Abstract
We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a priori bounded but the heat flux provides merely \(L^1\)-coercivity. Applying higher differentiability techniques in space and time, choosing a special weighted norm (equivalent to the Euclidean norm in \(\mathbb {R}^d\)), incorporating finer properties of integrable functions and flux truncation techniques, we prove long-time and large-data existence and uniqueness of weak solution, with an \(L^1\)-integrable flux, to an initial spatially-periodic problem for all values of a positive model parameter. If this parameter is smaller than \(2/(d+1)\), where d denotes the spatial dimension, we obtain higher integrability of the flux. As the developed approach is not restricted to a scalar equation, we also present an analogous result for nonlinear parabolic systems in which the nonlinearity, being the gradient of a strictly convex function, gives an a-priori \(L^\infty \)-bound on the gradient of the unknown solution.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 Problem setting and main result
This paper concerns a parabolic-like problem involving nonlinear elliptic operators that can be viewed as regularizations of the \(\infty \)-Laplacian. More precisely, for fixed \(L>0\) and \(T>0\) we set \(\Omega :=(0,L)^d\subset \mathbb {R}^d\) and \(Q:=(0,T)\times \Omega \) and investigate the following problem: for given \(\Omega \)-periodic functions \(g:[0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}\), \(u_0:\mathbb {R}^d\rightarrow \mathbb {R}\) and a given parameter \(a>0\), find an \(\Omega \)-periodic function \(u:[0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}\) and a vectorial \(\Omega \)-periodic function \(\varvec{q}:[0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}^d\) such that
The motivation for investigating such type of problems is given below. The main result of this paper is the following: for sufficiently smooth initial data \(u_0\), which satisfies a reasonable compatibility condition, and for sufficiently smooth right-hand side g, there exists a unique couple \((u,\varvec{q})\) solving (1.1) in the sense of distributions. To formulate the result precisely, we need to fix the notation, the appropriate function spaces and the concept of solution to (1.1). Since we are dealing with a spatially periodic problem, we recall the definition of periodic Sobolev spaces
where \(k\in \mathbb {N}_0\) and \(p\in [1,\infty )\) are arbitrary (note that \(L^2_{per}(\Omega )=L^2(\Omega )\) and that these spaces, as closed subspaces of reflexive Banach spaces, are reflexive as well provided that \(p\in (1,\infty )\)). The space \(W^{k,\infty }_{per}\) is then defined as
Throughout the paper, we use standard notation for Lebesgue, Sobolev and Bochner spaces equipped with the usual norms. Unless stated otherwise, bold letters, e.g. \(\varvec{q}\), are used for vector-valued functions to distinguish them from scalar functions. The symbol “\(\partial _t\)" stands for the partial derivative with respect to the time variable \(t\in (0,T)\), while the operators “\(\nabla \)" and “\(\mathop {\textrm{div}}\nolimits \)" take into account only the spatial variables \((x_1,\ldots , x_d)\in \Omega \). Later, we also use “\(\partial _j\)” to abbreviate partial derivative with respect to \(x_j\). The shortcut “a.e." abbreviates almost everywhere and “a.a." stands for almost all.
Next, we define the notion of a weak solution to (1.1) and formulate the main result.
Definition 1.1
Let \(u_0\in L^2(\Omega )\), \(g\in L^2(Q)\) and \(a>0\). We say that a couple \((u,\varvec{q})\) is a weak solution to problem (1.1) if
and
Theorem 1.2
Let \(a>0\), \(g\in L^2\left( 0,T; L^{2}(\Omega )\right) \) and \(u_0\in W_{per}^{1,\infty }(\Omega )\) satisfy
-
(i)
Then there exists a unique weak solution to problem (1.1) in the sense of Definition 1.1. Moreover, the solution satisfies
$$\begin{aligned} u\in L^2\left( 0,T;W^{2,2}_{per}(\Omega )\right) . \end{aligned}$$(1.4) -
(ii)
Furthermore, if \(g\in W^{1,2}\left( 0,T; L^2(\Omega )\right) \) and \(u_0\in W_{per}^{2,2}(\Omega )\), then the solution u to (1.1) fulfills \(u\in W^{1,\infty }(0,T; L^2(\Omega ))\). If, in addition, the parameter a satisfies
$$\begin{aligned} a\in \left( 0,\frac{2}{d+1}\right) , \end{aligned}$$(1.5)
then
The paper is structured in the following way. In the rest of this section, we describe the main novelties of our result in detail. We also add a physical motivation for studying such problems and show the key difficulties of the studied problem. Section 2 contains several auxiliary results needed in the proof of Theorem 1.2. In Sect. 3, we prove the uniqueness result. Sections 4 and 5 concern the existence result. In Sect. 4, we introduce a suitable \(\varepsilon \)-approximation of the problem (1.1), which is then treated by the standard Faedo-Galerkin method in combination with a cascade of energy estimates that helps to establish the existence of a weak solution to the \(\varepsilon \)-approximation for arbitrary fixed \(\varepsilon \in (0,1)\). Finally, we derive and summarize the whole cascade of estimates that are uniform with respect to \(\varepsilon \). Then, in Sect. 5, letting \(\varepsilon \rightarrow 0+\), we incorporate the flux truncation technique together with a special choice of weigthed scalar product (equivalent to the standard scalar product in \(\mathbb {R}^d\)) to identify a weak solution of the original problem. Section 6.2 is devoted to the proof of higher regularity (integrability) of the flux \(\varvec{q}\) for the values of a satisfying (1.5), which concludes the proof of the second part of Theorem 1.2. In the final section, we formulate a generalization of the results stated in Theorem 1.2.
1.2 State of the art and main novelties
In order to put our result in an appropriate context, we introduce nonlinear (quasilinear) elliptic and parabolic problems characterized by the presence of p-Laplacian or its generalizations of various forms. Thus, for \(d\in \mathbb {N}\), \(a>0\), \(\delta \in \{0,1\}\) and p satisfying \(1 < p \le \infty \), we define \(\varvec{f}_{\!p'}: \mathbb {R}^d\rightarrow \mathbb {R}^d\) by
Similarly, now for p satisfying \(1\le p < \infty \), we set \(\varvec{g}_p:\mathbb {R}^d \rightarrow \mathbb {R}^d\) as
Replacing the Eq. (1.1b) by
we obtain
while replacing (1.1b) by
we end up with
Next, let us first restrict ourselves to the case \(p\in (1,\infty )\). Then, the mappings \(\varvec{f}_{\!p'}\) and \(\varvec{g}_{p}\) are strictly monotone for all \(a>0\) and \(\delta \in \{0,1\}\). In addition, when \(\delta =0\), \(\varvec{f}_{\!p'}=(\varvec{g}_{p})^{-1}\) and (1.10) and (1.12) coincide. Note that when \(\delta =1\) the \((\varvec{q},\nabla u)\)-relations are smoothed out near zero (thus eliminating the degeneracy/singularity of the corresponding elliptic operator) and the problems (1.10) and (1.12) do not describe the same \((\varvec{q},\nabla u)\)-relation anymore. In all these cases the natural function spaces for the solution are as follows:
provided that the data satisfy \(u_0\in L^2_{per}(\Omega )\) and \(g\in L^{p'}(0,T; W^{1,p}_{per}\left( \Omega )^*\right) \). Within this functional setting, the existence and uniqueness theory for such problems is nowadays classical, see [20, 23] including and extending the monotone operator theory invented by Minty for the elliptic setting in Hilbert spaces (see [25]). It turns out that one can develop a rather complete theory for such problems and we refer to the classical monograph [15] for additional regularity results. Furthermore, one can introduce a much more general class of possible relationships between \(\varvec{q}\) and \(\nabla u\) that goes far beyond (1.9) or (1.11) and where \(\varvec{q}\) and \(\nabla u\) are related implicitly. This means that instead of (1.1b) one considers the equation \(\varvec{g}(\varvec{q}, \nabla u) = \varvec{0}\) in Q with \(\varvec{g}:\mathbb {R}^{d}\times \mathbb {R}^d \rightarrow \mathbb {R}^d\) continuous. Under suitable assumptions imposed on \(\varvec{g}\), providing among others p-coercivity for \(\nabla u\) and \(p'\)-coercivity for \(\varvec{q}\), a self-contained large-data mathematical theory within the above functional setting has been recently developed, also for the systems, in [12] (including, but also extending the results established in [10, 11] in the context of fluid mechanics).
