Abstract
The existence of global weak solutions to a parabolic energy-transport system in a bounded domain with no-flux boundary conditions is proved. The model can be derived in the diffusion limit from a kinetic equation with a linear collision operator involving a non-isothermal Maxwellian. The evolution of the local temperature is governed by a heat equation with a source term that depends on the energy of the distribution function. The limiting model consists of cross-diffusion equations with an entropy structure. The main difficulty is the nonstandard degeneracy, i.e., ellipticity is lost when the gas density or temperature vanishes. The existence proof is based on a priori estimates coming from the entropy inequality and the \(H^{-1}\) method and on techniques from mathematical fluid dynamics (renormalized formulation, div-curl lemma).
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1 Introduction
This paper is concerned with the global existence analysis of a degenerate diffusion system governing the evolution of the particle density \(\rho (x,t)\) and temperature \(\theta (x,t)\):
where \(E=\theta + \frac{3}{2}\rho \theta \) is the energy density, supplemented by no-flux boundary and initial conditions,
and \(\Omega \subset {{\mathbb {R}}}^3\) is a bounded domain. The equations describe a rarefied gas that exchanges heat with the background, coupled through the energy exchange. They can be formally derived from a collisional kinetic equation, coupled to a heat equation for the background temperature governed by a Fourier law, and they are written in dimensionless form. We refer to Sect. 2 for modeling details.
A major difficulty of system (1) is the derivation of suitable a priori estimates. This issue will be tackled by exploiting the entropy structure of the system. This means that Eq. (1) can be written in the cross-diffusion form
where
The so-called Onsager matrix B is symmetric and positive semidefinite. However, B becomes indefinite when \(\rho =0\) or \(\theta =0\), showing that (4) is of degenerate type. The Gibbs free energy
defines the
-
chemical potential \(\mu =\partial G/\partial \rho =\theta (\log (\rho /\theta ^{3/2})+\frac{5}{2})\),
-
the (mathematical) entropy \(h=\partial G/\partial \theta =\rho \log (\rho /\theta ^{3/2}) -\log \theta \), and
-
the energy density \(E=G-\theta \partial G/\partial \theta =(1+\frac{3}{2}\rho )\theta \).
We reveal the formal gradient-flow structure for (4) by defining the thermo-chemical potential \(\phi =\partial h/\partial \rho =\mu /\theta \) and the negative inverse temperature \(\partial h/\partial E=-1/\theta \) (interpreting h as a function of \((\rho ,E)\)) such that
where \(\mathrm {D}h\) is the vector with components \(\partial h/\partial \rho \) and \(\partial h/\partial E\). Furthermore, the entropy h is a Lyapunov functional along solutions to (4):
since M is positive semidefinite. In particular, we obtain a priori estimates for \(\nabla (\mathrm {D}h)^T {B}\) \(\times \nabla \mathrm {D}h\) in \(L^1(\Omega )\), from which we conclude gradient estimates for \(\sqrt{\rho \theta }\) and \(\log \theta \) in \(L^2(\Omega )\) (see below).
Still, this approach is not sufficient. Indeed, because of the degeneracy at \(\theta =0\), we cannot expect to achieve any control on the gradient of \(\rho \), and moreover, the bounds from the entropy estimate are not sufficient to conclude. Our idea, detailed below, is to apply well-known tools from mathematical fluid dynamics like \(H^{-1}\) estimates and compensated compactness. The originality of this work consists in the combination of these tools and entropy methods, which allows us to treat non-standard degeneracies. We remark that degenerate mobilities were also treated in Cahn–Hilliard equations; see, e.g., [1, 11].
1.1 State of the art
Equation (1) belong to the class of energy-transport models which have been investigated particularly in semiconductor theory [14]. The first energy-transport model for semiconductors was presented by Stratton [17]. First existence results were concerned with models with very particular diffusion coefficients (being not of the form (1)) [2, 3] or with uniformly positive definite diffusion matrices [8]. Existence results for physically more realistic diffusion coefficients were shown in [6], but only for situations close to equilibrium. A degenerate energy-transport system with a simplified temperature equation was analyzed in [15]. Energy-transport models do not only appear in semiconductor theory. For instance, they have been used to model self-gravitating particle clouds [4] and the dynamics in optical lattices [5].
In [19], the global existence of weak solutions to the model
in a bounded domain \(\Omega \) with no-flux boundary conditions was proved. At first glance, Eq. (1) look simpler than (6) because of the additional diffusion in the energy equation. However, the ideas in [19] cannot be easily applied to (1). Indeed, the key idea in [19] was to introduce the variables \(u=\rho \theta \) and \(v=\rho \theta ^2\) and to apply the Stampacchia trunction method to a time-discretized version of
The functionals \(\int \nolimits _\Omega \rho ^2\theta ^b dx\) turn out to be Lyapunov functionals along solutions to (7) for suitable values of \(b\in {{\mathbb {R}}}\), leading to uniform gradient estimates. However, the additional term in the energy equation of (1) complicates the derivation of a priori estimates. Thus, the proof in [19] seems to be rather specific to system (6) and is not generalizable. Our idea is to treat (1) by combining entropy methods and tools from mathematical fluid dynamics, which may be also applied to other cross-diffusion systems.
1.2 Mathematical key ideas
As explained before, the first key idea is to exploit, in contrast to [19], the entropy structure of (1). Indeed, recalling the mathematical entropy density
a formal computation (which is made rigorous for an approximate scheme; see (23)) gives the entropy dissipation equation
which provides \(H^1(\Omega )\) estimates for \(\sqrt{\rho \theta }\) and \(\log \theta \). Moreover, this estimate implies that \(\theta >0\) a.e. (but not \(\rho >0\)).
