 \end{aligned}$$has a unique solution defined on J that stays in K.

(b)
Suppose that \(\pm g\) are subtangential to \(\partial K\), i.e.,
$$\begin{aligned} \pm g(s,x) \in T_{\partial K}(x), \quad \forall \, s \in J, \,\,\,\forall \, x \in \partial K. \end{aligned}$$Then, the sets K, \(\partial K\) and \(K{\setminus } \partial K\) are flow invariant.
See Appendix 2 for more on flow invariance and Lemma 2.1.
Now we state the hypothesis on the velocity field v:

(H1)
v belongs to \(C^0([0,\infty )\times {\bar{\Omega }};{{\mathbb {R}}}^3)\cap C^1([0,\infty )\times \Omega ;{{\mathbb {R}}}^3)\); v is Lipschitz continuous in x on \([0,\infty )\times {\bar{\Omega }}\); v is three times partially differentiable in x; all of the partial derivatives of v belong to \(C^0([0,\infty )\times \Omega ;{{\mathbb {R}}}^3)\),

(H2)
\(\pm v(s,x) \in T_{\partial \Omega }(x)\) for all \(s \in [0,\infty ),\,\, x \in \partial \Omega \).
We remark that the xLipschitz continuity mentioned in (H1) implies that
The upcoming nonlinear modification of the linear transport equation requires \(C^3\)smoothness of v in x so that its characteristic ODEs are properly defined; due to Lemma 2.1, \({\bar{\Omega }}\), \(\partial \Omega \) and \(\Omega \) are flow invariant with respect to the flow X of (1.2); \(X(s,\tau ,\cdot )\) is continuous on \( {\bar{\Omega }}\) and \(C^3\)smooth in \(\Omega \). If \({\bar{\Omega }}\) is a cube, for instance, (H2) implies: at each vertex, v must be equal to zero, while on each edge, v may take nonzero values parallel to the edge. In Sect. 3, we relax the regularity assumption of (H1) as
 (H1)\('\):

v belongs to \(C^0([0,\infty )\times {\bar{\Omega }};{{\mathbb {R}}}^3)\cap C^1([0,\infty )\times \Omega ;{{\mathbb {R}}}^3)\); v is Lipschitz continuous in x on \([0,\infty )\times {\bar{\Omega }}\).
2.1 Case 1: problem with levelset being away from \(\partial \Omega \)
Let \(\Sigma (0)\subset \Omega \) be a closed \(C^2\)smooth surface. Let \(\phi ^0:\Omega \rightarrow {{\mathbb {R}}}\) be a \(C^2\)smooth function such that
Let f be the solution of the original levelset equation (1.3). We keep the notation and configuration in (1.5) and (1.6), where we repeat
Note that the second equality in the line above holds for all \(t\ge 0\) due to (H2). For each \(t\ge 0\), let \(\Sigma _\varepsilon (t)\) with \(\varepsilon >0\) be the \(\varepsilon \)neighborhood of \(\Sigma (t)\), i.e.,
where we always consider \(\varepsilon >0\) such that \(\Sigma _\varepsilon (t)\subset \Omega \). We say that a set \(\Theta \subset [0,\infty )\times \Omega \) is a (tglobal) tubular neighborhood of the levelset \(\{\Sigma (t)\}_{t\ge 0}\), if \(\Theta \) contains
and there exists a nonincreasing function \(\varepsilon :[0,\infty )\rightarrow {{\mathbb {R}}}_{>0}\) such that
The problem under consideration is to find a tubular neighborhood \(\Theta \) of \(\{\Sigma (t)\}_{t\ge 0}\) and a \(C^2\)function \(\phi \) satisfying
Note that this problem makes sense with \(\phi ^0\) being defined only in a neighborhood of \(\Sigma (0)\), e.g., \(\phi ^0\) is given as the local signed distance function of \(\Sigma (0)\). We state the first main result of this paper.
Theorem 2.2
Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be \(C^2\)smooth satisfying (2.2). Then, there exists a tubular neighborhood \(\Theta \) of the levelset \(\{\Sigma (t)\}_{t\ge 0}\) for which (2.3) admits a unique \(C^2\)solution \(\phi \) satisfying
Before starting our proof of Theorem 2.2, we introduce a system of ODEs for the method of characteristics. We treat the PDE in (2.3) as the Hamilton–Jacobi equation
generated by the Hamiltonian \(H: [0,\infty )\times \Omega \times {{\mathbb {R}}}^3\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) defined as
Set
where we also use the symbol “\(\textrm{D} \varphi \)” for the derivative or Jacobian matrix of \(\varphi \). The characteristic ODEs of (2.4) are given as (see Appendix 1 for more details)
Note that B(v(t, x), p) is still \(C^1\)smooth in (x, p) because of (H1), which is required for the method of characteristics. We sometimes use the notation \(x(s;\xi ),p(s;\xi ),\Phi (s;\xi ) \) to specify the initial point. The upcoming proof is based on the investigation (a priori estimates) of the variational equations of the characteristic ODEs (2.5)–(2.7) for each \(\xi \in \Sigma (0)\) to ensure the invertibility of \(x(s;\cdot )\) in a small neighborhood of \(\Sigma (s)\) for each \(s\ge 0\). In a general argument of the method of characteristics, such invertibility is proven only within a small time interval. Below, we will show an iterative scheme to extend the time interval in which the invertibility holds with a shrinking neighborhood of \(\Sigma (s)\) as s becomes larger.
Proof of Theorem 2.2
Our proof consists of four steps.
Step 1. We demonstrate a priori estimates for the above system of ODEs. For each \(\xi \in \Sigma (0)\), as long as \(x(s;\xi ),p(s;\xi ),\Phi (s;\xi ) \) exist, it holds that
where we note that due to (H2) applied to (2.10), the above equalities (2.9)–(2.12) hold for all \(s\ge 0\) together with
Hence, for each \(\xi \in \Sigma (0)\), there exist \(\frac{\partial x}{\partial \xi }(s)=\frac{\partial x}{\partial \xi }(s;\xi )\) and \(\frac{\partial \Phi }{\partial \xi }(s)=\frac{\partial \Phi }{\partial \xi }(s;\xi )\) for all \(s\ge 0\) satisfying
where
Hereafter, \(V_0,V_1,V_2,V_3,V_4\) will denote constants depending only on v, \(\sup _{\Sigma (0)}\nabla \phi ^0<\infty \) and \(\inf _{\Sigma (0)}\nabla \phi ^0>0\). Due to (2.1), it holds that for any \(\xi \in \Sigma (0)\),
Observe that for each \(\xi \in \Sigma (0)\), we have from (2.14),
For each \(i=1,2,3\) and \(\xi \in \Sigma (0)\), we have from (2.13),
Gronwall’s inequality implies
It follows from (2.15) that
Due to the continuity in (2.5)–(2.8) and (2.15)–(2.16), we find \(\varepsilon _1>0\) such that
Fix a number \(t_*\in (0,1]\) such that
Step 2. We show the injectivity of \(\xi \mapsto x(s;\xi )\), i.e.,
First, we give an auxiliary explanation on how to prove (2.19). In order to prove the injectivity of \(x(s;\cdot ):\Sigma _{\varepsilon }(0)\rightarrow x(s;\Sigma _{\varepsilon }(0))\) for each fixed \(s\in [0,s_*]\) (\(\varepsilon >0\) and \(s_*>0\) are some constants), we would take one of the following strategies:

(i)
Fix \(\varepsilon >0\) and take a sufficiently small \(s_*>0\),

(ii)
Fix \(s_*>0\) and take a sufficiently small \(\varepsilon >0\),
where we note that \(\det \frac{\partial x}{\partial \xi }(s;\xi )\ne 0\) everywhere on \(\Sigma _{\varepsilon }(0)\) is not enough for the injectivity in general. In our proof of Theorem 2.2, we need to repeat the argument from \([0,s_*]\) within \([s_*,2s_*]\) to demonstrate infinite iteration with the fixed \(s_*>0\). Hence, we take the strategy (ii) with \(s_*=t_*\) given from (2.18) and sufficiently small \(\varepsilon ={\tilde{\varepsilon }}_1\in (0,\varepsilon _1]\). Then, what we need to show is that
If the line segment joining \(\zeta \) and \({\tilde{\zeta }}\) is included in \(\Sigma _{\varepsilon }(0)\), we may immediately apply Taylor’s approximation to have
and get \({\tilde{\zeta }}=\zeta \) via \(\det \Lambda >0\), where \(t_*\) in (2.18) is given so that \(\det \Lambda >0\) holds in \(\Sigma _{\varepsilon _1}(0)\). However, it is not a priori clear if the line segment joining \(\zeta \) and \({\tilde{\zeta }}\) is included in \(\Sigma _{\varepsilon }(0)\) for each \(s\in [0,t_*]\). This is a major difficulty to prove (2.19) directly.
Now, we prove (2.19) by contradiction. Suppose that (2.19) does not hold. Then, we find a sequence \(\{\delta _j\}_{j\in {{\mathbb {N}}}}\subset (0,\varepsilon _1]\) with \(\delta _j\rightarrow 0\) as \(j\rightarrow \infty \) and \(\{t_j\}_{j\in {{\mathbb {N}}}}\subset (0,t_*]\) for which
Define \(d_j:=\sup \{ {\tilde{\zeta }}\zeta \,\, (2.20) \text{ holds } \}\). We prove that \(d_j\rightarrow 0\) as \(j\rightarrow \infty \). If not, we find \(\eta >0\) and a subsequence of \(\{d_j\}_{j\in {{\mathbb {N}}}}\) (still denoted by the same symbol) such that \(d_j\ge \eta \) for all j. We further take out subsequences so that \(t_j\rightarrow s\in [0,t_*]\) as \(j\rightarrow \infty \). Then, for each \(j\in {{\mathbb {N}}}\), there exist \(\zeta _j\) and \({\tilde{\zeta }}_j\) such that
Taking subsequences if necessary, we see that \(\zeta _j\), \({\tilde{\zeta }}_j\) converge to some \(\xi ,{\tilde{\xi }}\in \Sigma (0)\) as \(j\rightarrow \infty \), respectively, where \(\xi {\tilde{\xi }}\ge \frac{\eta }{2}\). The limit \(j\rightarrow \infty \) in \(x(t_j;\zeta _j)=x(t_j;{\tilde{\zeta }}_j)\) implies \(x(s;\xi )=x(s;{\tilde{\xi }})\) with \(\xi ,{\tilde{\xi }}\in \Sigma (0)\) and \(\xi \ne {\tilde{\xi }}\). This is a contradiction, because \(x(s;\cdot )_{\Sigma (0)}\) is given by the flow of (1.2). Consequently, \(d_j\rightarrow 0\) as \(j\rightarrow \infty \). Hence, for all j sufficiently large, we find that \(\zeta ,{\tilde{\zeta }}\) in (2.20), denoted by \(\zeta _j\), \({\tilde{\zeta }}_j\), are included in the \(\varepsilon _1\)neighborhood of some \(\xi _j\in \Sigma (0)\); for \(i=1,2,3\), Taylor’s approximation within the \(\varepsilon _1\)neighborhood of \(\xi _j\) yields
Therefore, by (2.17), we obtain
which leads to \({\tilde{\zeta }}_j=\zeta _j\). This is a contradiction and we conclude (2.19). Note that we also obtain \(\det \frac{\partial x}{\partial \xi }(s;\xi )>0\) for all \(s\in [0,t_*]\) and \(\xi \in \Sigma _{{\tilde{\varepsilon }}_1}(0)\); this follows from \( \frac{\partial x}{\partial \xi }(s;\xi )= \frac{\partial x}{\partial \xi }(0;\xi )+ \frac{\partial x}{\partial \xi }(s;\xi ) \frac{\partial x}{\partial \xi }(0;\xi )=I+[\frac{\textrm{d}}{\textrm{d} s} \frac{\partial x_k}{\partial \xi _l}(\lambda _{kl}s;\xi )s]\), \(\lambda _{kl}\in (0,1)\).
Step 3. We construct a local in space solution within \([0,t_*]\). Define
The invertibility of \(\xi \mapsto x(s;\xi )\) and \(\det \frac{\partial x}{\partial \xi }(s;\xi )>0\) imply that \(\psi _1\) and \(\psi _1(t,\cdot )\) are \(C^1\)diffeomorphic and that there exists \(\varepsilon _2>0\) such that
Let \(\varphi _1: O_{{\tilde{\varepsilon }}_1}(t_*)\rightarrow \Sigma _{{\tilde{\varepsilon }}_1}(0)\) be defined as
where we note that \(\varphi _1(t,\Sigma (t))=\Sigma (0)\). Then, it follows from the general results of the method of characteristics (see Appendix 1) that the function
is \(C^2\)smooth and satisfies
Step 4. We demonstrate an iteration of Step 1–3 with the common \(t_*\). The argument up to now, in particular Step 1, has used the \(C^2\)smoothness of \(\phi ^0\) on a neighborhood of \(\Sigma (0)\) and the upper/lower bound of \(\nabla \phi ^0\) on \(\Sigma (0)\), where we note that \(\nabla \phi _1(t_*,\cdot )\) on \(\Sigma (t_*)\) has exactly the same upper/lower bound as \(\nabla \phi ^0\) on \(\Sigma (0)\). Therefore, we may replace \(\phi ^0\) with \(\phi _1(t_*,\cdot )\) to demonstrate the same kind of estimates in terms of the above constants \(V_0,V_1,V_2,V_3,V_4\) and \(t_*\) as well as \(\varepsilon _2 >0\) appropriately chosen in (2.21) for the characteristic ODEs (2.5)–(2.7) for \(s\in [t_*,2t_*]\) with the initial condition
Furthermore, we find a constant \({\tilde{\varepsilon }}_2\in (0,\varepsilon _2]\) such that
are \(C^1\)diffeomorphic. Also, there exists \(\varepsilon _3>0\) such that
Let \(\varphi _2: O_{{\tilde{\varepsilon }}_2}(2t_*)\rightarrow \Sigma _{{\tilde{\varepsilon }}_2}(t_*)\) be defined as
where we note that \(\varphi _2(t,\Sigma (t))=\Sigma (t_*)\). Then, the function
is \(C^2\)smooth and satisfies
Note that \(\phi _1\) and \(\phi _2\) are smoothly connected at \(t=t_*\). With the common constant \(t_*\), we may repeat this process with \(\varepsilon _1,{\tilde{\varepsilon }}_1, \varepsilon _2,{\tilde{\varepsilon }}_2,,\varepsilon _3, {\tilde{\varepsilon }}_3, \cdots \) (note that it is possible that \(\varepsilon _k,{\tilde{\varepsilon }}_k,\rightarrow 0\) as \(k\rightarrow \infty \)). We conclude the proof with defining \(\Theta := \cup _{l\in {{\mathbb {N}}}} O_{{\tilde{\varepsilon }}_l}(lt_*)\). \(\square \)
If we choose \(\phi ^0\) which coincides with the (local) signed distance function of \(\Sigma (0)\), then the solution \(\phi \) obtained in Theorem 2.2 satisfies
2.2 Case 2: problem with levelset touching \(\partial \Omega \)
We consider the problem (2.3) with the levelset touching the boundary of \(\Omega \). Let \(K\subset {{\mathbb {R}}}^3\) be a bounded connected open set such that \(K\cap \Omega \ne \emptyset \), \(\partial K\cap \partial \Omega \ne \emptyset \) and \(\partial K\) is a closed \(C^2\)smooth surface. Define
Let \(\phi ^0\) be a \(C^2\)smooth \({{\mathbb {R}}}\)valued function defined in an open set containing \({\bar{K}}\cup {\bar{\Omega }}\) such that
Then, we have
Let f be the solution of the original levelset equation (1.3). Define
where we note that \(\partial \Omega \) is invariant under the flow X of (1.2), and hence \(\Sigma (t)\) always touches \(\partial \Omega \). If we follow the same argument as given in Sect. 2.1, we would face nontrivial issues at/near \(\Sigma (t)\cap \partial \Omega \) coming from the behavior of the variational equations of the characteristic ODEs on \(\partial \Omega \). Hence, we modify the reasoning of Sect. 2.1 so that \(\Sigma (t)\cap \partial \Omega \) is not involved.
Let \(\{\Omega ^k\}_{k\in {{\mathbb {N}}}}\) be a monotone approximation of \(\Omega \), i.e., each \(\Omega ^k\) is an open subset of \(\Omega \); \(\Omega ^k\subset \Omega ^{k+1}\) for all \(k\in {{\mathbb {N}}}\); for any \(G\subset \Omega \) compact, there exists \(k=k(G)\) such that \(G\subset \Omega ^k\). Introduce
where for each \(k\in {{\mathbb {N}}}\) we have \(\varepsilon >0\) depending on k such that
Now, we may follow the same argument as given in Sect. 2.1 with \(\Sigma ^k(0)\) in place of \(\Sigma (0)\) to obtain the following objects:
The method of characteristics implies that \(\phi ^k\equiv \phi ^{k'}\) on \(\Theta ^k\cap \Theta ^{k'}\) for every \(k,k'\in {{\mathbb {N}}}\). Therefore, setting
we obtain a unique \(C^2\)solution \(\phi \) of (2.3). We remark that
We summarize the result:
Theorem 2.3
Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be \(C^2\)smooth satisfying (2.22). Then, there exists a tubular neighborhood \(\Theta \) in the sense of (2.23) of the levelset \(\{\Sigma (t)\}_{t\ge 0}\) touching \(\partial \Omega \) for which (2.3) admits a unique \(C^2\)solution \(\phi \) satisfying
Let us note in passing that another method to investigate a problem with the levelset touching \(\partial \Omega \) would run via smooth extension of v outside \(\Omega \). The following steps would suffice:
Step 1. Extend the velocity field v to \({{\mathbb {R}}}\times {{\mathbb {R}}}^3\) as a \(C^3\)function by means of Whitney’s extension theorem [31] or the extension operators in Sobolev spaces (see, e.g., Chapter 5 of [2]) together with the Sobolev embedding theorem, where additional conditions on v and \(\Omega \) are required accordingly.
Step 2. Extend the flow \(X(t,\tau ,\xi )\) to \(t,\tau \in {{\mathbb {R}}}\), \(\xi \in {{\mathbb {R}}}^3\) and the problem (2.3) to a tubular neighborhood \({\tilde{\Theta }}\) of the levelset \(\{ X(t,0,\partial K) \}_{t\ge 0}\).
Step 3. Solve the extended problem in the same way as Sect. 2.1, where each characteristic curve starting at a point of \(\Omega \) stays inside \(\Omega \) forever due to the flow invariance of \({\bar{\Omega }}\) and \(\partial \Omega \) under X.
The restriction of the solution obtained in Step 3 to \(\Theta :={\tilde{\Theta }}\cap ([0,\infty )\times {\bar{\Omega }})\) then is the desired object.
2.3 Simpler nonlinear modification
The nonlinear modification in (2.3) is designed to preserve \(\nabla \phi \) along each characteristic curve on the levelset. If we relax the requirement, i.e., if we only ask for an a priori bound of \(\nabla \phi \) on the levelset, we may use a much simpler modification.
We take the same configuration of the levelset considered in Sect. 2.1. Let \(\alpha >0\) be a constant such that
With a constant \(\beta >\alpha \), we consider
which is seen as the Hamilton–Jacobi equation (2.4) with
The characteristic ODEs in this case are given as
As long as the above characteristic ODEs have solutions, \(p(s)^2\) evolves as
which leads to
If \(\beta \alpha \le \nabla \phi ^0\le \beta +\alpha \) on \(\Sigma (0)\), we have for each \(\xi \in \Sigma (0)\),
In fact, suppose that there exists \(\tau >0\) such that \(p(\tau )>\beta +\alpha \); set \(s^*:=\sup \{ s\le \tau \,\, p(s)=\beta +\alpha \}\); the continuity of \(p(\cdot )\) implies that \(p(s^*)=\beta +\alpha \) and \(p(s)> \beta +\alpha \) for all \(s\in (s^*,\tau ]\); then, we necessarily have
which is a contradiction; a fully analogous argument yields the lower bound. Therefore, we may apply the reasoning of Sect. 2.1 to the following problems: find a tubular neighborhood \(\Theta \) of \(\{\Gamma (t)\}_{t\ge 0}\) and a \(C^2\)function \(\phi \) satisfying
We obtain the following result.
Theorem 2.4
Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be \(C^2\)smooth satisfying (2.2) and \(\beta \alpha \le \nabla \phi ^0\le \beta +\alpha \) on \(\Sigma (0)\), where \(\Sigma (0)\) does not touch \(\partial \Omega \). Then, there exists a tubular neighborhood \(\Theta \) of the levelset \(\{\Sigma (t)\}_{t\ge 0}\) for which (2.24) admits a unique \(C^2\)solution \(\phi \) satisfying
Note that Theorem 2.4 is interesting for numerical purposes because it is usually not important to exactly keep \(\nabla \phi  \equiv 1\), but rather to stay away from extreme values of the gradient norm. Therefore, the problem (2.24) should be investigated in more detail from the numerical perspective in the future.
3 Viscosity solution on the whole domain
Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded connected open set. Let \(v=v(t,x)\) be a given function satisfying (H1)\('\) and (H2). Let \(\Sigma (0)\subset \Omega \) be a closed 2dimensional surface (topological manifold) and let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be a \(C^0\)function such that
where \(f(t,x):=\phi ^0(X(0,t,x))\) is not necessarily a classical solution of the original linear transport equation. We keep the notation and configuration in (1.5) and (1.6) with the current X and \(\phi ^0\), where we repeat
If we deal with the Hamilton–Jacobi equation in (2.3) on the whole domain \(\Omega \), the nonlinear term would suffer from singularity, i.e., \(\langle (\nabla v) p, p \rangle p^{2} \) cannot be continuous at \(p=0\). Hence, we remove the singularity by a multiplying cutoff with respect to p; furthermore, for a simpler structure near \(\partial \Omega \), we also introduce a smooth cutoff with respect to x so that the nonlinearity is effective near the levelset but ineffective near \(\partial \Omega \) (note that \(\nabla v\) is not required on \(\partial \Omega \) after cutoff). Apparently, such a cutoff with respect to x requires a priori information on the unknown levelset (i.e., distance between the levelset and \(\partial \Omega \)). However, we will see that it is sufficient to introduce a cutoff only with the information of the original levelset \(\{\Sigma (t)\}_{t\ge 0}\) defined as (1.6).
We explain two ways to introduce suitable cutoff functions. The first way is simpler but available only for the problem within each finite time interval. Fix an arbitrary terminal time \(T>0\). For \(\varepsilon >0\), let \(K_{\varepsilon } \subset {\bar{\Omega }}\) denote the intersection of \({\bar{\Omega }}\) and the \(\varepsilon \)neighborhood of \(\partial \Omega \). Then, since \(\Sigma (t)\) never touches \(\partial \Omega \) within [0, T], we find \(\varepsilon >0\) such that
Let \(0<a_0<a_1\) be arbitrary constants. Define the continuous function \(R_T:[0,T]\times {\bar{\Omega }}\times {{\mathbb {R}}}^3\rightarrow {{\mathbb {R}}}\) as
This choice of cutoff is uncomplicated since we do not need detailed information on the asymptotics of dist\((\Sigma (t),\partial \Omega )\) as \(t\rightarrow \infty \); however, \(R_T\) depends on the terminal time and a time global analysis is impossible. The second way is to allow tdependency for the cutoff. Since \(\Sigma (t)\) never touches \(\partial \Omega \) within \([0,\infty )\), we find a smooth nonincreasing function \(\varepsilon :[0,\infty )\rightarrow {{\mathbb {R}}}_{>0}\) (possibly \(\varepsilon (t)\rightarrow 0\) as \(t\rightarrow \infty \)) such that
Then, we take a smooth function \(\eta _1:[0,\infty )\times {\bar{\Omega }}\rightarrow [0,1]\) such that
where we omit an explicit formula of such \(\eta _1\), and define the continuous function R as
where \(\eta _2\) is the same as in \(R_T\) above.
Due to (2.1), there exists a constant \(V_0>0\) such that
In the rest of the paper, we take R given as (3.2). Note that all upcoming results hold also for \(R_T\) as long as the terminal time T is unchanged. For an arbitrary \(T>0\), we discuss existence of a unique viscosity solution \(\phi \) of
Before going on with (3.4), we recall the definition of viscosity (sub/super)solutions of a general first order Hamilton–Jacobi equation of the form
where \(O\subset {{\mathbb {R}}}^N\) is an open set, \(G=G(z,u,q):O\times {{\mathbb {R}}}\times {{\mathbb {R}}}^N\rightarrow {{\mathbb {R}}}\) is a given continuous function and \(u:O\rightarrow {{\mathbb {R}}}\) is the unknown function. Our evolutional Hamilton–Jacobi equation is also seen in this form with \(z=(t,x)\). Let us note in passing that (3.5) is often treated as a typical example of degenerate second order PDEs (i.e., the second order term is completely degenerate to be 0) in the literature, e.g., in the main reference [11]. To state the definition, we introduce the upper semicontinuous envelope \(u^*:O\rightarrow {{\mathbb {R}}}\) and the lower semicontinuous envelope \(u_*:O\rightarrow {{\mathbb {R}}}\) of a locally bounded function \(u:O\rightarrow {{\mathbb {R}}}\) as
Note that \(u^*\) is upper semicontinuous and \(u_*\) is lower semicontinuous; if u is upper semicontinuous (resp. lower semicontinuous), we have \(u=u^*\) (resp. \(u=u_*\)).
Definition. A function \(u:O\rightarrow {{\mathbb {R}}}\) is a viscosity subsolution (resp. supersolution) of (3.5), provided