A natural and interesting question is what happens when \(p\rightarrow 1^+\) or \(p\rightarrow \infty \). In the case \(\delta =0\), we formally obtain from (1.11) for \(p=1\) that
Then, the governing equation for the time-independent (stationary) problem being of the form \(\ -\mathop {\textrm{div}}\nolimits (\nabla u/|\nabla u|) = g\) formally represents the Euler-Lagrange equation corresponding to the minimization of the total variation functional. Analogously, and again for \(\delta =0\), it follows from (1.9) that for \(p=\infty \) (i.e. \(p'=1\)) one has
which, together with the governing equation \(-\mathop {\textrm{div}}\nolimits \varvec{q} = g\), corresponds to the so-called \(\infty \)-Laplacian, see also Fig. 1.
Both limiting cases have attracted attention in the scientific community. Not only is the understanding of these limiting cases interesting as a mathematical problem per se, but also the total variation equation or \(\infty \)-Laplacian are frequently used when studying sharp interface-like problems, image recovering, etc. Let us point out that, in the elliptic (i.e. stationary) setting, one faces serious difficulties with defining a proper concept of solution and usually one has to introduce a new one. While for \(p=1\) this has led to the theory of BV spaces, see e.g. [19], for \(p=\infty \) the concept of viscosity solution was introduced in [3]. In principle, one can say that the expected \(L^1\)-regularity for \(\nabla u\) (when \(p=1\)) or the \(L^1\)-regularity for \(\varvec{q}\) (when \(p=\infty \)) must be relaxed and one is led to work in the “weak\(^{*}\) closure of \(L^1\)" or, more precisely, in the space of Radon measures. In the parabolic setting, there is a certain mollification effect coming from the presence of the time derivative and therefore the case \(p=1\) is not so difficult to treat provided that the initial data are sufficiently regular, see e.g. [2]. However, for \(p=\infty \), one seems to be forced to keep the notion of a viscosity solution, see [1, 26]. Furthermore, it is also well known that the viscosity solution is in principle the best object one can deal with, which is well documented by the existence of a singular solution (see [4] or the monograph [22]).
The above discussion was focused on the case \(\delta =0\), which leads to certain singular behaviour near zero. For a mollified problem with \(\delta =1\), the limiting cases take the form
which may have better properties since both equations represent strictly monotone mapping unlike the case \(\delta =0\), see also Fig. 2. Nevertheless, even in this regularized case, one encounters difficulties. The most famous example concerns the case \(a=2\) and \(p=1\), i.e. the minimal surface problem. Due to Finn’s counterexample (see [16]), it is known that even for smooth data one can obtain an irregular solution that is not a Sobolev function. However, such a singularity appears only on (the Dirichlet part of) the boundary. This follows from two results: the interior regularity established for the stationary problem with \(p=1\) and \(a\le 2\) in [9] and the existence result established in [8] showing that the solution of the Neumann problem (for \(p=1\) and \(a>0\) arbitrary) is indeed a Sobolev function and there is no need to involve BV spaces. As this paper documents, a similar situation occurs the problems with \(p=\infty \) and \(\delta =1\).
Apparently, one could follow the procedure developed for \(\infty \)-Laplacian and try to treat the problem with the notion of viscosity solution. However, it is not clear how to adopt the theory of viscosity solution to our setting since we are dealing with a different elliptic operator (compare the limiting behaviour for \(p=\infty \) and \(\delta =0\) or \(\delta =1\) depicted at Figs. 1 and 2). More importantly, it turns out (and this is one of the main messages of this paper) that we do not need to introduce the concept of viscosity solution as we are able to establish the existence of a standard weak solution. Our method builds on the approach developed in [13] and [6], where a similar elliptic problem arising in solid mechanics is analyzed. In this paper, we generalize the approach proposed in [6, 13] (and used in some sense also in [8]) and adopt it to the parabolic setting.
An interesting problem might be the study of the limit \(a\rightarrow \infty \). In such a case
and consequently (for \(\varvec{f}_{\!1}\) introduced in (1.7))
However, the limiting mapping is not strictly monotone (see Fig. 3) and the method developed in this paper cannot be applied.
To summarize and emphasize the novelty of our result once again, we show the existence of a weak solution to the evolutionary problem (1.1) for all \(a>0\) with no need to introduce the concept of viscosity solution and with \(\varvec{q}\) being an integrable function.
It is worth mentioning that our proof of Theorem 1.2, as presented below, is based on two properties of the nonlinear function \(\varvec{f}_{\!1}\) defined in (1.7), namely, its radial structure, i.e. \(\varvec{f}_{\!1}(\varvec{q}) = \alpha (|\varvec{q}|)\varvec{q}\), and the existence of strictly convex potential to \(\varvec{f}_{\!1}\). Consequently, the specific form of the Eq. (1.1b) is not essential and we can develop a satisfactory theory for a general class of relations behaving like mollified \(\infty \)-Laplacian (provided that there is a strictly convex potential behind). We state such a generalized result in Theorem 7.1 in Sect. 7 but do not provide the proof for simplicity here. However, an interested reader can compare our proof with the general methods invented in [7] for the elliptic setting. In fact, by adopting these methods and combining them with the proof of Theorem 1.2, one can prove Theorem 7.1.