Clearly, the entropy estimates are not sufficient to pass to the de-regularization limit in the approximate scheme. Further bounds are derived from the \(H^{-1}(\Omega )\) method, i.e., we use basically \((-\Delta )^{-1}\rho \) and \((-\Delta )^{-1}E\), respectively, as test functions in the weak formulation of (1) (second key idea). This method gives estimates for
Combining these bounds with those coming from the entropy inequality and the conservation laws leads to estimates for \(\nabla (\rho \theta )=\sqrt{\rho \theta }\nabla \sqrt{\rho \theta }\), \(\nabla \theta =\theta \nabla \log \theta \) and consequently for E in \(W^{1,1}(\Omega )\). Moreover, \(\partial _t E\) is bounded in some dual Sobolev space. This allows us to apply the Aubin–Lions lemma to E. Unfortunately, we do not obtain gradient estimates for \(\rho \).
To overcome this issue, we use tools from mathematical fluid dynamics (third key idea). Let \((\rho _\delta ,\theta _\delta )\) be approximate solutions to (1) (in a sense made precise in Sect. 3). First, we write the mass balance equation in the renormalized form
in the sense of distributions for smooth functions f with bounded derivatives. Let g another smooth function with bounded derivatives and introduce the vectors
We deduce from the properties of f and g and the a priori estimates that \({\text {div}}_{(t,x)}U_\delta \) and \({\text {curl}}_{(t,x)}V_\delta \) are uniformly bounded in \(L^1(\Omega \times (0,T))\) and hence relatively compact in \(W^{-1,r}(\Omega )\) for some \(r>1\). The div-curl lemma implies that \(\overline{U_\delta \cdot V_\delta }=\overline{U_\delta }\cdot \overline{V_\delta }\) a.e., where the bar denotes the weak limit of the corresponding sequence. Thus, \(\overline{f(\rho _\delta )g(\theta _\delta )}=\overline{f(\rho _\delta )}\; \overline{g(\theta _\delta )}\) a.e. A truncation procedure yields that \(\overline{\rho _\delta \theta _\delta }=\rho \theta \), where \(\rho \) and \(\theta \) are the weak limits of \((\rho _\delta )\) and \((\theta _\delta )\), respectively. As \((E_\delta )\) converges strongly, by the Aubin–Lions lemma, we are able to prove that \(\theta _\delta \rightarrow \theta \) and eventually \(\rho _\delta \rightarrow \rho \) a.e. These limits allow us to identify the weak limits and to pass to the limit \(\delta \rightarrow 0\) in the approximate equations. The approximate scheme contains additional terms which need to be treated carefully such that our arguments are more technical than presented here. In fact, we need three approximation levels; see Sect. 3 for details.
1.3 Main result
Our main result is as follows:
Theorem 1
(Existence of weak solutions). Let \(\Omega \subset {{\mathbb {R}}}^3\) be a bounded domain with \(\partial \Omega \in C^{1,1}\). Let \(\rho ^0\), \(\theta ^0\in L^1(\Omega )\) satisfy \(\rho ^0\ge 0\), \(\theta ^0\ge 0\) in \(\Omega \) and \(\rho ^0\theta ^0\), \(h(\rho ^0,\theta ^0)\in L^1(\Omega )\), where h is defined in (8). Let \(T>0\) and \(\Omega _T=\Omega \times (0,T)\). Then there exist \(\rho \), \(\theta \in L^\infty (0,T;L^1(\Omega ))\) such that
it holds that \(\rho \ge 0\) and \(\theta >0\) a.e. in \(\Omega _T\); \((\rho ,\theta )\) is a weak solution to (1)–(3) in the sense
for any test functions \(\psi _1\in L^4(0,T; W^{1,4}(\Omega ))\), \(\psi _2\in L^{6}(0,T; W^{2,4}(\Omega ))\); and the initial data (3) is satisfied in the sense of \(W^{1,4}(\Omega )'\) and \(W^{2,4}(\Omega )'\), respectively. Moreover, the total mass and energy are preserved:
In the theorem, we denote by \(X'\) the dual space of the Banach space X.
The paper is organized as follows. Equation (1) are formally derived from a relaxation-time kinetic model in Sect. 2, while the proof of Theorem 1 is presented in Sect. 3.
2 Formal derivation from a kinetic model
We consider a gas which is rarefied enough such that collisions between gas particles can be neglected, but there are thermalizing collisions at a fixed rate with a nonmoving background. This is modeled by sampling post-collisional velocities from a Maxwellian distribution with zero mean velocity and with the background temperature, which is determined from the assumptions of energy conservation as well as heat transport in the background governed by the Fourier law. These assumptions lead to the equations
which are written in dimensionless form with a diffusive macroscopic scaling with the scaled Knudsen number \(0<\varepsilon \ll 1\). The gas is described by the distribution function \(f_\varepsilon (x,v,t)\) with the velocity \(v\in {{\mathbb {R}}}^3\), and the temperature of the background is \(\theta _\varepsilon (x,t)\). The gradient and Laplace operators are meant with respect to the position variable x, and the Maxwellian is given by
Finally, the position density of the gas is defined by
The right-hand side of the heat equation (12) has been chosen such that the sum of the kinetic energy of the gas and the thermal energy of the background is conserved. In [12], the energy-transport system (1) has been derived formally from (11)–(12) in the macroscopic limit \(\varepsilon \rightarrow 0\). We repeat the argument here for completeness.
In the computations, the moments of the Maxwellian up to order 4 will be needed:
where \(v_i\), \(v_j\) denote the components of v (\(i,j=1,2,3\)). From (11)–(12), the local conservation laws for mass and energy,
can be derived by integration of (11) with respect to v and, respectively, by integration of (11) against \(|v|^2/2\) and adding to (12).
In a formal convergence analysis, we assume \(f_\varepsilon \rightarrow f\), \(\rho _\varepsilon \rightarrow \rho \), and \(\theta _\varepsilon \rightarrow \theta \) as \(\varepsilon \rightarrow 0\) and deduce from (11) that \(f=\rho M(\theta )\). With (14), we obtain for the kinetic energy density
The limit of the mass flux is obtained by multiplication of (11) by \(v/\varepsilon \), integration with respect to v, and passing to the limit, using again (14):
Analogously, we compute the flux of the kinetic energy,
for \(i=1,2,3\). Using these results in the limits of the conservation laws leads to (1).