\(u^*\) is bounded from above (resp. \(u_*\) is bounded from below);

If \((\varphi ,z)\in C^1(O;{{\mathbb {R}}})\times O\) satisfies
$$\begin{aligned} \max _{y\in O} (u^*(y)\varphi (y))=u^*(z)\varphi (z) \quad (\text{ resp. } \min _{y\in O} (u_*(y)\varphi (y))=u_*(z)\varphi (z)), \end{aligned}$$we have
$$\begin{aligned} G(z,u^*(z),\nabla _y \varphi (z))\le 0 \quad (\text{ resp. } G(z,u_*(z),\nabla _y \varphi (z))\ge 0). \end{aligned}$$
A function \(u:O\rightarrow {{\mathbb {R}}}\) is a viscosity solution of (3.5), if it is both a viscosity subsolution and supersolution of (3.5).
It is wellknown that \(\max ,\min \) in the definition can be replaced by the local (strict) maximum, local (strict) minimum, respectively.
3.1 Existence of viscosity solutions
We state the main result of this subsection.
Theorem 3.1
Suppose that v satisfy (H1)\('\) and (H2). Let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be continuous satisfying (3.1). Let \(T>0\) be arbitrary. Then, there exists a unique viscosity solution \(\phi \in C^0([0,T)\times {\bar{\Omega }};{{\mathbb {R}}})\) of (3.4), i.e., \(\phi \) satisfies the first equation in (3.4) in the sense of the above definition and the initial/boundary condition strictly. Furthermore, it holds that
Remark
Because \(T>0\) is arbitrary, Theorem 3.1implies the existence of a unique viscosity solution \(\phi \in C^0([0,\infty )\times {\bar{\Omega }};{{\mathbb {R}}})\) of (3.4)\(_{T=\infty }\) with \(\Sigma ^\phi (t)=\Sigma (t)\) for all \(t\in [0,\infty )\).
Proof of Theorem 3.1
Introduce the function \(S:[0,T]\times {\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) as
where \(\varepsilon (t)\ge \varepsilon (T)\) for all \(t\in [0,T]\), S is such that \(S\equiv 0\) near \(\partial \Omega \) for all \(t\in [0,T]\) and S is smooth except on \(\cup _{0\le t\le T}(\{t\}\times \Sigma (t))\). It holds that
where we note that \(\eta _1\equiv 0\) on the region on which S has positive transition from \(V_0\) to 0. As a candidate of a viscosity sub/supersolution, introduce the functions \(\rho ,{\tilde{\rho }}:[0,T)\times {\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) as
where \(\rho ,{\tilde{\rho }}\) are continuous (smooth except on the levelset), \(\rho (0,\cdot )={\tilde{\rho }}(0,\cdot )=\phi ^0\) and \(\rho (t,x)={\tilde{\rho }}(t,x)=\phi ^0(X(0,t,x))\) on \([0,T)\times \partial \Omega \).
Step 1. We prove that \(\rho \) (resp. \({\tilde{\rho }}\)) is a viscosity subsolution (resp. supersolution) of (3.4) satisfying the initial/boundary condition strictly. For this purpose, we first observe:
Lemma 3.2
Suppose that v satisfy (H1)\('\) and (H2). Let \(\phi ^0:{\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) be continuous. Then, \(f(t,x):=\phi ^0(X(0,t,x))\) satisfies \(\frac{\partial f}{\partial t}+v\cdot \nabla f=0\) on \((0,T)\times \Omega \) in the sense of viscosity solutions.
Proof
We first check that f is a viscosity subsolution. Fix any \((t,x)\in (0,T)\times \Omega \). Let \(\varphi \) be any test function satisfying the condition of the test for viscosity subsolutions at (t, x), i.e., \(f(s,y)\varphi (s,y) \le f(t,x)\varphi (t,x)\) for all (s, y) near (t, x). Then, we see that
i.e., \(\textrm{D}\varphi (t,x)=(\frac{\partial \varphi }{\partial s}(t,x),\nabla _y\varphi (t,x))\in \textrm{D}^+f(t,x)\), where \(\textrm{D}^+f(t,x)\) stands for the superdifferential^{Footnote 3} of f at (t, x). Since \(f(s,X(s,t,x))\equiv f(t,x)= \phi ^0(X(0,t,x))\) for all \(0\le s\le t\) and
(3.8) with \(y=X(s,t,x)\) and \(s\rightarrow t0\) implies that
from which we obtain
Therefore, we conclude that f is a viscosity subsolution. A similar reasoning shows that f is a viscosity supersolution, where we look at the subdifferential^{Footnote 4}\(\textrm{D}^f(t,x)\) of f. \(\square \)
We come back to \(\rho ,{\tilde{\rho }}\) in the proof of Theorem 3.1. Fix any \((t,x)\in (0,T)\times \Omega \). We check that \(\rho \) is viscosity subsolution of (3.4). Let \(\varphi \) be any test function satisfying the condition of the test for viscosity subsolutions at (t, x):
Case 1: \((t,x)\in \cup _{0< s<T}(\{s\}\times \Sigma (s))\). Then, \(\rho (s,X(s,t,x))=\rho (t,x)= 0\) for all \(0\le s\le t\). Hence, the same reasoning as the proof of Lemma 3.2 works, i.e., taking \((s,y)=(s,X(s,t,x)\) in (3.9) and sending \(s\rightarrow t0\), we obtain
This yields the desired inequality:
Case 2: \((t,x)\not \in \cup _{0< s<T}(\{s\}\times \Sigma (s))\). Setting \(r:=\rho (t,x)\varphi (t,x)\) in (3.9), we have
Since \(e^{\int _0^s S(s',X(s',s,y))ds'}>0\), (3.10)\(\times e^{\int _0^s S(s',X(s',s,y))ds'}\) yields
while \(0= \rho (t,x)(\varphi (t,x)+r) = \rho (t,x)e^{\int _0^t S(s',X(s',t,x))ds'}(\varphi (t,x)+r)e^{\int _0^t S(s',X(s',t,x))ds'}\). Hence, setting \(\psi (s,y):=(\varphi (s,y)+r)e^{\int _0^s S(s',X(s',s,y))ds'}\), we obtain
Since \(\psi (s,y)\) is \(C^1\)smooth near (t, x), \(\psi \) serves as a test function for \(f(s,y)=\phi ^0(X(0,s,y))\) at (t, x); since Lemma 3.2 confirms that f is a viscosity solution of \(\frac{\partial f}{\partial t}+v\cdot \nabla f=0\) on \((0,T)\times \Omega \), we obtain
Set \(g(s,y):=\int _0^s S(s',X(s',s,y))ds'\). Then, \(\psi =(\varphi +r) e^g\) and the left hand side of (3.11) is
A direct calculation yields \(\frac{\partial g}{\partial s}(t,x)+v(t,x)\cdot \nabla _y g(t,x)=S(t,x)\). Therefore, we obtain
It follows from (3.6) and (3.7) that, if \(x\in \Omega ^+(t)\) (resp. \(x\in \Omega ^(t)\)), \(\rho (t,x)>0\) and \(\rho (t,x)S(t,x)=\rho (t,x)V_0\le \rho (t,x)R(t,x,\nabla _y\varphi (t,x))\) (resp. \(\rho (t,x)<0\) and \(\rho (t,x)S(t,x)\le \rho (t,x)R(t,x,\nabla _y\varphi (t,x))\)). Thus, we obtain
We conclude that \(\rho \) is a viscosity subsolution. A similar reasoning shows that \({\tilde{\rho }}\) is a viscosity supersolution.
Note that one can also proceed in Case 2 of Step 1 based on mollification of \(\phi ^0\) and the fact that \(\rho \), \({\tilde{\rho }}\) satisfy \(\frac{\partial \rho }{\partial t}(t,x)+v(t,x)\cdot \nabla \rho (t,x)=\rho (t,x)S(t,x)\), \(\frac{\partial {\tilde{\rho }}}{\partial t}(t,x)+v(t,x)\cdot \nabla {\tilde{\rho }}(t,x)={\tilde{\rho }}(t,x)S(t,x)\), provided \(\phi ^0\) is smooth.
Step 2. We apply Perron’s method and comparison principle. Recall that \(\partial \Omega \) is invariant under X and that S is defined to satisfy \(S\equiv 0\) near \(\partial \Omega \). Hence, there exist \(\varepsilon _1,\varepsilon _2\) with \(0<\varepsilon _2\ll \varepsilon _1\ll \varepsilon (T)\) such that \(S\equiv 0\) on \([0,T)\times K_{\varepsilon _1}\) and \(X(s,t,x) \in K_{\varepsilon _1}\) for all \(s\in [0,t]\) and all \((t,x)\in [0,T)\times K_{\varepsilon _2}\). Therefore, the viscosity subsolution and supersolution \(\rho ,{\tilde{\rho }}\in C^0([0,T)\times {\bar{\Omega }})\) are such that
Applying Perron’s method (Theorem 3.1 in [23]), we obtain a viscosity solution \(\phi :(0,T)\times \Omega \rightarrow {{\mathbb {R}}}\) of the first equation in (3.4) such that
Since the levelsets of \(\rho (t,\cdot ),{\tilde{\rho }}(t,\cdot )\) are equal to \(\Sigma (t)\), the levelsets of \(\phi (t,\cdot )\) must be equal to \(\Sigma (t)\) as well. It is clear that \(\phi \) can be continuously extended up to \(([0,T)\times \partial \Omega )\cup (\{0\}\times {\bar{\Omega }})\) satisfying
the same holds for \(\phi ^*,\phi _*\), the lower/upper semicontinuous envelop of \(\phi \):
By definition, we have \(\phi _*\le \phi ^*\). On the other hand, \(\phi \) being both a viscosity subsolution and supersolution implies that \(\phi ^*\) is an upper semicontinuous viscosity subsolution (note that \((\phi ^{*})^{*}=\phi ^*\)) and \(\phi _*\) is a lower semicontinuous viscosity supersolution (note that \((\phi _{*})_*=\phi _*\)); the comparison principle (Theorem 8.2 in [11]) implies that \(\phi ^*\le \phi _*\) on \([0,T)\times \Omega \). Here, we remark that (3.4) in the form of (3.5) does not directly satisfy the monotonicity property (\(u\mapsto G(z,u,q)\) must be nondecreasing), but (3.3) implies that one can verify the monotonicity property through the change of variable \(u=e^{V_0t}{\tilde{u}}\) (see the following subsection and Chapter 2 of [20]). Thus, we conclude that \(\phi \) is continuous on \([0,T)\times {\bar{\Omega }}\) satisfying the initial/boundary condition strictly. Furthermore, such a viscosity solution is unique. In fact, if \({\tilde{\phi }}\) is a viscosity solution of (3.4) in the current sense, the comparison principle implies \(\phi \le {\tilde{\phi }}\) by regarding \(\phi \) as a viscosity subsolution and \({\tilde{\phi }}\) as a viscosity supersolution; \(\phi \ge {\tilde{\phi }}\) by regarding \(\phi \) as a viscosity supersolution and \({\tilde{\phi }}\) as a viscosity subsolution. \(\square \)
We remark that the result and reasoning of this subsection hold also for the Hamilton–Jacobi equation corresponding to (2.24), where we need to add suitable cutoff to the nonlinearity in (2.24). To see this, observe that the mapping \(u\mapsto u(\beta p)\) is not monotone for all \(p\in {{\mathbb {R}}}^3\); hence, we define
to confirm \(\sup R\le {\tilde{V}}_0:=2\beta +3\alpha \). Then, with this R, Theorem 3.1 still holds.
3.2 \(C^2\)regularity of viscosity solution near the levelset
We prove that the viscosity solution of (3.4) under the stronger regularity assumptions on v and \(\phi ^0\) coincides with the classical solution of (2.3) in a tglobal tubular neighborhood of the levelset. Note that \(\nabla \phi \) is nicely controlled on the levelset in (2.3) and there exists a tubular neighborhood \(\Theta \) in which \(\eta _1(t,x)\eta _2(\nabla \phi (t,x))\equiv 1\), i.e., the solution of (2.3) satisfies the Hamilton–Jacobi equation in (3.4) within \(\Theta \) and local in space comparison of the solutions to (2.3) and (3.4) makes sense.
Our proof is motivated by the technique known as doubling the number of variables, which is standard in proofs of comparison principles for two viscosity (sub/super)solutions; more precisely, we adapt the localized version of this technique to our situation with an unusual choice of a penalty function, where the reasoning can be much simpler due to the fact that one of the two viscosity solutions is smooth.
The outcome of standard “localized doubling the number of variables” (see, e.g., Theorem 3.12 of [6]) for a Hamilton–Jacobi equation
states the following:

Let \(u,{\tilde{u}}\) be \(C^0\)viscosity solutions of (3.14) defined in a cone
$$\begin{aligned} {\mathcal {C}}:= \{ (t,x)\in [0,b]\times {{\mathbb {R}}}^N\,\, xz\le C(bt) \}, \end{aligned}$$where \(C>0\) is a constant such that
$$\begin{aligned}&H(t,x,p)H(t,x,q)\le Cpq ,\\&H(t,x,p)H(s,y,p)\le C(1+p)(t,x)(s,y). \end{aligned}$$If \(u(0,\cdot )={\tilde{u}}(0,\cdot )\) on \(\{x\,\,xz\le Cb\}\) (the bottom of \({\mathcal {C}}\)), then \(u\equiv {\tilde{u}}\) on \({\mathcal {C}}\).
The cone is the region of dependence for general first order Hamilton–Jacobi equations, i.e., the value u(b, z) is determined by the information only on the bottom of \({\mathcal {C}}\) (the speed of propagation is finite). If we directly apply the result to our case, we have to take such a cone contained in the tubular neighborhood \(\Theta \), which implies that the time interval [0, b] must be small. This is the nontrivial aspect of this subsection. In order to overcome the difficulty, we will introduce an unusual penalty function (i.e., \(h({\bar{u}}^2)\) below) that consists of the classical solution itself. In this sense, our technique is specialized for local in space comparison of a viscosity solution and a classical solution. We emphasize that if both solutions are defined on the whole domain, the issue is obvious, but otherwise not.
Suppose that v satisfies (H1) and (H2). We consider (3.4) with initial data \(\phi ^0\) belonging to \(C^2({\bar{\Omega }};{{\mathbb {R}}})\) and satisfying (2.2) with \(a_1<\nabla \phi ^0\) on \(\Sigma (0)\) (this ensures \(\eta _1\eta _2\equiv 1\) near the levelset; \(a_1\) appears in the definition of \(\eta _2\)), where \(\Sigma (0)\subset \Omega \) is assumed to be a closed \(C^2\)smooth surface. Then, Theorem 2.2 yields a tubular neighborhood \(\Theta \) of the levelset \(\{\Sigma (t)\}_{t\ge 0}\) and a unique \(C^2\)solution \(\phi :\Theta \rightarrow {{\mathbb {R}}}\) of (2.3) with \(\eta _1(t,x)\eta _2(\nabla \phi (t,x))\equiv 1\) in \(\Theta \) (hence, from here, we rewrite (2.3) with R given in (3.2)), while Theorem 3.1 yields a unique continuous viscosity solution \({\tilde{\phi }}:[0,\infty )\times {\bar{\Omega }}\rightarrow {{\mathbb {R}}}\) of (3.4). We fix an arbitrary \(T>0\) and set \(\Theta _T:=\Theta _{0\le t\le T}\).
We need to convert the Hamilton–Jacobi equation in (3.4) into
with G such that \(u\mapsto G(t,x,p,u)\) is nondecreasing. This is done by the change of the functions as
with the constant \(V_0\) given in (3.3). Since \(\phi \) is a \(C^2\)solution of the original Hamilton–Jacobi equation, it is clear that w satisfies the new Hamilton–Jacobi equation
Note that w satisfies (3.17) also in the sense of viscosity solutions. In the case of \({\tilde{\phi }}\) being a viscosity solution, one can also show that \({\tilde{w}}:=e^{V_0t}{\tilde{\phi }}\) is a viscosity solution of (3.17). For the readers’ convenience, we briefly explain how to do it: suppose that a test function \(\psi \) is such that \({\tilde{w}}\psi \) has a local maximum at \((t_0,x_0)\); then, setting the constant \(r:={\tilde{w}}(t_0,x_0)\psi (t_0,x_0)\), we have
from which we obtain
since \({\tilde{\phi }}\) is a viscosity subsolution, it holds that
hence, noting that \(\psi (t_0,x_0)+r={\tilde{w}}(t_0,x_0)\) and \({\tilde{\phi }}(t_0,x_0)=e^{V_0t_0}{\tilde{w}}(t_0,x_0)\), we obtain
therefore, we conclude that \({\tilde{w}}\) is a viscosity subsolution of (3.17); similar argument shows that \({\tilde{w}}\) is a viscosity supersolution of (3.17).
We rewrite (3.17) in the form of (3.15) with
Due to the condition of v and the definition of R (see (3.2)), there exists a constant \(C>0\) such that
In order to have \(\phi \) defined also on the boundary of a tubular neighborhood, we consider another tubular neighborhood \({\tilde{\Theta }}\subset \Theta \) of \(\{\Sigma (t)\}_{t\ge 0}\) such that \(\phi \) is defined on the closure of \({\tilde{\Theta }}\), and (partly) describe \({\tilde{\Theta }}_T:={\tilde{\Theta }}_{0\le t \le T}\) as a family of cylinders, i.e., a foliation:

Let \(\alpha >0\) be a constant such that
$$\begin{aligned} \alpha 2V_0C\sup _{{\tilde{\Theta }}_T}\nabla w>0\,\,\, \text{ with } w(t,x)=e^{V_0t}\phi (t,x); \end{aligned}$$(3.19) 
Consider the function
$$\begin{aligned} {\bar{u}}(t,x):= e^{\alpha t}w(t,x), \end{aligned}$$where \({\bar{u}}\) solves in the classical sense
$$\begin{aligned} \frac{\partial u}{\partial t}+v\cdot \nabla u+u{\tilde{R}}(t,x,e^{\alpha t}\nabla u)= \alpha u ,\quad u(0,\cdot )=\phi ^0; \end{aligned}$$(3.20) 
With a constant \(m_0>0\), define
$$\begin{aligned} A_m&:=\{ (t,x)\in [0,T]\times \Omega \,\,{\bar{u}}(t,x)=m\},\quad m_0\le m\le m_0,\quad \\ \Gamma _T&:= \bigcup _{m_0\le m\le m_0} A_m. \end{aligned}$$Since \(\nabla {\bar{u}}\ne 0\) near \(\cup _{0\le t\le T} (\{t\}\times \Sigma (t))\), we may choose \(m_0>0\) (possibly very small) so that \(\Gamma _T\) is contained in \({\tilde{\Theta }}_T\) and \(\partial \Gamma _T{\setminus } \Gamma _T_{t=0,T}=(A_{m_0}\cup A_{m_0})_{0<t<T}\), while \(\Gamma _T\) contains the [0, T]part of a tubular neighborhood of \(\{\Sigma (t)\}_{t\ge 0}\). Note that such an \(m_0\) depends, in general, on T.
Theorem 3.3
Suppose that v satisfies (H1) and (H2). Let \(\phi ^0\) be a function belonging to \(C^2({\bar{\Omega }};{{\mathbb {R}}})\) and satisfying (2.2) with \(a_1<\nabla \phi ^0\) on \(\Sigma (0)\), where \(a_1\) appears in the cutoff \(\eta _2\) and \(\Sigma (0)\subset \Omega \) is a closed \(C^2\)smooth surface. Then, for the \(C^2\)solution \(\phi \) of (2.3) and the continuous viscosity solution \({\tilde{\phi }}\) of (3.4), it holds that \(\phi \equiv {\tilde{\phi }}\) on \(\Gamma _T\), where \(T>0\) is arbitrary.
Remark
\(\Gamma _T\) would become narrower as T gets larger. In order to find a more optimal region on which \(\phi \equiv {\tilde{\phi }}\), one can iterate Theorem 3.3 for \(T=T_0\), \(T=2T_0\), \(T=3T_0\), \(\ldots \), with a small \(T_0>0\).
Proof of Theorem 3.3
We proceed by contradiction. Suppose that the assertion does not hold. We assume
From now on, we deal with \(w,{\tilde{w}}\) given as (3.16). Then, we find an interior point \((t^*,x^*)\) of \(\Gamma _T\) such that
Since the levelset of w and that of \({\tilde{w}}\) are identical, we see that \((t^*,x^*)\not \in A_0=\cup _{0\le t\le T}(\{t\}\times \Sigma (t))\). Let \(m^*\in (m_0,m_0) \) be such that
Let \(\delta >0\) be a constant such that
Take a constant \(M>0\) such that
and a monotone increasing \(C^1\)function \(h: {{\mathbb {R}}}\rightarrow [0,3M]\) such that
With this setting, one can proceed with doubling the number of variables in terms of the function \(F_{\varepsilon ,\lambda }:\Gamma _T\times \Gamma _T\rightarrow {{\mathbb {R}}}\) defined as
However, since w is now a classical solution, we may simplify the argument of doubling the number of variables as follows. We redefine M as
For each \(\lambda >0\), define \(F_\lambda :\Gamma _T\rightarrow {{\mathbb {R}}}\) as
Let \((t_0,x_0)=(t_0(\lambda ),x_0(\lambda ))\in \Gamma _T\) be such that
Step 1. We claim that there exists a sufficiently small \(\lambda >0\) for which \((t_0,x_0)\) is away from the “lateral surface” of \(\Gamma _T\) (i.e., \((t_0,x_0)\not \in A_{\pm m_0}\), or equivalently, \({\bar{u}}(t_0,x_0)\ne m_0\)) and \(t_0\ne 0\). We prove the claim. Since \(F_{\lambda }(t_0,x_0)\ge F_{\lambda }(t^*,x^*)\) and \(h({\bar{u}}(t^*,x^*)^2)=h((m^*)^2)=0\) by definition, we may fix \(\lambda >0\) small enough to obtain
Suppose that \((t_0,x_0)\) is on the “lateral surface” of \(\Gamma _T\), i.e., \((t_0,x_0)\in A_{\pm m_0}\), or equivalently, \({\bar{u}}(t_0,x_0)=m_0\). Then, with the definition of h, we see that
which contradicts to (3.21). Since \(w\equiv {\tilde{w}}\) on \(\Gamma _T_{t=0}\), (3.21) implies that \(t_0=0\) is impossible.
Step 2. Fixing \(\lambda >0\) as mentioned in Step 1, we demonstrate the test at \((t_0,x_0)\) for w being a viscosity subsolution (this is an obvious issue since w is a classical solution) and \({\tilde{w}}\) being a viscosity supersolution with the test function \(\psi \) given as
Note that this makes sense, because the point \((t_0,x_0)\) is either an interior point of \(\Gamma _T\), or \(t_0=T\) and \((T,x_0)\not \in A_{\pm m_0}\) (see Lemma in Section 10.2 of [18] for a remark on the case \(t_0=T\)). Since w is a classical solution, we have
For the above \(\psi \), we see that \({\tilde{w}}(t,x)\psi (t,x)\) takes a minimum at \((t_0,x_0)\). Hence, by the test for \({\tilde{w}}\) being a viscosity supersolutions, we obtain
that is,
Since G(t, x, p, u) is nondecreasing with respect to u, (3.21) and (3.23) yield
By (3.22) and (3.24), we obtain
Since \({\bar{u}}\) solves (3.20), we obtain with (3.18),
Since \({\tilde{R}}\le 2V_0\), \({\bar{u}}(t,x)=e^{\alpha t}w(t,x)\) and \(h'\ge 0\), we find that
Due to the choice of \(\alpha \) in (3.19), we reach a contradiction.
The case of \(\max _{\Gamma _T}({\tilde{\phi }}\phi )>0\) can be treated in the same way. \(\square \)
We remark that the result and reasoning of this subsection hold also for the problems (2.24) and (3.4) with R given in (3.13) and \(C^2\)initial data \(\phi ^0\) such that \(\beta \alpha \le \nabla \phi ^0\le \beta +\alpha \).
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
This modification has first been introduced by Ilia Roisman in the lecture entitled Implicit surface method for numerical simulations of moving interfaces given at the international workshop on ‘Transport Processes at Fluidic Interfaces—from Experimental to Mathematical Analysis’, Aachen, Germany, December 2011.
Similar results are valid if \(g(\cdot ,x)\) is merely measurable for every x; cf. the references given in Appendix 2.