1.3 A fluid mechanics problem motivating this study
Consider an incompressible fluid with constant density flowing, at a uniform temperature, in a three-dimensional domain. In the absence of external body forces, unsteady flows of such a fluid are described by the following set of equations for the unknown velocity field \(\varvec{v}= (v_1,v_2,v_3)\) and the pressure p:
where \(\mathbb {S}\), the deviatoric part of the Cauchy stress tensor, enters the additional (so-called constitutive) equation relating \(\mathbb {S}\) to the symmetric part of the velocity gradient denoted by \(\mathbb {D}\) and characterizing the material properties of a particular class of fluids. While for the Newtonian fluids one has \(\mathbb {S} = 2\nu _* \mathbb {D}\), where \(\nu _*>0\) is the kinematic viscosity, there are many viscous fluids and fluid-like materials in which the relation between \(\mathbb {S}\) and \(\mathbb {D}\) is nonlinear. There are fluids (see for example [17, 18, 21, 27, 29]) in which the constitutive relation capable of describing experimental data can be of the form
The general goal is to understand mathematical properties associated with the system of partial differential Eqs. (1.13)–(1.14). A possible natural approach is to look first at a geometrically simplified version of the problem. For example, one can investigate simple shear flows taking place between two infinite parallel plates located at \(x_2=0\) and \(x_2=L\). Time-dependent simple shear flows are characterized by the velocity field of the form \(\varvec{v}(t,x_1,x_2,x_3)=(u(t,x_2),0,0)\). Note that such velocity field fulfills \(\mathop {\textrm{div}}\nolimits \varvec{v}=0\). We also infer that the only nontrivial components of \(\mathbb {D}\) are \(\mathbb {D}_{12}=\mathbb {D}_{21}=\frac{1}{2} \partial _2 u\). Hence it follows from (1.14) that also all components of \({\mathbb {S}}\) other than \(\mathbb {S}_{12}=\mathbb {S}_{21}=:\sigma =\sigma (t,x_2)\) vanish. Then the second equation in (1.13) together with (1.14) leads to:
It follows from the second and the third equation in (1.15a) that \(p=p(t,x_1)\). After inserting this piece of information into the first equation of (1.15a) we can decompose this equation and obtain
for some function g depending only on time. When studying the unsteady Poiseuille flow, the function g, corresponding to the pressure drop, must be given. Then the first equation in (1.16) together with (1.15b) represents a one-dimensional version of the governing equations of the problem (1.1) studied in this paper (with the caveat that in (1.1) the function g may also depend on the spatial variable).
1.4 Difficulties and main idea
As mentioned above, the key difficulty is due to a weak a priori estimate for \(\varvec{q}\) compensating the fact that \(\nabla u\) is bounded a priori. To be more explicit, let us recall the definition (1.7) with \(\delta =1\), i.e. \(\varvec{f}_{\!1}(\varvec{q}):= \frac{\varvec{q}}{(1+ |\varvec{q}|^a)^{\frac{1}{a}}}\). Obviously, \(\left| {\varvec{f}_{\!1}(\varvec{q})}\right| =\frac{\left| {\varvec{q}}\right| }{(1+\left| {\varvec{q}}\right| ^a)^{\frac{1}{a}}}<1\) for all \(\varvec{q}\in \mathbb {R}^d\). This directly yields that \(\nabla {u}\in L^{\infty }(Q;\mathbb {R}^d)\), but it also brings the restriction that the inverse function of (the injective function) \(\varvec{f}_{\!1}\) cannot be defined outside of the unit ball in \(\mathbb {R}^d\) and hence we may not simply write \(\varvec{q}\) as a function of \(\nabla {u}\) and directly apply the Faedo–Galerkin approximation method.
Next, standard energy estimates are not sufficient to establish the existence of a weak solution. Indeed, multiplying the linear Eq. (1.1a) by the solution u, integrating by parts with respect to the spatial variables (the spatial periodicity ensures that the boundary terms vanish) and substituting for \(\nabla u\) from (1.1b) we conclude that
However, this implies merely that \(\varvec{q}\) belongs to \(L^1(Q;\mathbb {R}^d)\) which is not a reflexive Banach space (it does not even have a predual). Hence, when constructing a solution, we may not identify a weak limit of a subsequence of \(\left\{ \varvec{q}^n\right\} _{n=1}^{\infty }\), a sequence of some approximations bounded in \(L^1(Q;\mathbb {R}^d)\). Similar difficulties occur if one aims to investigate the limiting behaviour when converging from the p-Laplacian to the \(\infty \)-Laplacian, i.e. when studying the limit \(p'\rightarrow 1+\) in (1.7).
At this point one might consider a priori estimates involving higher derivatives. Let us denote by s a general time or spatial variable, i.e. s can represent \(t, x_1, \ldots , x_d\). Let us differentiate the Eq. (1.1a) with respect to s, multiply the result by \(\partial _s u\) and integrate over \(\Omega \). Finally, in the integral involving \(\varvec{q}\), we integrate by parts and obtain
Hence, if the data are sufficiently regular, one can hope for an a priori estimate for \(\varvec{q}\) of the form
Let us now focus on the information coming from (1.17) for general \(\varvec{f}_{\!p'}\) with \(p'\in [1,\infty )\). Using (1.7) (cf. Lemma 2.1) one obtains
For \(p'>1\) we have \(p'-2>-1\) and we can employ the Cauchy–Schwarz inequality for the last term to obtain the estimate
where \(C:=\min \{p'-1,1\}>0\) and this can be exploited to control \(\partial _s\varvec{q}\) in \(L^s(Q;\mathbb {R}^d)\) for some \(s>1\). However, in the critical case \(p'=1\), there is a sudden loss of information as one then deduces merely the estimate
Consequently, the power of \(\left| {\varvec{q}}\right| \) in this weighted estimate drops by a. For small values of a, namely for those satisfying (1.5), it can be deduced from (1.17) and (1.19) using Sobolev embedding that \(\varvec{q}\) is bounded in \(L^b(Q;\mathbb {R}^d)\) for some \(b>1\), see (1.6). This is shown in the proof of the second part of Theorem 1.2. However, for large values of a, the estimate (1.19) seems to be useless at the first glance. We will however show that it implies almost everywhere convergence for a selected subsequence of \(\{\varvec{q}^m\}\). This is still not sufficient to take the limit in the governing equation (due to \(L^1\)-integrability of \(\{\varvec{q}^m\}\)). This is why we truncate a suitable m-approximating problem with respect to the flux \(\varvec{q}^m\) and then, in order to take the limit from the truncated formulation of the approximate problem to the weak formulation of the original problem, we shall work directly with the quantity \(\partial _s\varvec{q}\cdot \partial _s(\nabla u)\) (or more precisely with the right-hand side of (1.18)), which in some sense still generates an estimate for \(\partial _s \varvec{q}\) in some scalar product in \(\mathbb {R}^d\) induced by \(\varvec{q}\) itself.
2 Preliminaries
Here and in the remaining parts of this text we set, for \(a>0\),
The aim of this section is to collect basic properties of \(\varvec{f}\) as well as its \(\varepsilon \)-approximation \(\varvec{f}^{\varepsilon }\) defined, for \(\varepsilon >0\), as:
Lemma 2.1
The following assertions hold true:
-
(i)
\(\varvec{f}\), \(\varvec{f}^{\varepsilon }\in C^1(\mathbb {R}^d;\mathbb {R}^d)\) and for all \(i,j=1,\ldots , d\) and arbitrary \(\varvec{q}\in \mathbb {R}^d\) there holds:
$$\begin{aligned}{} & {} \left( \nabla _{\!\varvec{q}}\varvec{f} (\varvec{q})\right) _{ij}:= \frac{\partial f_i(\varvec{q})}{\partial q_j} =\frac{(1+\left| {\varvec{q}}\right| ^a)\delta _{ij}-\left| {\varvec{q}}\right| ^{a-2}q_iq_j}{(1+\left| {\varvec{q}}\right| ^a)^{1+\frac{1}{a}}} \quad \text { and } \nonumber \\{} & {} \left( \nabla _{\!\varvec{q}} \varvec{f}^{\varepsilon } (\varvec{q})\right) _{ij} = \left( \nabla _{\!\varvec{q}} \varvec{f} (\varvec{q})\right) _{ij} + \varepsilon \delta _{ij}, \end{aligned}$$(2.3)where \(\delta _{ij}\) is the Kronecker delta.