3 Proof of Theorem 1
We approximate Eq. (1) in the following way. The time derivative is replaced by the implicit Euler discretization with parameter \(\tau >0\). This is needed to avoid issues related to the time regularity. A higher-order \(H^4\) regularization for \(\phi =\partial h/\partial \rho \) in the mass balance equation with parameter \(\varepsilon >0\) gives \(H^2(\Omega )\) regularity and compactness in \(W^{1,4}(\Omega )\). Furthermore, \(H^2(\Omega )\) and \(W^{1,4}(\Omega )\) regularizations for \(\log \theta \) with the same parameter are added to the energy balance equation. The \(W^{1,4}(\Omega )\) regularization is needed to derive estimates when using both \(\log \theta \) and \(-1/\theta \) as test functions in (1). Furthermore, we add an additional \(H^2(\Omega )\) regularization for \(\phi \) in the mass balance equation with parameter \(\delta >0\), which removes the degeneracy of the diffusion matrix M in (4). Finally, we add the artificial heat flux \(\Delta \theta ^{3}\) in the energy density equation with the same parameter \(\delta \) to obtain gradient estimates for the temperature, and we add the term \(\theta ^{-N}\log \theta \) for some \(N>0\) to achieve an estimate for \(\theta ^{-(N+1)}\).
After having proved the existence of solutions to the approximate problem and some a priori estimates coming from the entropy inequality, we perform the limits \(\varepsilon \rightarrow 0\), \(\tau \rightarrow 0\), and \(\delta \rightarrow 0\) (in this order).
3.1 Solution of the approximate problem
We wish to solve a system which approximates (1) and is formulated in the variables \(\phi \) and \(w=\log \theta \), similarly as in (4). We interpret \(\rho \) and \(E=\theta (1+\frac{3}{2}\rho )\) as functions of \((\phi ,w)\), i.e.
In this notation, the diffusion coefficients become
Let \(T>0\) and let the approximation parameters \(\tau >0\) (such that \(T/\tau \in {{\mathbb {N}}}\)), \(\varepsilon >0\), and \(\delta >0\) be given. Furthermore, let \(0<N<5\) be a number needed for the approximation \(\theta ^{-N}\log \theta \) in the energy balance equation.
We wish to find \((\phi ^k,w^k)\in H^2(\Omega ;{{\mathbb {R}}}^2)\) such that, with \(\rho ^k=\rho (\phi ^k,w^k)\), \(E^k=E(\rho ^k,w^k)\),
for all \((\psi _1,\psi _2)\in H^2(\Omega ;{{\mathbb {R}}}^2)\), and \(M_{ij}^k\) are given by (15) with \((\rho ,w)\) replaced by \((\rho ^k,w^k)\). The existence of solutions to (16)–(17) is shown in two steps.
Step 1: solution of the linearized approximated problem. In the following, we drop the superindex k. Let \(({{\widetilde{\phi }}},{\widetilde{w}})\in W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\) be given and set \({{\widetilde{\rho }}}=\rho ({{\widetilde{\phi }}},{\widetilde{w}})\), \({\widetilde{E}}=E({\widetilde{\phi }},{\widetilde{w}})\). We wish to find \((\phi ,w)\in H^2(\Omega ;{{\mathbb {R}}}^2)\) such that
for all \((\psi _1,\psi _2)\in H^2(\Omega ;{{\mathbb {R}}}^2)\), where \(\sigma \in [0,1]\) and
where \(\widetilde{M}_{ij}\) is given by (15) with \((\rho ,w)\) replaced by \(({\widetilde{\rho }},{\widetilde{w}})\). The bilinear forms \(a_1\) and \(a_2\) are coercive on \(H^2(\Omega )\) since, by the generalized Poincaré inequality [18, Chap. 2, Sect. 1.4],
for some constant \(C>0\). The linear forms \(F_1\) and \(F_2\) are continuous on \(H^2(\Omega )\) since, by the continuous embedding \(W^{1,4}(\Omega )\hookrightarrow L^\infty (\Omega )\), \({\widetilde{\phi }}\) and \({\widetilde{w}}\) are \(L^\infty (\Omega )\) functions such that \({\widetilde{\rho }}\), \({\widetilde{E}}\in L^\infty (\Omega )\) too. The Lax–Milgram lemma implies the existence of a unique solution \((\phi ,w)\) to (18) such that \(\rho =\rho (\phi ,w)>0\) and \(E=E(\phi ,w)>0\). This defines the fixed-point operator \(S:W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\times [0,1]\rightarrow W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\), \(S({\widetilde{\phi }},{\widetilde{w}},\sigma )=(\phi ,w)\), where \((\phi ,w)\) solves (18).
Step 2: solution of the approximate problem. We wish to apply the Leray–Schauder fixed-point theorem. It holds that \(S({\widetilde{\phi }},{\widetilde{w}},0)=0\). Standard arguments show that \(S:W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\rightarrow H^2(\Omega ;{{\mathbb {R}}}^2)\) is continuous. Since \(H^2(\Omega ;{{\mathbb {R}}}^2)\) is compactly embedded into \(W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\), \(S:W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\rightarrow W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\) is compact. It remains to show that there exists a uniform bound in \(W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\) for all fixed points.