\(\textrm{D}^+f(z):=\left\{ a\in {{\mathbb {R}}}^4\,\Big \, \limsup _{{\tilde{z}}\rightarrow z}\frac{f({\tilde{z}})\rho (z) a\cdot ({\tilde{z}}z) }{{\tilde{z}}z}\le 0\right\} \) with \(z=(t,x)\).
\(\textrm{D}^f(z):=\left\{ a\in {{\mathbb {R}}}^4\,\Big \, \liminf _{{\tilde{z}}\rightarrow z}\frac{f({\tilde{z}}){\tilde{\rho }}(z) a\cdot ({\tilde{z}}z) }{{\tilde{z}}z}\ge 0\right\} =\textrm{D}^+(f)(z)\) with \(z=(t,x)\).
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Acknowledgements
The authors thank Prof. Qing Liu in Okinawa Institute of Science and Technology for his pointing out Theorem 3.12 of [6]. They also thank the anonymous reviewers for their valuable comments, which helped to improve Sect. 3. Dieter Bothe and Mathis Fricke are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—ProjectID 265191195—SFB 1194. This work was done mostly during Kohei Soga’s oneyear research stay at the department of mathematics, Technische Universität Darmstadt, Germany, with the grant Fukuzawa Fund (Keio Gijuku Fukuzawa Memorial Fund for the Advancement of Education and Research). Kohei Soga is also supported by JSPS Grantinaid for Young Scientists #18K13443 and JSPS GrantsinAid for Scientific Research (C) #22K03391.
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Appendices
Appendix 1: Method of characteristics
The proofs above require the method of characteristics for Hamilton–Jacobi equations of contact type, i.e. for Hamilton–Jacobi equations generated by a Hamiltonian function H depending also on \(\Phi \) as below. Since we are not aware of a reference about the details of the method of characteristics in this case and for only \(C^1\)regularity of H in time, we include a brief explanation.
Consider a function \(H=H(t,x,p,\Phi )\) such that
Note that the upcoming argument is available also for a function H defined on \([0,\infty )\times K\) with an open subset \(K\subset {{\mathbb {R}}}^{2N+1}\). Let \(w:{{\mathbb {R}}}^n\rightarrow {{\mathbb {R}}}\) be a given \(C^2\)function. We will construct a unique \(C^2\)solution u of
where \(O\subset {{\mathbb {R}}}\times {{\mathbb {R}}}^N\) is a neighborhood of a point \((0,\xi )\in {{\mathbb {R}}}\times {{\mathbb {R}}}^N\). We remark that \(\xi \) can be arbitrary, but the size of O depends on \(\xi \); there exists \(\varepsilon >0\) such that \([\varepsilon ,\varepsilon ]\times B_\varepsilon (\xi )\subset O\) (here, \(B_\varepsilon (\xi )\subset {{\mathbb {R}}}^N\) is the \(\varepsilon \)ball with the center \(\xi \)) and (A.1) is solvable for negative time; O is determined by the inverse function theorem around the point \((0,\xi )\); if one wants to solve (A.1) in O larger than an open set coming from the inverse map theorem around a single point, one needs more arguments, in addition to the upcoming discussion, to confirm injectivity.
We consider autonomization by introducing
with the extended configuration variable (t, x). Then, we see that (A.1) can be seen as
The characteristic ODEsystem of (A.1) is given as the autonomous (contact) Hamiltonian system generated by \({\tilde{H}}(t,x,r,p,\Phi )\), i.e., setting \(Q:=(t,x)\), \(P=(r,p)\), \({\tilde{H}}={\tilde{H}}(Q,P,\Phi )\),
where the solution is written as \(Q(s;\xi ),P(s;\xi ),\Phi (s;\xi )\). This system is equivalent to
Observe that
(A.7) implies that \(t(s)=s\) serves as the nonautonomous factor in (A.2), (A.3),(A.4),(A.6); hence, our regularity assumption on H is sufficient to provide the smooth dependency with respect to initial data (note that this is apparently not clear if we only look at the autonomous system with \({\tilde{H}}(Q,P,\Phi )\), as \(C^2\)regularity in Q is missing). (A.8) and (A.9) imply that
The map \(F(s,\xi ):=Q(s;\xi )=(s,x(s;\xi ))\) is \(C^1\)smooth as long as the characteristic ODEs have solutions. Furthermore, it holds that
where \(\textrm{D}_{(s,\xi )}F\) stands for the Jacobian matrix of F. Therefore, for each \(\xi \in {{\mathbb {R}}}^N\), the inverse map theorem guarantees that there exist two sets, a neighborhood \(O_1\) of \((0,\xi )\) and a neighborhood \(O_2\) of \(F(0,\xi )=(0,\xi )\), such that \(F:O_1\rightarrow O_2\) is a \(C^1\)diffeomorphism. We obtain the inverse map of F as
where \(\xi =\varphi (t,x)\) is the point for which \(x(s;\xi )\) passes through x at \(s=t\). The essential point is to obtain open sets \(O_1\) and \(O_2=F(O_1)\) such that \(F:O_1\rightarrow O_2\) is a \(C^1\)diffeomorphism, no matter how we find them. The easiest way is to use the inverse map theorem around a single point \((0,\xi )\) based on (A.11). The invertibility of F, or the invertibility of \(\xi \mapsto x(s;\xi )\) for fixed s, on a wider region requires injectivity, which will be an additional issue to be verified.
We proceed, assuming that \(O_1\subset {{\mathbb {R}}}\times {{\mathbb {R}}}^N\) is a neighborhood of a subset of \(\{0\}\times {{\mathbb {R}}}^N\) and \(F:O_1\rightarrow O_2=F(O_1)\) is a \(C^1\)diffeomorphism. By (A.10), we have
We claim that
In fact, it holds that
inserting (A.12) in \(\frac{\partial \Phi }{\partial s}\), we get
inserting the characteristic ODEs in \(\frac{\partial Q}{\partial s}\) and \(\frac{\partial P}{\partial s}\), we get
by (A.10), we get
since \(G_i(0,\xi )=0\), we conclude
Hence, it holds that
Now we define
which, apparently, is only \(C^1\)smooth. Observe that
Therefore, with (A.7) and (A.10), we see that u is indeed \(C^2\)smooth and satisfies
The uniqueness of (A.1) follows from the uniqueness of the characteristic ODEs.
As a conclusion, all one needs to do to solve (A.1) by the method of characteristics are