-
(ii)
Introducing the scalar functions \(f(s):=\frac{s}{(1+s^a)^{\frac{1}{a}}}\) and \(f_{\varepsilon }(s):= f(s) +\varepsilon s\) we have the following “radial" representations for \(\varvec{f}\) and \(\varvec{f}^{\varepsilon }\):
$$\begin{aligned} \varvec{f} (\varvec{q})=f(\left| {\varvec{q}}\right| )\frac{\varvec{q}}{\left| {\varvec{q}}\right| } \quad \text { and }\quad \varvec{f}^{\varepsilon }(\varvec{q})=f_{\varepsilon }(\left| {\varvec{q}}\right| )\frac{\varvec{q}}{\left| {\varvec{q}}\right| } \quad \text { for every } \varvec{q}\ne \varvec{0}. \end{aligned}$$(2.4) -
(iii)
For \(\varepsilon > 0\) the function \(\varvec{f}^{\varepsilon }\) is a diffeomorphism from \(\mathbb {R}^d\) onto \(\mathbb {R}^d\), while \(\varvec{f}\) is a diffeomorphism from \(\mathbb {R}^d\) onto the open unit ball \(B_1(0)\subset \mathbb {R}^d\).
Proof
For \(\varvec{q}\ne \varvec{0}\) we have
This result can be easily extended to \(\varvec{q}=\varvec{0}\). Indeed, the above formula for partial derivatives is clearly continuous on \(\mathbb {R}^d\setminus \{\varvec{0}\}\) and since \(a>0\) and \(\left| {q_iq_j}\right| \le \left| {\varvec{q}}\right| ^2\) for all \(i,j\in \{1\dots ,d\}\), we conclude \(\left| {\varvec{q}}\right| ^{a-2}q_iq_j\rightarrow 0\) as \(\varvec{q}\rightarrow \varvec{0}\). Thus \(\varvec{f}, \varvec{f}^{\varepsilon }\in C^1(\mathbb {R}^d;\mathbb {R}^d)\). This proves the first assertion.
As the vectors \(\varvec{q}\) and \(\varvec{f}^{\varepsilon }(\varvec{q})\) have the same direction, the formulae (2.4) follow. Furthermore, \(\lim _{s\rightarrow 0^+} f (s)=0\), \(\lim _{s\rightarrow \infty }f (s)=1\) and \(f'(s)=(1+s^a)^{-\frac{1+a}{a}} >0\). Consequently, f is a strictly increasing \(C^1\)-function mapping \([0,\infty )\) onto [0, 1) and, for any \(\varepsilon >0\), \(f_{\varepsilon }\) is a strictly increasing \(C^1\)-function mapping \([0,\infty )\) onto \([0,\infty )\). Hence the functions
are well defined inverse functions of \(\varvec{f}\) and \(\varvec{f}^{\varepsilon }\), respectively. It is straightforward to check that \(\varvec{f}^{-1}\) and \(\left( \varvec{f}^{\varepsilon }\right) ^{-1}\) are continuously differentiable, which completes the proof of (ii) and (iii). \(\square \)
Next, we set
and we focus on its (finer) properties. (In (2.5), \(\mathbb {I}\) stands for the identity matrix and \((\varvec{q}\otimes \varvec{q})_{ij} = q_iq_j\).)
Lemma 2.2
(Scalar product generated by \(\nabla _{\!\varvec{q}} \varvec{f}(\varvec{q})\)) Let \(\varvec{q}\in \mathbb {R}^d\) be arbitrary. The bilinear form on \(\mathbb {R}^d\) given by
is a scalar product on \(\mathbb {R}^d\) satisfying
The corresponding quadratic form fulfills
Hence, is for fixed \(\varvec{q}\in \mathbb {R}^d\) the norm on \(\mathbb {R}^d\) equivalent to the Euclidean norm \(|\cdot |\).
Proof
The proof follows from the definition of \(\varvec{f}\), the formula (2.3) for its derivatives, (2.6) and the Cauchy-Schwarz inequality. The inequalities in (2.8) are direct consequences of (2.6). \(\square \)
The last essential property we need in the proof is the strict monotonicity of \(\varvec{f}\), the strong monotonicity of \(\varvec{f}^{\varepsilon }\) and, consequently, the Lipschitz continuity of its inverse function \((\varvec{f}^{\varepsilon })^{-1}\).
Lemma 2.3
The mappings \(\varvec{f}, \varvec{f}^{\varepsilon }:\mathbb {R}^d\rightarrow \mathbb {R}^d\) defined in (2.1) and (2.2) satisfy, for all \(\varepsilon \in (0,1)\),
Moreover, for any \(\varepsilon >0\), the inverse function \(\left( \varvec{f}^{\varepsilon }\right) ^{-1}\) is uniformly Lipschitz continuous on \(\mathbb {R}^d\), namely,
Proof
We first observe, using also (2.5), that (for \(\varvec{q}_1\ne \varvec{q}_2\))
which gives the strong monotonicity of \(\varvec{f}^{\varepsilon }\) and strict monotonicity of \(\varvec{f}\). Since
we conclude from the last two inequalities that \(\varepsilon \left| {\varvec{q}_1-\varvec{q}_2}\right| \le \left| {\varvec{f}^{\varepsilon }(\varvec{q}_1)-\varvec{f}^{\varepsilon }(\varvec{q}_2)}\right| \), which is equivalent to (2.11). \(\square \)
3 Proof of uniqueness
In this short section, we shall prove that there is at most one weak solution to the problem (1.1).
Let us assume that there are two weak solutions \((u_1,\varvec{q}_1)\) and \((u_2,\varvec{q}_2)\) to the problem (1.1) with the same initial value \(u_0\in L^2(\Omega )\) and the same right-hand side \(g\in L^2(Q)\). Note that the constitutive Eq. (1.2b) implies that \(\nabla u_1, \nabla u_2 \in L^{\infty }(Q)\) and consequently \(u_1\) and \(u_2\) are admissible test function in (1.2a). Subtracting (1.2a) for \((u_2,\varvec{q}_2)\) from the same equation for \((u_1,\varvec{q}_1)\) and taking \(\varphi =u_1(t,\cdot )-u_2(t,\cdot )\) as a test function, we obtain
By (1.2b), \(\nabla u_1-\nabla u_2 = \varvec{f}(\varvec{q}_1) - \varvec{f}(\varvec{q}_2)\). Inserting this relation into (3.1), we obtain
Integrating this with respect to time \(t\in (0, T]\) and using \(u_1(0,x) - u_2(0,x) = 0\) a.e. in \(\Omega \) we arrive at
By taking \(t=T\) and using the strict monotonicity of \(\varvec{f}\), see (2.9), the second term leads to the conclusion that \(\varvec{q}_1=\varvec{q}_2\) a.e. in \((0,T)\times \Omega \). The first term then implies that, for all \(t\in (0,T]\), \(u_1(t,\cdot ) = u_2(t,\cdot )\) a.e. in \(\Omega \). This completes the proof of uniqueness.
4 \(\varepsilon \)-approximations and their properties
In this section, we introduce, for any \(\varepsilon \in (0,1)\), an \(\varepsilon \)-approximation of the problem (1.1) and show, by means of the Galerkin method and regularity techniques performed at the Galerkin level, that this \(\varepsilon \)-approximation admits a unique weak solution with second spatial derivatives in \(L^2(Q)\).