Let \(\sigma \in (0,1]\) and let \((\phi ,w)\) be a fixed point of \(S(\cdot ,\cdot ,\sigma )\). It is a solution to (16)–(17) with \(\phi =\phi ^k\), \(w=w^k\), \(\rho =\rho ^k\), and \(E=E^k\). We use the test functions \(\psi _1=\phi \) and \(\psi _2=1-e^{-w}\) in (16) and (17), respectively, and add both equations. (We use \(1-e^{-w}\) instead of \(-e^{-w}\) as a test function in order to be able to treat the term \(\varepsilon \int \nolimits _\Omega (1+e^w)w\psi _2 dx\) and to obtain the entropy and energy balance in one single equation.) Then
To estimate the first integral \(I_1\), we use the entropy density (8), formulated in terms of the variables \((\rho ,E)\),
The function \({\widetilde{h}}\) in the variables \((\rho ,E)\) is convex, since the determinant of its Hessian,
equals \((1+\frac{3}{2}\rho )/(\rho E^{2})\), which is positive. This implies that
for any \((\rho _1,E_2)\), \((\rho _2,E_2)>0\), and consequently,
The second integral \(I_2\) is nonnegative since
The integrals \(I_3\), \(I_7\), and \(I_8\) are estimated according to
Therefore, we obtain from (19)
where \(C>0\) is here and in the following a generic constant independent of \(\tau \), \(\varepsilon \), and \(\delta \). This gives a uniform \(H^2(\Omega )\) estimate for \(\phi \) and w, independent of \(\sigma \) (but depending on \(\varepsilon \) and \(\delta \)), and hence the desired uniform estimate for \((\phi ,w)\) in \(W^{1,4}(\Omega ;{{\mathbb {R}}}^2)\). By the Leray–Schauder fixed-point theorem, there exists a solution \((\phi ^k,w^k):=(\phi ,w)\in H^2(\Omega ;{{\mathbb {R}}}^2)\) to (16)–(17) with \(\sigma =1\), \(\rho ^k=\rho (\phi ^k,w^k)\), and \(E^k=E(\phi ^k,w^k)\). Moreover, this solution satisfies (20) with \(\sigma =1\).
We reformulate Eqs. (16)–(17) by inserting definition (15) of the diffusion coefficients and computing (we drop the superindex k)
Therefore, \((\phi ^k,\rho ^k,\theta ^k,w^k)\) solves
for test functions \(\psi _1\), \(\psi _2\in H^2(\Omega )\).
3.2 Uniform estimates
Set \(\theta ^{k-1}=\exp (w^{k-1})\) and \(\theta ^k=\exp (w^k)\). In the following, we drop again the superindex k to simplify the notation. We reformulate inequality (20) to obtain gradient estimates for expressions depending on \(\rho \) and \(\theta \). We estimate the second integral in (20):
We infer from (20) with \(\sigma =1\) the reformulated discrete entropy inequality
There exists \(c\in (0,1)\) such that \(x-\log x\ge c(x+|\log x|)\) for all \(x>0\). Therefore,
This provides the following uniform estimates independent of \((\delta ,\varepsilon ,\tau )\):
3.3 Limit \(\varepsilon \rightarrow 0\)
Let \(\phi _\varepsilon =\phi ^k\), \(w_\varepsilon =w^k\) be a solution to (16)–(17). We set \(\rho _\varepsilon =\rho (\phi _\varepsilon ,w_\varepsilon )\), \(E_\varepsilon =E(\rho _\varepsilon ,w_\varepsilon )\), \(\theta _\varepsilon =\exp (w_\varepsilon )\), and \(\phi _\varepsilon =\log (\rho _\varepsilon /\theta _\varepsilon ^{3/2})+5/2\). We deduce from (23) and (24) the following bounds which are independent of \(\varepsilon \) and \(\delta \) (but not of \(\tau \)):
These bounds allow us to derive further estimates. By the Poincaré inequality, we have
This gives \(\varepsilon \)-uniform bounds for \(\theta _\varepsilon \) and \(\log \theta _\varepsilon \) in \(H^1(\Omega )\):
The \(L^1(\Omega )\) bound for \(\rho _\varepsilon \theta _\varepsilon \) and the \(L^2(\Omega )\) bound for \(\nabla \sqrt{\rho _\varepsilon \theta _\varepsilon }\) imply that
These estimates provide a uniform bound for the energy. Indeed, we deduce from the Sobolev embedding \(H^1(\Omega )\hookrightarrow L^6(\Omega )\) that \(\nabla (\rho _\varepsilon \theta _\varepsilon )=2\sqrt{\rho _\varepsilon \theta _\varepsilon }\nabla \sqrt{\rho _\varepsilon \theta _\varepsilon }\) is uniformly bounded in \(L^{3/2}(\Omega )\). This shows that \((E_\varepsilon )\) is bounded in \(W^{1,3/2}(\Omega )\).
We know that \((\log \theta _\varepsilon )\) and \((\phi _\varepsilon )\) are bounded in \(H^1(\Omega )\). Consequently, \(\log \rho _\varepsilon =\phi _\varepsilon + \frac{3}{2}\log \theta _\varepsilon -\frac{5}{2}\) is bounded in \(H^1(\Omega )\) too, i.e.
The previous uniform bounds are sufficient to perform the limit \(\varepsilon \rightarrow 0\). There exist subsequences which are not relabeled such that, as \(\varepsilon \rightarrow 0\),
where \(1<p<6\) and Y, Z are functions in \(H^1(\Omega )\). Up to a subsequence, we have \(\log \rho _\varepsilon \rightarrow Y\) and \(\log \theta _\varepsilon \rightarrow Z\) a.e. in \(\Omega \). Thus, \(\rho _\varepsilon \rightarrow e^Y=:\rho \) and \(\theta _\varepsilon \rightarrow e^Z=:\theta \) a.e. in \(\Omega \). In particular, \(\rho >0\) and \(\theta >0\) a.e. in \(\Omega \). It follows from
for any \(R>1\) that \((\rho _\varepsilon )\) is equi-integrable. Vitali’s convergence theorem implies that \(\rho _\varepsilon \rightarrow \rho \) strongly in \(L^1(\Omega )\). Furthermore, possibly for a subsequence, \(\sqrt{\rho _\varepsilon \theta _\varepsilon }\rightarrow \sqrt{\rho \theta }\) a.e. in \(\Omega \). The \(H^1(\Omega )\) bound for \((\sqrt{\rho _\varepsilon \theta _\varepsilon })\) then yields
where \(1<p<6\). Furthermore, we have
We deduce from the strong convergence of \((\phi _\varepsilon )\), \((\rho _\varepsilon )\), and \((\theta _\varepsilon )\) as well as from the a.e. positivity of \(\rho \) and \(\theta \) that \(\phi =\log \rho -\frac{3}{2}\log \theta +\frac{5}{2}\) a.e. in \(\Omega \).