Analyze (A.2), (A.3) and (A.4) with \(t(s)=s\), where (A.5) and (A.6) are not explicitly necessary;

For the map \(F(s,\xi ):=(s,x(s,\xi ))\), find an open set \(O_1\subset {{\mathbb {R}}}\times {{\mathbb {R}}}^N\) with \(O_1_{t=0}\ne \emptyset \) such that \(F:O_1\rightarrow O_2=F(O_1)\) is a \(C^1\)diffeomorphism;

Define \(u(t,x):=\Phi (t;\varphi (t,x)):O_2\rightarrow {{\mathbb {R}}}\) with \(F^{1}(t,x)=(t,\varphi (t,x))\).
Appendix 2: ODEs on closed sets and flow invariance
Let \(J \subset {{\mathbb {R}}}\) be an open interval, \(V \subset {{\mathbb {R}}}^N\) open and \(g: J \times V \rightarrow {{\mathbb {R}}}^N\) continuous. By Peano’s theorem, the initial value problem (IVP for short)
has a local (classical) solution for every \((t_0, x_0) \in J \times V\), i.e. there is \(\varepsilon =\varepsilon (t_0, x_0) >0\) and a \(C^1\)function \(x: I_\varepsilon \rightarrow {{\mathbb {R}}}^N\), with \(I_\varepsilon =(t_0 \varepsilon , t_0 + \varepsilon )\), such that (A.13) is satisfied in every point.
In this situation, a closed set \(K \subset V\) is said to be positive (negative) flow invariant for (A.13), if every solution of (A.13) that starts in \(t=t_0\) at a point \(x_0 \in K\) stays inside K for all (admissible) \(t > t_0\) (\(t < t_0\)). In this case, one also speaks of forward (backward) invariance of K for the righthand side g. A closed set K is called flow invariant, or just invariant, for (A.13), if K is both positive and negative flow invariant for (A.13).
Since classical solutions for (A.13) with (only) continuous righthand side need not be unique, it might happen that, for given closed \(K \subset V\), one solution starting in \(x_0\) stays in K, while another solution leaves K. The autonomous standard example for nonuniqueness, namely \(g(x)=2\sqrt{x}\) on \(V={{\mathbb {R}}}\), already shows this behavior with \(K:=\{0\}\) and \(x_0=0\). One therefore calls a closed \(K \subset V\) weakly (positively or negatively) flow invariant for (A.13), if \(t_0 \in J\) and \(x_0 \in K\) implies the existence of one solution staying in K (for \(t>t_0\) or \(t<t_0\), respectively).
We are interested in situations of unique solvability of (A.13), which holds true if g is jointly continuous and locally Lipschitz continuous in x, i.e. for every \((t_0, x_0) \in J \times V\) there is \(\delta = \delta (t_0, x_0) >0\) and \(L=L(t_0, x_0)>0\) such that
In this case, the PicardLindelöf theorem yields unique solvability of (A.13). Therefore, weak flow invariance then is the same as flow invariance. Let us note in passing that forward (backward) unique solvability of (A.13) for continuous g holds under weaker additional assumptions such as onesided Lipschitz continuity in x. More precisely, if for every \((t_0, x_0) \in J \times V\) there is \(\delta =\delta (t_0, x_0) >0\) and a \(k=k_{t_0,x_0} \in L^1(J)\) such that
then forward uniqueness holds for (A.13).
Evidently, flow invariance of K requires conditions on g. The extreme case is \(K=\{x_0\}\), where \(g(t, x_0)=0\) for \(t \in I_\varepsilon \) is required. In the general case, if \(x_0 \in K\) and \(x(t) \in K\) on \([t_0, t_0 +\varepsilon )\) holds, then
with a remainder term \(e(\cdot ; t_0, x_0)\), which satisfies
Hence
Consequently, the condition
with the socalled “tangent cone”
is a necessary condition, where \(X={{\mathbb {R}}}^N\). Actually, it turns out that the apparently weaker condition
with the socalled Bouligand contingent cone
is sufficient for positive flow invariance of the closed set \(K \subset V\); as above, \(X={{\mathbb {R}}}^N\) here. This is a direct consequence of the following result on existence of solutions for ordinary differential equations on closed sets.
Theorem A1
Let \(J=(a,b) \subset {{\mathbb {R}}}\), \(K\subset {{\mathbb {R}}}^N\) closed and \(g:J \times K \rightarrow {{\mathbb {R}}}^N\) continuous. Then, given any \((t_0, x_0) \in J \times K\), the IVP (A.13) has a local (classical) forward solution if and only if g satisfies the subtangential condition (A.16).
Proofs of Theorem A1 can be found in [3, 14] or [26]. We call elements from \(T_K(x)\) subtangential vectors (to K at x). Observe that in Theorem A1 the righthand side g(t, x) is only defined for \(x \in K\), hence the constraint \(x(t) \in K\) is incorporated into the domain of definition of g. If g is defined on the larger domain \(J \times V\) with \(V \supset K\), then Theorem A1 yields weak positive flow invariance of K as only continuity of g is assumed. A solution which stays inside K is then also called viable [5]. Positive flow invariance follows if forward uniqueness for (A.13) holds. Since backward solutions of (A.13) are equivalent to forward solutions for \({\widetilde{g}} (t,x):= g(2t_0t,x)\), we also obtain a characterisation of negative flow invariance. Together, this is contained in
Theorem A2
Let \(J=(a,b) \subset {{\mathbb {R}}}\), \(K \subset {{\mathbb {R}}}^N\) closed and \(g: J \times K \rightarrow {{\mathbb {R}}}^N\) continuous. Then the following statements hold true:

(a)
The IVP (A.13) admits local solutions on some \(I_\varepsilon \) (i.e. forward and backward) iff
$$\begin{aligned} g(t,x) \in T_K(x) \text { and } g(t,x) \in T_K(x) \text { on } J \times K. \end{aligned}$$ 
(b)
If g also satisfies the onesided Lipschitz condition (A.15) and satisfies (A.16), then IVP (A.13) has a unique local forward solution for every \(t_0 \in J\) and \(x_0 \in K\).

(c)
If g is locally Lipschitz continuous in x (in the sense of (A.14) above), with \(\pm g(t,x) \in T_K(x)\) for all \(t \in J\), \(x \in K\), then IVP (A.13) has a unique local solution on some \(I_\varepsilon \) (i.e., forward and backward) for every \(t_0 \in J\) and \(x_0 \in K\). If g is defined on \(J \times V\) with some open \(V \supset K\), then K is positive flow invariant for (A.13) in the situation of (b), while K is flow invariant for (A.13) in the situation as described in (c).
Theorems A1 and A2 have been generalized in several directions. See [14] for extensions to ODEs in Banach space, [15] for extensions to differential inclusions (in \({{\mathbb {R}}}^N\) as well as in real Banach spaces), [7] for timedependent constraints, i.e. IVP (A.13) with the additional condition that \(x(t) \in K(t)\) is to hold and [8, 9] for extensions to more general evolution problems. Requirements on the regularity of the function g in t have also been relaxed to measurability; see, e.g., [7, 9, 15]. Due to Theorems A1 and A2, information on flow invariance of given sets can be—to a large extent—obtained from properties of the set of subtangential vectors and their calculus.
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Bothe, D., Fricke, M. & Soga, K. Mathematical analysis of modified levelset equations. Math. Ann. (2024). https://doi.org/10.1007/s0020802402868y
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DOI: https://doi.org/10.1007/s0020802402868y