Let \(\varepsilon \in (0,1)\) and \(a>0\). We say that a couple of \(\Omega \)-periodic functions \((u,\varvec{q})=(u^{\varepsilon },\varvec{q}^{\varepsilon })\) solves the \(\varepsilon \)-approximation of the problem (1.1) if
In accordance with the assumptions of Theorem 1.2, we assume that \(u_0 \in W^{1,\infty }_{per}(\Omega )\) satisfies (1.3) and \(g\in L^2(Q)\). We say that a couple \((u,\varvec{q}) = (u^{\varepsilon },\varvec{q}^{\varepsilon })\) is weak solution to (4.1) if
and
Uniqueness of such a solution follows from the same argument as in Sect. 3. To establish the existence of the solution, we apply the Galerkin method combined with higher differentiability estimates that we will perform at the level of Galerkin approximations. These estimates and the limit from the Galerkin approximation to the continuous level represent the core of this section. In Sect. 4.6, we establish and summarize the estimates that are uniform with respect to \(\varepsilon \).
4.1 Galerkin approximations
Consider the basis \(\left\{ \omega _r\right\} _{r=1}^{\infty }\) in \(W_{per}^{1,2}(\Omega )\) consisting of solutions of the following spectral problem:
It is well-known (see e.g. [28] or [24, Appendix A.4]) that there is a non-decreasing sequence of (positive) eigenvalues \(\{\lambda _r\}_{r=1}^{\infty }\) and a corresponding set of eigenfunctions \(\left\{ \omega _r\right\} _{r=1}^{\infty }\) that are orthogonal in \(W^{1,2}_{per}(\Omega )\) and orthonormal in \(L^2_{per}(\Omega )\). Moreover, the projections \({\mathcal {P}}^N\) defined through \({\mathcal {P}}^N(u)=\sum _{i=1}^N \left( \int _{\Omega } u \omega _i\mathop {}\!\textrm{d}x\right) \omega _i\) are continuous both as mappings from \(L^2_{per}(\Omega )\) to \(L^2_{per}(\Omega )\) and from \(W^{1,2}_{per}(\Omega )\) to \(W^{1,2}_{per}(\Omega )\). Also, due to \(\Omega \)-periodicity and elliptic regularity, the \(\Omega \)-periodic extensions of \(\omega _r\) belong to \(C^{\infty }(\mathbb {R}^d)\).
Before introducing the Galerkin approximations of the problem (4.3) we recall, referring to Lemma 2.1, that the relation \(\nabla u = \varvec{f}^{\varepsilon }(\varvec{q})\) is equivalent to \(\varvec{q}= (\varvec{f}^{\varepsilon })^{-1}(\nabla u)\) where \((\varvec{f}^{\varepsilon })^{-1}\) is a Lipschitz mapping from \(\mathbb {R}^d\) to \(\mathbb {R}^d\).
For an arbitrary, fixed \(N\in \mathbb {N}\), we look for \(u^N\) in the form
where the coefficients \(c_r^N\), \(r=1, \dots , N\), are determined as the solution of the system of ordinary differential equations of the form
The local-in-time well-posedness of the above problem (4.5) directly follows from Caratheodory theory (recall here that \(\left( \varvec{f}^{\varepsilon }\right) ^{-1}\) is a Lipschitz mapping). In addition, thanks to the first uniform estimates established in the next subsection, we deduce that the Galerkin system (4.5) is well-posed on (0, T].
4.2 First uniform estimates
Multiplying the r-th equation in (4.5a) by \(c_r\) and summing these equations up for \(r=1,\dots ,N\), we obtain
Using the one-to-one correspondence between \(\varvec{q}^N\) and \(\nabla u^N\), see (4.5a), the second term on the left-hand side can be evaluated explicitly and the above equation takes the form
Integrating over time, using then the Gronwall inequality and the fact that \(\Vert {\mathcal {P}}^N u_0\Vert _{L^2(\Omega )}\le \Vert u_0\Vert _{L^2(\Omega )}\), we obtain
In addition, it also directly follows from \(\nabla u^N = \varvec{f}^{\varepsilon }(\varvec{q}^N)\) (see the second equation in (4.5a)) and the above estimates on \(\varvec{q}^N\) that
Indeed, the second equation in (4.5a), see also the Eq. (4.1b), implies that
where we used the facts that \(\varepsilon ^2 < \varepsilon \) and \((1+\left| {\varvec{q}^N}\right| ^a)^{-\frac{1}{a}} \le 1\). By integrating the obtained inequality over \((0,T) \times \Omega \) and using (4.6), we get (4.7).
4.3 Time derivative estimate (uniform with respect to N)
Multiplying the r-th equation in (4.5a) by \(\mathop {}\!\frac{\textrm{d}}{\textrm{d}t}c_r\) and summing these equations up for \(r=1,\dots ,N\), we obtain
Applying Young’s inequality to the term on the right-hand side, we get
Next, we focus on the second term on the left-hand side. Since \(\nabla u^N = \varvec{f}^{\varepsilon }(\varvec{q}^N)\), it follows from the definition of \(\varvec{f}^{\varepsilon }\) that
Inserting the outcome of this computation into (4.8), integrating the result over (0, T) and using the fact that the function
is increasing (implying that \(\left| {\varvec{q}^N}\right| (1+\left| {\varvec{q}^N}\right| ^a)^{-\frac{1}{a}}-{s}{(1+s^a)^{-\frac{1}{a}}} \ge 0\) on \((0, \left| {\varvec{q}^N}\right| \))), we obtain
Noticing that \(\left| {\varvec{q}^N}\right| (1+\left| {\varvec{q}^N}\right| ^a)^{-\frac{1}{a}}-{s}{(1+s^a)^{-\frac{1}{a}}} \le 1\) on \((0, \left| {\varvec{q}^N}\right| )\) we conclude that
where
Consequently,
which implies that
The fact that \(\Vert {\mathcal {P}}^N (u_0)\Vert _{W^{1,2}_{per}(\Omega )} \le \Vert u_0\Vert _{W^{1,2}_{per}(\Omega )}\) thus finally yields
4.4 Spatial derivative estimates
This time, we multiply the r-th equation in (4.5a) by \(\lambda _r c_r\) and sum the obtained identities up for \(r=1,\dots ,N\). Since, due to (4.4) and the smoothness of \(\omega ^r\),
we get
Hence,
Since \(\nabla u^N = \varvec{f}^{\varepsilon }(\varvec{q}^N)\), recalling (2.5) we get
Hence, by Lemma 2.2, we get
and also, by means of the Cauchy-Schwarz inequality and (2.8),
which, using \(\varepsilon ^2 < \varepsilon \), implies that
Incorporating (4.13) and (4.14) into (4.12), integrating the result with respect to time and using Young’s inequality and the continuity of \({\mathcal {P}}^N\) in \(W^{1,2}_{per}(\Omega )\), we arrive at estimates that are uniform with respect to both N and \(\varepsilon \):
4.5 Limit \(N\rightarrow \infty \)
Due to the reflexivity and separability of the underlying function spaces and the Aubin-Lions compactness lemma, it follows from the estimates (4.6), (4.7), (4.11) and (4.15) that there is a subsequence of \(\left\{ (u^N, \varvec{q}^N)\right\} _{N=1}^{\infty }\) (which we do not relabel) such that
Letting \(N\rightarrow \infty \) in (4.5), it is simple to conclude from the above convergence results that
Since \(u^N(0, \cdot )={\mathcal {P}}^N(u_0)\), \({\mathcal {P}}^N(u_0) \xrightarrow {N\rightarrow \infty } u_0\) in \(L^2(\Omega )\) and \(u\in C\left( [0,T];L^2(\Omega )\right) \), we observe that (4.3c) holds.