The uniform bounds for \(w_\varepsilon \) are sufficient to pass to the limit \(\varepsilon \rightarrow 0\) in the \(\varepsilon \)-terms,
as well as in the \(\delta \)-terms. The most difficult term is \(\delta \int \nolimits _\Omega e^{-N w_\varepsilon }w_\varepsilon \psi _2 dx\). It follows from \(\sqrt{\theta }|\log \theta |^{(2N+1)/(2N)}\le C\) for \(\theta \le 1\) and \(\theta ^{-(N+1)}\sqrt{\theta }|\log \theta |^{(2N+1)/(2N)}\le C\) for \(\theta >1\) as well as from (24) that
Since \(\delta e^{-Nw_\varepsilon }w_\varepsilon \rightarrow \delta \theta ^{-N}\log \theta \) a.e. in \(\Omega \), we conclude that this limit also holds strongly in \(L^1(\Omega )\). Therefore, we can perform the limit \(\varepsilon \rightarrow 0\) in (21)–(22) (now writing the superindex k) leading to
for any test functions \(\psi _1\in W^{1,3}(\Omega )\), \(\psi _2\in W^{1,6}(\Omega )\).
3.4 Limit \(\tau \rightarrow 0\)
We introduce the piecewise constant functions in time \(\rho _\tau (x,t) =\rho ^k(x)\), \(\theta _\tau (x,t)=\theta ^k(x)\), \(\phi _\tau (x,t)=\phi ^k(x)\), and \(E_\tau (x,t)=E^k(x)\) for \(x\in \Omega \), \(t\in ((k-1)\tau ,k\tau ]\). Furthermore, let \((\pi _\tau u)(x,t) =u^{k-1}(x)\) for \(x\in \Omega \), \(t\in ((k-1)\tau ,k\tau ]\) be the shift operator for piecewise constant functions u. We reformulate (26)–(27):
for piecewise constant test functions in time \(\psi _1\), \(\psi _2\in L^2(0,T;W^{1,6}(\Omega ))\). By density [16, Prop. 1.36], these formulations hold for all test functions in \(L^2(0,T;W^{1,6}(\Omega ))\). We collect the uniform estimates from the discrete entropy inequality (23):
where the constant \(C>0\) does not depend on \(\tau \) or \(\delta \). In the following, we show some additional estimates for \((\rho _\tau ,\theta _\tau )\).
Lemma 2
(Mass and energy control). It holds for any \(t\in (0,T)\) that
Proof
Using \(\psi _1=1\) in (26) and summing from \(k=1,\ldots ,n\) gives
where \(n\le N\). We infer from bound (33) for \((\phi _\tau )\) that
The second statement follows after choosing \(\psi _2=1\) in (27) and using (25). \(\square \)
Lemma 3
(Higher integrability). It holds that
where \((\alpha ,\beta )\in \{(1,2),(1,3),(\frac{3}{2},3),(2,1),(2,2),(2,3)\}\).
Proof
The proof is based on the \(H^{-1}(\Omega )\) method, i.e., we use test functions of the type \((-\Delta )^{-1}\rho _\tau \) and \((-\Delta )^{-1}E_\tau \). More precisely, let \(\Psi _1\), \(\Psi _2\in L^\infty (0,T;H^1(\Omega ))\) be the unique solutions to, respectively,
where \(\fint udx = {\text {meas}}(\Omega )^{-1}\int \nolimits _\Omega udx\).
Step 1: uniform bounds for \(\Psi _2\). We use the test function \(\Psi _2\) in the weak formulation of the second equation in (34) and take into account the energy control. Then
It follows from Sobolev’s embedding and the Poincaré–Wirtinger inequality that
and so
We proceed by bootstrapping this result. Elliptic regularity for
gives (here, we need the boundary regularity \(\partial \Omega \in C^{1,1}\))
Since \((E_\tau )\) is bounded in \(L^{\infty }(0,T;L^1(\Omega ))\), an interpolation shows that
We deduce from the embedding \(L^6(0,T;W^{2,6/5}(\Omega ))\hookrightarrow L^6(\Omega _T)\) that
Step 2: Test functions \(\Psi _1\) and \(\Psi _2\). We choose \(\Psi _1\) and \(\Psi _2\) as test functions in (28) and (29), respectively:
We estimate the first integral in (36). Since \(\Psi _1\) has zero spatial average and \(\nabla \Psi _1\cdot \nu =0\) on \(\partial \Omega \), it follows from (34) that
The function \(\Psi _1\) is piecewise constant in time. We write \(\Psi _1(x,t)=\Psi _1^k(x)\) for \(x\in \Omega \), \(t\in ((k-1)\tau ,k\tau ]\). Then, using Young’s inequality,
We conclude that
In a similar way, we have
Inserting these inequalities into (36) and (37), respectively, and adding both inequalities, we find that
We start with the last integral. It follows from (35) that
The first norm is estimated according to
where the last inequality follows from the condition \(N<5\) (and hence \(6N/5<N+1\)). Because of (33), this leads to
Therefore, we infer that
Since \(E_\tau =\theta _\tau +\frac{3}{2}\rho _\tau \theta _\tau \), the right-hand side can be controlled (for sufficiently small \(\delta >0\)) by the last two integrals on the left-hand side of (38).
Next, we consider the following term appearing in \(J_3\):
where the last inequality follows from the fact that \(z\mapsto z\log z\) is bounded from below. Furthermore, we deduce from Lemma 2, bound (33) for \(\phi _\tau \), and the Poincaré–Wirtinger inequality that
This shows that
The first integral on the right-hand side can be controlled by the last integral on the left-hand side of (38). The last integral on the right-hand side is controlled after applying Gronwall’s inequality. The integrals \(J_4\), \(J_5\), and \(J_6\) can be controlled by the expressions on the left-hand side of (38). We conclude that
We deduce from this estimate and Young’s inequality that
This proves the lemma. \(\square \)
Step 3: Strong convergence of \((\rho _\tau )\) and \((\theta _\tau )\). First, we prove a gradient bound for the particle density.
Lemma 4
(Gradient estimate). There exist \(N\in (0,5)\), \(m\in (\frac{1}{2},1)\), and \(\alpha \in (\frac{2}{3},1)\) such that
where \(C(\delta )>0\) does not depend on \(\tau \), \(p\ge 1/m\), and \(3q/(3-q)>1/m\) (or equivalently, \(q>3/(3m+1)\)). Moreover, with a constant \(C>0\) independent of \(\tau \) and \(\delta \),
The condition \(q>3/(3m+1)\) guarantees that \(W^{1,q}(\Omega )\hookrightarrow L^{1/m}(\Omega )\). This is needed below for the application of the nonlinear Aubin–Lions lemma.
Proof
It follows from Lemma 3 that \((\rho _\tau \theta _\tau ^{1/2})\) is bounded in \(L^2(\Omega _T)\), while estimate (33) implies that \((\theta _\tau ^{-1/2})\) is bounded in \(L^{2(N+1)}(\Omega _T)\). Consequently, \(\rho _\tau = \rho _\tau \theta ^{1/2}\theta _\tau ^{-1/2}\) is uniformly bounded in \(L^{r}(\Omega _T)\), where \(r:=2(N+1)/(N+2)>1\). Together with the \(L^\infty (0,T;L^1(\Omega ))\) bound for \((\rho _\tau )\), an interpolation with \(1/c=(1-\alpha )/1 + \alpha /r\) and \(b\ge 1\) gives
A simple computation shows that \(c=r/(\alpha +(1-\alpha )r)\). We choose \(b=r/\alpha \) and use the \(L^r(\Omega _T)\) bound for \((\rho _\tau )\):
Let \(\frac{1}{2}<m<1\). Then
We know from (32) and (33) that \(\nabla \log \rho _\tau = \nabla \phi _\tau + \frac{3}{2}\nabla \log \theta _\tau \) is uniformly bounded in \(L^2(\Omega _T)\) (but not uniformly in \(\delta \)). It follows that \(\nabla \rho _\tau ^m = m\rho _\tau ^m\nabla \log \rho _\tau \) is uniformly bounded in \(L^p(0,T;L^q(\Omega ))\), where \(p,q\ge 1\) satisfy
We deduce from the Poincaré–Wirtinger inequality and the \(L^\infty (0,T;L^1(\Omega ))\) bound for \((\rho _\tau )\) that
We claim that there exist \(N\in (0,5)\), \(m\in (\frac{1}{2},1)\), and \(\alpha \in (0,1)\) such that
where p and q are given by (40). A straightforward computation shows that these inequalities are equivalent to
We choose \(r=2\alpha m/(2m-1)\) (recall that \(m>1/2\)) such that the first inequality is satisfied. With this choice, the second inequality is equivalent to \(m<1/(3(1-\alpha ))\). Since we want \(m<1\), we need to choose \(\alpha >2/3\). Then \(\frac{1}{2}<m<1<1/(3(1-\alpha ))\). By definition of r,
Thus, it remains to prove that \(N\in (0,5)\) can be chosen such that this identity holds for some \(\alpha >\frac{2}{3}\) and \(m\in (\frac{1}{2},1)\). Equation (41) is equivalent to
and the requirement \(N<5\) gives \(m>6/(12-7\alpha )\). The right-hand side is smaller than one if \(\alpha <\frac{6}{7}\). This is compatible with the previous constraint \(\alpha >\frac{2}{3}\) and proves the claim.
To finish the proof of the lemma, we observe that (31) and Lemma 3 imply that
Moreover, we deduce from (32) and Lemma 3 that
Thus, \((E_\tau )\) is bounded in \(L^1(0,T;W^{1,1}(\Omega ))\), and the proof is finished. \(\square \)
Lemma 5
(Bounds for the discrete time derivative). There exists a constant \(C>0\) which does not depend on \(\tau \) such that
Proof
We infer from (42) and (33) that
Furthermore,
Taking into account Lemma 3, the first three terms on the right-hand side are uniformly bounded. Since \(N<5\), the last term can be estimated from above by \(\delta \Vert \theta _\tau ^{-(N+1)}\Vert _{L^1(\Omega _T)}^{5/6}\) which is bounded because of (33). This finishes the proof. \(\square \)
Lemmas 4 and 5 allow us to apply the Aubin–Lions lemma in the version of [7, Theorem 3]. This is possible since \(p\ge 1/m\) and \(W^{1,q}(\Omega )\hookrightarrow L^{1/m}(\Omega )\) (the last fact is a consequence of \(q>3/(3m+1)\)). We infer the existence of a subsequence which is not relabeled such that, as \(\tau \rightarrow 0\),
Concerning \((E_\tau )\), Lemmas 4 and 5 allow us to apply the Aubin–Lions lemma in the version of [9] (or Theorem 3 in [7] with \(m=1\)) to obtain a subsequence of \((E_\tau )\) (not relabeled) such that, as \(\tau \rightarrow 0\),
In fact, because of the \(L^2(\Omega _T)\) bound for \((E_\tau )\) from Lemma 3, this convergence holds in \(L^\eta (\Omega _T)\) for any \(\eta <2\). Up to subsequences, we know that \(\rho _\tau \rightarrow \rho \) and \(E_\tau \rightarrow E\) a.e. in \(\Omega _T\). Thus,
In particular, \(E=\theta +\frac{3}{2}\rho \theta \). The bound for \((\theta _\tau )\) in \(L^4(\Omega _T)\) (not uniform in \(\delta \)) shows that the previous convergence holds in \(L^\eta (\Omega _T)\) for any \(\eta <4\). We deduce from the \(L^2(\Omega _T)\) bounds for \(\log \theta _\tau \) and \(\log \rho _\tau =\phi _\tau +\frac{3}{2}\log \theta _\tau -\frac{5}{2}\) that \(\log \theta \) and \(\log \rho \) are integrable and thus, \(\rho >0\), \(\theta >0\) a.e. in \(\Omega _T\). Furthermore, \(\phi _\tau \rightarrow \log \rho -\frac{3}{2}\log \theta +\frac{5}{2}=:\phi \) a.e. in \(\Omega _T\) and, because of (33), weakly in \(L^2(0,T;H^1(\Omega ))\).