By virtue of (4.16c) there is a subsequence (that we again do not relabel) so that
As \(\left( \varvec{f}^{\varepsilon }\right) ^{-1}\) is (Lipschitz) continuous, it follows from the second equation in (4.5a) and (4.18) that
Since the weak limit in \(L^2(Q)\) coincides with the pointwise limit a.e. in Q (provided that these limits exist), we conclude that
Thus, the existence and uniqueness of a weak solution to the \(\varepsilon \)-approximation (4.1) in the sense of definition (4.3) is completed.
In the next subsection, we establish and summarize the estimates associated with the \(\varepsilon \)-approximation (4.1) that are uniform with respect to \(\varepsilon \).
4.6 \(\varepsilon \)-independent estimates for \((u^{\varepsilon }, \varvec{q}^{\varepsilon })\)
Observing that \(u^{\varepsilon }\) is an admissible test function in (4.17), we set \(\varphi = u^{\varepsilon }\) in (4.17). Then, proceeding step by step as at the Galerkin level, we obtain
It is easy to conclude from the boundedness of the second term, by applying Hölder’s inequality, that
Further estimates are obtained in this subsection by taking the limit \(N\rightarrow \infty \) in the estimates obtained at the Galerkin level. In order to simplify the notation, we drop the \(\varepsilon \) label for objects depending also on N. In particular, if such an object appears in a statement together with another one depending only on \(\varepsilon \), they both correspond to the same \(\varepsilon \).
We define \({\varvec{q}_0^{\varepsilon }}\) through the equation
As \(\nabla {\mathcal {P}}^N(u_0) = \varvec{f}^{\varepsilon }(\varvec{q}^N(0,\cdot ))\), see (4.10), \(\nabla {\mathcal {P}}^N(u_0) \rightarrow \nabla u_0\) strongly in , and \((\varvec{f}^{\varepsilon })^{-1}\) is Lipschitz continuous, we conclude that
Consequently, we can take the limit \(N\rightarrow \infty \) in (4.9) and conclude, using also the weak lower semicontinuity of the \(L^2\)-norm together with (4.16b), that
It follows from (1.3) and (4.22) that
This implies that
As \(U<1\) (see (1.3)), we get
The bound \({\mathcal {C}}(a,U{,|\Omega |})\) diverges as \(a\rightarrow 0+\), \(U\rightarrow 1-\) or \(\left| {\Omega }\right| \rightarrow \infty \). Inserting (4.24) into (4.23), we get
Finally, we let \(N\rightarrow \infty \) in (4.15). Recalling (4.16d) and also (4.18) together with (4.19), we have
This implies (see the next subsection for the proof in a slightly more general setting) that
Consequently, letting \(N\rightarrow \infty \) in (4.15), we get
4.7 Weak lower semicontinuity of the weighted \(L^2\)-norm
Here, we shall prove the following statement: if
then
To prove it, we first recall that , where \(\mathbb {A}\) is introduced in (2.5). Observing that
we get
Since \(|\mathbb {A}(\varvec{q}^n)|\le C(d)\) and (4.28) holds, Lebesgue’s dominated convergence theorem implies that
Furthermore, noticing that
we see that, as \(n\rightarrow \infty \), \(I^n_2\) vanishes by virtue of (4.27). To conclude that \(I^n_1\) vanishes as well, we first apply Hölder’s inequality to get that
and then we notice that \(\Vert \varvec{z}^n - \varvec{z}\Vert _{L^2(Q;\mathbb {R}^d)}\) is bounded due to (4.27) and the last integral vanishes again by Lebesgue’s dominated convergence theorem. Thus, \(\lim _{n\rightarrow \infty } (I^n_1 + I^n_2) = 0 \) and the assertion (4.29) follows from (4.30)-(4.32).
5 Limit \(\varepsilon \rightarrow 0+\)
5.1 The attainment of \(\nabla u = \varvec{f}(\varvec{q})\) a.e. in Q
In Sect. 4, assuming that \(u_0\in W^{1,\infty }_{per}(\Omega )\) satisfies (1.3) and \(g\in L^2(Q)\), we established, for any \(a>0\) and \(\varepsilon \in (0,1)\), the existence of unique weak solution to (4.1) satisfying (4.3). Furthermore, particularly in Sect. 4.6, we showed that \(\{(u^{\varepsilon }, \varvec{q}^{\varepsilon })\}_{\varepsilon \in (0,1)}\) satisfies the estimates (4.20), (4.21), (4.25) and (4.26). As a consequence of these estimates (that are uniform w.r.t. \(\varepsilon \)) and the Aubin-Lions compactness lemma, one can find \(\varepsilon _m\rightarrow 0\) and the corresponding sequence \((u^m,\varvec{q}^m):= (u^{\varepsilon _m}, \varvec{q}^{\varepsilon _m})\) such that
and also, using (5.1d) and Egoroff’s theorem on one side and (4.21) and Chacon’s biting lemma (see [5]) on the other side, there is a \({\varvec{q}\in L^1(Q; \mathbb {R}^d)}\) such that for each \(\delta >0\) there exists a \({\tilde{Q}}_{\delta }\subset Q\) fulfilling \({\tilde{Q}}_{\delta _2}\subset {\tilde{Q}}_{\delta _1}\) if \(\delta _1\le \delta _2\) as well as \(|Q{\setminus } {\tilde{Q}}_{\delta }|\le \delta \) such that
Further, we denote
and it follows from (5.2) that
Hence, using the (strict) monotonicity of \(\varvec{f}\), see Lemma 2.3, the facts that \(\varvec{f}(\varvec{q})\in L^{\infty }(Q;\mathbb {R}^d)\) and \(\varvec{f}(\varvec{q}^m) = \nabla u^m - \varepsilon _m \varvec{q}^m\), see (4.3b), the convergence properties (5.2), the obvious relation \(Q_{\delta }\subset {\tilde{Q}}_{\delta }\), and the fact that \(\varvec{q}\) is bounded (depending on \(\delta \)) on \(Q_{\delta }\), we observe that
This implies that there is a subsequence (that we again denote by \(\varvec{q}^m\)) such that
As \(\varvec{f}\) is strictly monotone, we conclude (referring for example to Lemma 6 in [14]) that
However, as \(\delta >0\) is arbitrary and \(|Q{\setminus } Q_\delta |\le C\delta \), this yields
As \(\varvec{f}\) is continuous, letting \(m\rightarrow \infty \) in \(\varvec{f}(\varvec{q}^m) = \nabla u^m - \varepsilon _m \varvec{q}^m\) (valid a.e. in Q) and using (5.1d) and (5.3), we conclude that (1.2b) holds.
5.2 Limit in the governing evolutionary equation
It remains to show that (1.2a) holds. Towards this goal, we “truncate" the Eq. (4.3a) for \(\varepsilon _m\)-approximation with the help of smooth, compactly supported approximations of unity denoted by \(\tau _k\), which are the functions of \(|\varvec{q}^m|\). The required Eq. (1.2a) is then obtained by a careful study of the limiting process as \(m\rightarrow \infty \) and \(k\rightarrow \infty \).