The previous bounds and the strong convergences of \((\rho _\tau )\) and \((\theta _\tau )\) allow us to pass to the limit \(\tau \rightarrow 0\) in (28)–(29). For this, we observe that, by (42),
Furthermore, by Lemma 3,
The strong convergence of \((\theta _\tau )\) to \(\theta \), the uniform bounds on \((\theta _\tau )\), and the a.e. positivity of \(\theta \) imply that
Finally, by Lemma 5,
Then (28)–(29) become in the limit \(\tau \rightarrow 0\),
for any test functions \(\psi _1\), \(\psi _2\in C_0^2(\Omega _T)\).
3.5 Limit \(\delta \rightarrow 0\)
In this subsection, we need some tools from mathematical fluid dynamics, in particular the concept of renormalized solutions and the div-curl lemma. In the following, we denote by \(\overline{u_\delta }\) the weak or distributional limit of a sequence \((u_\delta )\) whenever it exists. Let \((\rho _\delta , E_\delta )\) be a weak solution to (43)–(44) and set \(\phi _\delta =\log (\rho _\delta /\theta _\delta ^{3/2})+\frac{5}{2}\), \(E_\delta =\theta _\delta +\frac{3}{2}\rho _\delta \theta _\delta \).
Step 1: Renormalized mass balance equation. We compute the renormalized form of (43). Let \(f\in C^2([0,\infty )) \cap L^\infty (0,\infty )\) satisfy \(|f'(s)|\le C(1+s)^{-1}\) and \(|f''(s)|\le C(1+s)^{-2}\) for \(s\ge 0\). Furthermore, let \(\xi \in C_0^\infty (\Omega _T)\). Choosing \(\psi _1=f'(\rho _\delta )\xi \) in (43), we find that
This computation can be made rigorous (such that \(\psi _1\) is an admissible test function) by using renormalization techniques; see, e.g., [13, Section 10.18]. The previous equation can be rewritten as
Step 2: Application of the div-curl lemma. We apply the div-curl lemma to the vector fields
where f is as before and \(g\in C^1([0,\infty ))\cap L^\infty (0,\infty )\) satisfies \(|g'(s)|\le C(1+s)^{-1}\) for \(s>0\). We know from (31) that \((\nabla \sqrt{\rho _\delta \theta _\delta })\) and \((\sqrt{\delta }\nabla \phi _\delta )\) are bounded in \(L^2(\Omega _T)\) and from Lemma 3 that \((\sqrt{\rho _\delta \theta _\delta })\) is bounded in \(L^4(\Omega _T)\). Consequently,
is uniformly bounded in \(L^{4/3}(\Omega _T)\). Thus, \((U_\delta )\) is bounded in \(L^{4/3}(\Omega _T)\). Because of the properties of g, \((V_\delta )\) is trivially bounded in \(L^\infty (\Omega _T)\).
The left-hand side of (45) equals \(-{\text {div}}_{(t,x)}U_\delta \). We wish to bound the right-hand side of (45). For this, we observe that, thanks to (33), the first term \(\delta f'(\rho _\delta )\phi _\delta \) is uniformly bounded in \(L^2(\Omega _T)\). We rewrite the second term as
Since \(\rho _\delta |f''(\rho _\delta )|\le C\rho _\delta /(1+\rho _\delta )^2\le C\) and \((\sqrt{\theta _\delta }\nabla \sqrt{\rho _\delta })\), \((\sqrt{\rho _\delta \theta _\delta })\) are bounded in \(L^2(\Omega _T)\) by (31), expression (46) is bounded in \(L^1(\Omega _T)\). In order to bound the last term in (45), we observe that, by (32) and (33),
is uniformly bounded in \(L^2(\Omega _T)\). Then
is uniformly bounded in \(L^1(\Omega _T)\). We infer that the right-hand side of (45) and consequently also \(-{\text {div}}_{(t,x)}U_\delta \) are uniformly bounded in \(L^1(\Omega _T)\). By Sobolev’s embedding, it follows that \({\text {div}}_{(t,x)}U_\delta \) is relatively compact in \(W^{-1,r}(\Omega _T)\) for some \(r>1\).
It follows from the uniform \(L^2(\Omega _T)\) bound for \((\nabla \log \theta _\delta )\) (see (32)) and \(\theta _\delta |g'(\theta _\delta )|\le C\theta _\delta /(1+\theta _\delta )\le C\) that
is uniformly bounded in \(L^2(\Omega _T)\). By Sobolev’s embedding, this expression is relatively compact in \(W^{-1,r}(\Omega _T;{{\mathbb {R}}}^{3\times 3})\) for some \(r>1\).
The div-curl lemma [13, Theorem 10.21] implies that \(\overline{U_\delta \cdot V_\delta }=\overline{U_\delta }\cdot \overline{V_\delta }\) a.e. in \(\Omega _T\), which means that
for all \(f\in C^2([0,\infty ))\cap L^\infty (0,\infty )\) and \(g\in C^1([0,\infty )) \cap L^\infty (0,\infty )\) satisfying \(|f'(s)|\le C(1+s)^{-1}\), \(|f''(s)|\le C(1+s)^{-2}\), and \(|g'(s)|\le C(1+s)^{-1}\) for \(s>0\).
Step 3: Proof of \(\overline{\rho _\delta \theta _\delta }=\rho \theta \). We wish to relax the assumptions on the functions f and g. To this end, we introduce the truncation function \(T_1\in C^2([0,\infty ))\) by \(T_1(s)=s\) for \(0\le s<1\), \(T_1(s)=2\) for \(s>3\), and \(T_1\) is nondecreasing and concave in \([0,\infty )\). Then we define \(T_k(s)=k T_1(s/k)\) for \(s>0\) and \(k\in {{\mathbb {N}}}\). It is possible to choose \(f=T_k\) in (47). Together with Fatou’s lemma and the boundedness of g, we infer that
Furthermore, we deduce from (30) that
such that we obtain for any \(k\ge 2\),
Then the limit \(k\rightarrow \infty \) implies that
for any \(g\in C^1([0,\infty ))\cap L^\infty (0,\infty )\) satisfying \(|g'(s)|\le C(1+s)^{-1}\) for \(s>0\). We choose \(g=T_k\) which leads to
We claim that both terms on the right-hand side converge to zero as \(k\rightarrow \infty \). Indeed, it follows from Fatou’s lemma and the \(L^1(\Omega _T)\) bound for \((\rho _\delta \theta _\delta ^2)\) from Lemma 3 that
while we deduce from Fatou’s lemma and the \(L^2(\Omega _T)\) bound for \((\theta _\delta )\), again from Lemma 3, that
We infer from (49) that for any \(k\ge 1\),
which implies, in the limit \(k\rightarrow \infty \), that
Step 4: Pointwise convergence of \((\theta _\delta )\). We prove via the Aubin–Lions lemma that \(E_\delta =\theta _\delta +\frac{3}{2}\rho _\delta \theta _\delta \) is strongly convergent. We know from Lemma 4 that \((E_\delta )\) is bounded in \(L^1(0,T;W^{1,1}(\Omega ))\). For the time derivative of \(E_\delta \), we estimate (44) for \(\psi _2\in C_0^\infty (\Omega _T)\):
Taking into account estimate (39) and again using Lemma 3, we infer that
We apply the Aubin–Lions lemma to \((E_\delta )\) to obtain the existence of a subsequence which is not relabeled such that, as \(\delta \rightarrow 0\), \((E_\delta )\) converges strongly in \(L^\eta (\Omega _T)\) for \(\eta <2\). Since \((1+\theta _\delta )^{-1}\) converges weakly in \(L^\eta (\Omega _T)\) for any \(\eta <\infty \), we find that
We choose \(g(s)=s(1+s)^{-1}\) in (48) and recall (50):
Using these expressions, we deduce from (51) that
This means that
We apply [13, Theorem 10.19] to the strictly decreasing function \(s\mapsto (1+s)^{-1}\) for \(s\ge 0\) to conclude that
The strict convexity of \(s\mapsto (1+s)^{-1}\) then implies, by [13, Theorem 10.20], that \(\theta _\delta \rightarrow \theta \) a.e. in \(\Omega _T\). We deduce from the \(L^2(\Omega _T)\) bound for \((\theta _\delta )\) from Lemma 3 that this convergence is in fact strong in \(L^1(\Omega _T)\).
Step 5: Limit \(\delta \rightarrow 0\) in equations (43)–(44). We know from (42) that \((\nabla (\rho _\delta \theta _\delta ))\) is bounded in \(L^{4/3}(\Omega _T)\). Thus, up to a subsequence, \(\nabla (\rho _\delta \theta _\delta )\rightharpoonup \zeta _1\) weakly in \(L^{4/3}(\Omega _T)\) for some \(\zeta _1\in L^{4/3}(\Omega _T)\). Since \(\rho _\delta \theta _\delta \rightharpoonup \rho \theta \) weakly in \(L^1(\Omega _T)\), by (50), we infer that \(\zeta _1=\nabla (\rho \theta )\), i.e.
We know from Lemma 3 that \((\rho _\delta \theta _\delta ^2)\) is bounded in \(L^{3/2}(\Omega _T)\), so that up to a subsequence, \(\rho _\delta \theta _\delta ^2\rightarrow \zeta _2\) weakly in \(L^{3/2}(\Omega _T)\). We deduce from the strong convergence of \((\theta _\delta )\) and the boundedness of \(s\mapsto (1+s^2)^{-1}\) that \((1+\theta _\delta ^2)^{-1}\rightarrow (1+\theta ^2)^{-1}\) strongly in \(L^\eta (\Omega _T)\) for any \(\eta <\infty \). Therefore,
An application of (48) with \(g(s)=s^2(1+s^2)^{-1}\) together with the strong convergence of \((\theta _\delta )\) leads to
Hence, \(\zeta _2=\rho \theta ^2\) a.e. in \(\Omega _T\) and
Furthermore, it follows from (33), Lemma 3, and (39) that
For any \(\psi _1\in L^4(0,T;W^{1,4}(\Omega ))\), we have
Hence, up to subsequences,
We deduce from the bound for \((\log \theta _\delta )\) in \(L^\infty (0,T;L^1(\Omega ))\) that \(\theta >0\) a.e. in \(\Omega _T\).
We claim that \((\rho _\delta )\) also converges strongly. Indeed, the a.e. convergence of \((E_\delta )\) and \((\theta _\delta )\) imply that \(\rho _\delta =\frac{2}{3}(E_\delta /\theta _\delta -1)\rightarrow \rho \) a.e. in \(\Omega _T\). The \(L^\infty (0,T;L^1(\Omega ))\) bound for \((\rho _\delta \log \rho _\delta )\) from (30) shows that \((\rho _\delta )\) is equi-integrable, and together with its a.e. convergence, we conclude from the de la Vallée–Poussin theorem [10, Chap. 8, Sect. 1.7, Corollary 1.3] that
The positivity of \(\rho _\delta \) implies that \(\rho \ge 0\) a.e. in \(\Omega _T\). Note, however, that we cannot conclude that \(\rho >0\) a.e., since the control on \(\phi _\delta \) is now lost.
Convergences (52)–(55) allow us to perform the limit \(\delta \rightarrow 0\) in (43)–(44) showing that \((\rho ,\theta )\) solves (9)–(10). Theorem 1 is proved.
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The first three authors acknowledge partial support from the FWF, the Austrian Science Fund (FWF), grants F65 and W1245. The second author has been additionally supported by the grants P30000 and P33010 of the FWF. The fourth author acknowledges support from the Alexander von Humboldt Foundation.
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Favre, G., Jüngel, A., Schmeiser, C. et al. Existence analysis of a degenerate diffusion system for heat-conducting gases. Nonlinear Differ. Equ. Appl. 28, 41 (2021). https://doi.org/10.1007/s00030-021-00700-z
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DOI: https://doi.org/10.1007/s00030-021-00700-z