It follows from (4.3a) that, for all \(m\in \mathbb {N}\),
In order to make use of these relations in the absence of weak convergence of \(\varvec{q}^m\) in \(L^1(Q)\), we consider
and set as a test function \(\varphi \) in (5.4)
where \(\tau _k\), \(k\in \mathbb {N}\), “approximates unity”, i.e. \(\tau _k\in C_0^{\infty }\left( [0,\infty )\right) \) satisfies for all \(k\in \mathbb {N}\) the following conditions: \(0\le \tau _k(s)\le 1\) for all \(s\in [0,\infty )\), \(\tau _k(s)= 1\) on [0, k], \(\tau _k(s)=0\) on \([ k+1, \infty )\) and \(-2\le \tau _k'(s)\le 0\) for all \(s\in (k,k+1)\). Note that, for fixed m, the test function specified in (5.5) is an admissible test function due to (4.26).
Inserting (5.5) into (5.4), we obtain
Letting \(m\rightarrow \infty \) and \(k\rightarrow \infty \) in (5.6), our aim is to show that we can remove m and replace \(\tau _k\) by 1 in the first three integrals, while the last integral vanishes. Each term requires a special treatment.
To treat the term involving the time derivative, we first observe that
By Hölder’s inequality, (4.25), (5.3) and Lebesgue’s dominated convergence theorem, we observe that
By (5.1b), \(J_2^{m,k} \rightarrow 0\) as \(m\rightarrow \infty \). Using Levi’s monotone convergence theorem we also get
Consequently, it follows from (5.7) and the above arguments that
Even simpler arguments give
Since, by (5.3) and Lebesgue’s dominated convergence theorem,
we also observe (again using Levi’s monotone convergence theorem) that
It remains to show that the last term in (5.6) tends to zero as \(m,k \rightarrow \infty \). To prove this, we will incorporate the weighted \(L^2\)-estimates for \(\nabla \varvec{q}^m\), see (4.26). Before starting to treat this term, we introduce, for every \(k\in \mathbb {N}\), an auxiliary function \(G_k\) through
and observe that \(G_k(t) =0\) on [0, k] and
Let us now rewrite the last term in (5.6) in the following manner:
To show that \(J_4^{m,k}\) vanishes as \(m\rightarrow \infty \) and \(k\rightarrow \infty \), we first employ, for any fixed k, (5.3) and Lebesgue’s dominated convergence theorem (noticing that \(\left| {\nabla \psi \cdot \varvec{f}(\varvec{q}^m)}\right| G_k(\left| {\varvec{q}^m}\right| ) \le 2^{\frac{a+1}{a}} (1+k)\Vert \nabla \psi \Vert _{L^{\infty }(Q;\mathbb {R}^d)}\)) and obtain
Since \(G_k(t)= 0\) on [0, k], we conclude from the estimate (5.9) and the fact that \(\varvec{q}\in L^1(Q;\mathbb {R}^d)\) that
Hence, \(\lim _{k\rightarrow \infty }\lim _{m\rightarrow \infty } J_4^{m,k} = 0\).
In order to show that also the term \(J_5^{m,k}\) vanishes as \(m\rightarrow \infty \) and \(k\rightarrow \infty \) we need to proceed more carefully. First, recalling (2.5), we notice that
where \(\varvec{e}_s\in \mathbb {R}^d\) is the sth vector of the canonical basis in \(\mathbb {R}^d\), \(s=1, \dots , d\). This allows us to rewrite and estimate \(J^{m,k}_5\) introduced in (5.10) as follows:
Letting \(m\rightarrow \infty \) in the last term, using (5.3), (5.9) and Lebesgue’s dominated convergence theorem, we get
However, as \(G_k(s)= 0\) on [0, k] and (5.9) holds, we further observe that
Hence \(\lim _{k\rightarrow \infty }\lim _{m\rightarrow \infty } \left| J_5^{m,k}\right| = 0\) and, taking all computations starting at (5.10) into consideration, the last term in (5.6) vanishes as \(m\rightarrow \infty \) and \(k\rightarrow \infty \). The proof of the first part of Theorem 1.2 is complete.
6 Improved time derivative estimates and higher integrability of the flux for \(a\in (0, 2/(d+1))\)
In order to prove the second part of Theorem 1.2, we will combine the uniform spatial derivative estimates established in (4.26) for \((u^\varepsilon ,\varvec{q}^\varepsilon )\) together with the uniform time derivative estimates that we are going to prove next.
6.1 Improved time derivative estimates
Consider, for any \(\varepsilon \in (0,1)\), the unique weak solution \((u^\varepsilon ,\varvec{q}^\varepsilon )\) to (4.1) satisfying (4.2) and (4.3). It follows from (4.3a), (4.3c) and the assumption \(u_0\in W^{1,2}_{per}(\Omega )\) that, for \(\tau \in (0,T]\),
By setting \(\varphi = u^\varepsilon \) in (4.3a), followed by integration over time between 0 and \(\tau \), we also have
Step 1. For any \(z:[0,T]\times \Omega \rightarrow \mathbb {R}\) and for \(\tau \in \mathbb {R}\) such that \(t+\tau \in [0,T]\), we set
Taking the weak formulation (4.3a) at \(t+\tau \), followed by subtracting (4.3a) at t, and taking then \(\varphi = \frac{1}{\tau } \delta _{\tau }u^\varepsilon \) as a test function in the resulting equation, we obtain
Using (4.3b) (or (4.1b)) and (2.5), we observe that
where
Inserting (6.4) into (6.3) and using the Cauchy-Schwarz inequality to estimate the right-hand side and the Gronwall lemma, we conclude that for a.a. \(t\in (0,T]\) the following estimates holds:
This would lead to the required \(\varepsilon \)-independent estimates provided that we can control \(\Vert \delta _{\tau } u^\varepsilon (0,\cdot )\Vert _{L^2(\Omega )}^2\) uniformly w.r.t. \(\varepsilon \).
Step 2. Towards this aim, we start by noticing that trivially
and
Inserting (6.1) and (6.2) into (6.6) we get
This can be rewritten as
where \(\varvec{q}^\varepsilon (0,\cdot )\) is defined, in accordance with Sect. 4.6, through
Since \(\nabla u^\varepsilon = \varvec{f}^{\varepsilon }(\varvec{q}^\varepsilon )\) and \(\varvec{f}^{\varepsilon }\) is monotone, the second term at the left-hand side of (6.7) is nonnegative. Introducing the notation
and
we conclude from (6.7), using Hölder’s inequality, that
This (together with relabelling s on v and \(\tau \) on s) implies that
Since \(y(0)=0\), integrating (6.10) over \((0,\tau )\) and using then Hölder’s inequality, we get
which implies that
Using this to estimate the last term in (6.9), it follows from (6.9) that
Recalling the definition of A, (6.11) leads to (using also \(1/\tau ^2 \le 1/s^2\))
This finally gives
As \(g\in W^{1,2}\left( 0,T;L^2(\Omega )\right) \) and \(W^{1,2}\left( 0,T;L^2(\Omega )\right) \hookrightarrow C([0,T];L^2(\Omega ))\), the second and the third terms on the right-hand side are bounded.Footnote 1 Hence, we finally get
In order to estimate \(\Vert \nabla \varvec{q}^{\varepsilon }(0,\cdot )\Vert _{L^2(\Omega ;\mathbb {R}^{d\times d})}\), we first recall that it follows from (1.3) and (6.8) that
which implies that
Next, applying the partial derivative w.r.t. \(x_j\) to (6.8) and using also (2.5) we get
Taking the scalar product of this identity with \(\partial _j \varvec{q}^\varepsilon (0,\cdot )\) and summing the result over j, \(j=1,\dots ,d\), we arrive at
By virtue of (2.8), this leads to
which implies that
Consequently, using (6.5) and (6.12), we conclude that
Step 3. Letting \(\tau \rightarrow 0\) in (6.14) (\(\varepsilon \in (0,1)\) being fixed) we claim that
While the limits in the first and third terms of (6.14) are standard and are based on weak lower semicontinuity of the \(L^2\)-norm, the limit in the second term follows from the facts that, as \(\tau \rightarrow 0\),
followed by the convergence arguments established in Sect. 4.7. Thus, (6.15) holds. Consequently, we conclude that \(\partial _t u \in L^\infty (0,T; L^2(\Omega ))\), which is the first statement of part (ii) of Theorem 1.2.
6.2 Higher integrability result
It follows from (4.26) and (6.15) that
Introducing the time-spatial gradient \(\nabla _{t,x}u:= (\partial _t u, \partial _j u, \dots , \partial _d u)\), we can rewrite the last estimate as
Using the last inequality in (2.8), it implies thatFootnote 2
and by simple manipulation also
Hence, using also (4.21), we conclude that, for \(a\in (0,1)\),
and it then follows from Sobolev embedding that
where \(p<\infty \) is arbitrary if \(d=1\) and \(p=\frac{2(d+1)}{d-1}\) if \(d>1\). Thus if \(d=1\) and \(a<1\) we have a bound in any Lebesgue space. In the case of \(d>1\), the above computation gives that
which improves the integrability of \(\{\varvec{q}^\varepsilon \}\), uniformly w.r.t. \(\varepsilon \), provided that
For \(\varepsilon _m \rightarrow 0\), this piece of information is preserved. Thus, the second assertion of Theorem 1.2 is established.
7 Generalization to systems of nonlinear parabolic equations
Finally, we generalize our problem and formulate the existence and uniqueness results for such a generalization. A detailed proof is not provided as it follows from the combination of the arguments developed in the proof of Theorem 1.2 above and from the arguments used when proving the result established in [7], where the stationary case is treated.
Theorem 7.1
Let \(\Omega \), Q be as before and let \(F:\mathbb {R} \rightarrow \mathbb {R}_+\) be a strictly convex \({C}^{1,1}\) function fulfilling \(F(0)=0\). Assume in addition that there exists a positive constant C such that for all \(s\in \mathbb {R}\) there holds
For \(N\in \mathbb {N}\) arbitrary, set
Let , \(u_0\in W^{1,2}_{per}\left( \Omega ;\mathbb {R}^N\right) \) and there exist a compact set \(K\subset \mathbb {R}^{d\times N}\) such that
Then, there exists a unique couple \((u,\varvec{q})\) such that
and
Notes
Note that it would be sufficient to assume that \(g\in W^{\beta ,2}\left( 0,T;L^2(\Omega )\right) \) for some \(\beta >1/2\).
\({\mathcal {C}}^*\) is a generic constant, whose value can change from line to line.
References
Akagi, G., Juutinen, P., Kajikiya, R.: Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. Math. Ann. 343(4), 921–953 (2009)
Andreu-Vaillo, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223. Birkhäuser Verlag, Basel (2004)
Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6(551–561), 1967 (1967)
Aronsson, G.: Construction of singular solutions to the \(p\)-harmonic equation and its limit equation for \(p=\infty \). Manuscr. Math. 56(2), 135–158 (1986)
Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107(3), 655–663 (1989)
Beck, L., Bulíček, M., Málek, J., Süli, E.: On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth. Arch. Rational. Mech. Anal. 225(2), 717–769 (2017)
Beck, L., Bulíček, M., Maringová, E.: Globally Lipschitz minimizers for variational problems with linear growth. ESAIM Control Optim. Calc. Var. 24(4), 1395–1413 (2018)
Beck, L., Bulíček, M., Gmeineder, F.: On a Neumann problem for variational functionals of linear growth. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). to appear (2020)
Bildhauer, M., Fuchs, M.: On a class of variational integrals with linear growth satisfying the condition of \(\mu \)-ellipticity. Rend. Mat. Appl. 22, 249–274 (2002)
Bulíček, M., Gwiazda, P., Málek, J., . Świerczewska Gwiazda, A.: On steady flows of incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var. 2(2), 109–136 (2009)
Bulíček, M., Gwiazda, P., Málek, J., Świerczewska Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012)
Bulíček, M., Maringová, E., Málek, J.: On nonlinear problems of parabolic type with implicit constitutive equations involving flux. Math. Models Methods Appl. Sci. 31(10), 2039–2090 (2021)
Bulíček, M., Málek, J., Süli, E.: Analysis and approximation of a strain-limiting nonlinear elastic model. Math. Mech. Solids 20(1), 92–118 (2015)
Dal Maso, G., Murat, F.: Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. Nonlinear Anal. 31(3–4), 405–412 (1998)
DiBenedetto, E.: Degenerate Parabolic Equations. Springer-Verlag, New York (1993)
Finn, R.: Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature. J. Analyse Math. 14, 139–160 (1965)
Galindo-Rosales, F.J., Rubio-Hernández, F.J., Sevilla, A.: An apparent viscosity function for shear thickening fluids. J. Nonnewton. Fluid Mech. 166(5), 321–325 (2011)
Galindo-Rosales, F.J., Rubio-Hernández, F.J., Sevilla, A., Ewoldt, R.H.: How Dr. Malcom M. Cross may have tackled the development of “an apparent viscosity function for shear thickening fluids’’. J. Nonnewton. Fluid Mech. 166(23), 1421–1424 (2011)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, NN.: Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., (1968)
Lee, Y.S., Wetzel, E.D., Wagner, N.J.: The ballistic impact characteristics of kevlar®woven fabrics impregnated with a colloidal shear thickening fluid. J. Mater. Sci. 38(13), 2825–2833 (2003)
Lindqvist, P.: Notes on the infinity Laplace equation. In: Springer Briefs in Mathematics BCAM Basque Center for Applied Mathematics, Bilbao. Springer, Cham (2016)
Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod; Gauthier-Villars, Paris (1969)
Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996)
Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)
Portilheiro, M., Vázquez, J.L.: Degenerate homogeneous parabolic equations associated with the infinity-Laplacian. Calc. Var. Partial. Differ. Equ. 46(3–4), 705–724 (2013)
Srivastava, A., Majumdar, A., Butola, B.S.: Improving the impact resistance of textile structures by using shear thickening fluids: a review. Crit. Rev. Solid State Mater. Sci. 37(2), 115–129 (2012)
Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (1995)
Wagner, N.J., Brady, J.: Shear thickening in colloidal dispersions. Phys. Today 62(10), 27–32 (2009)
Funding
Open access publishing supported by the National Technical Library in Prague.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by László Székehylidi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
M. Bulíček’s work is supported by the project 20-11027X financed by Czech science foundation (GAČR). J. Málek acknowledges the support of the project No. 18-12719S financed by Czech Science Foundation (GAČR). M. Bulíček and J. Málek are members of the Nečas Center for Mathematical Modeling. The PhD position of D. Hruška is funded by the German Science Foundation DFG in context of the Priority Program SPP 2026 “Geometry at Infinity”.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bulíček, M., Hruška, D. & Málek, J. On evolutionary problems with a-priori bounded gradients. Calc. Var. 62, 188 (2023). https://doi.org/10.1007/s00526-023-02524-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-023-02524-4