Global weak solutions in nonlinear 3D thermoelasticity

Abstract

Here we study a nonlinear thermoelasticity hyperbolic-parabolic system describing the balance of momentum and internal energy of a heat-conducting elastic body, preserving the positivity of temperature. So far, no global existence results in such a natural case were available. Our result is obtained by using thermodynamically justified variables which allow us to obtain an equivalent system in which the internal energy balance is replaced with entropy balance. For this system, a concept of weak solution with defect measure is introduced, which satisfies entropy inequality instead of balance and has a positive temperature almost everywhere. Then, the global existence, consistency and weak–strong uniqueness are shown in the cases where heat capacity and heat conductivity are both either constant or non-constant. Let us point out that this is the first result concerning global existence for large initial data in nonlinear thermoelasticity where the model is in full accordance with the laws of thermodynamics.

Appendix

Lemma

Let $\Omega$ be a Lipschitz domain and let $p\in (1,\infty ]$. Assume that $\theta \in L^p(0,T;{\mathcal {M}}(\Omega ))$, and that ${\textbf{u}}\in L^p(0,T;H^1(\Omega ))$ with ${\textbf{u}}_t \in L^\infty (0,T;L^2(\Omega ))$ satisfies the following equation

$$\begin{aligned} \int _0^T\int _\Omega {\textbf{u}}_t \cdot \varvec{\varphi }_t - \int _0^T\int _\Omega {\mathbb {D}}({\textbf{u}}): \nabla \varvec{\varphi }+ \mu \int _0^T{}_{{\mathcal {M}}(\Omega )}\langle \theta , \nabla \cdot \varvec{\varphi }\rangle _{C_0(\Omega )} =-\int _\Omega {\textbf{v}}_0\cdot \varvec{\varphi }, \end{aligned}$$

for all test function $\varvec{\varphi }\in H^1(0,T;C^{\infty }_0(\Omega ))$, ${\varvec{\varphi }}(T)=0$, where ${\textbf{v}}_0\in L^2(\Omega )$ is given. Then, ${\textbf{u}}_t \in C_w(0,T;L^2(\Omega ))$.

Proof

First, let us prove that. Since $\theta \in L^p(0,T;{\mathcal {M}}(\Omega ))$, one has from the equation for every $\varvec{\varphi }\in C_0^\infty ((0,T)\times \Omega )$ that

$$\begin{aligned} \left| \int _0^T\int _\Omega {\textbf{u}}_{tt}\cdot \varvec{\varphi }\right| \le \big (||{\textbf{u}}||_{L^p(0,T;H^1(\Omega ))}+||\theta ||_{L^p(0,T;{\mathcal {M}}^+(\Omega ))}\big )||\varvec{\varphi }||_{L^{p^*}(0,T; W^{2,\infty }(\Omega ))}, \end{aligned}$$

so ${\textbf{u}}_{tt} \in L^{p^*}(0,T; [C_0^\infty (\Omega )]')$, and consequently ${\textbf{u}}_t\in C(0,T; [C_0^\infty (\Omega )]')$.

Now, for an arbitrary $\varvec{\varphi }\in L^{2}(\Omega )$, let $\varvec{\varphi }_n\in C_0^\infty (\Omega )$ be such that $||\varvec{\varphi }_n-\varvec{\varphi }||_{L^{2}(\Omega )}\rightarrow 0$, as $n\rightarrow \infty$. Then, for almost all $t_1, t_2$, one has

$$\begin{aligned}{} & {} \int _\Omega ({\textbf{u}}_t(t_1)-{\textbf{u}}_t(t_2))\cdot \varvec{\varphi }\\{} & {} \quad =\int _\Omega ({\textbf{u}}_t(t_1)-{\textbf{u}}_t(t_2))\cdot (\varvec{\varphi }-\varvec{\varphi }_n)+\int _\Omega ({\textbf{u}}_t(t_1)-{\textbf{u}}_t(t_2))\cdot \varvec{\varphi }_n, \end{aligned}$$

so

$$\begin{aligned}{} & {} \lim \limits _{t_2\rightarrow t_1}\left| \int _\Omega ({\textbf{u}}_t(t_1)-{\textbf{u}}_t(t_2))\cdot \varvec{\varphi }\right| \\{} & {} \quad \le \lim \limits _{t_2\rightarrow t_1}\left| \int _\Omega ({\textbf{u}}_t(t_1)-{\textbf{u}}_t(t_2))\cdot (\varvec{\varphi }-\varvec{\varphi }_n)\right| + \underbrace{\lim \limits _{t_2\rightarrow t_1}\left| \int _\Omega ({\textbf{u}}_t(t_1)-{\textbf{u}}_t(t_2))\cdot \varvec{\varphi }_n\right| }_{=0}\\{} & {} \quad \le 2||{\textbf{u}}_t||_{L^\infty (0,T;L^2(\Omega ))}||\varvec{\varphi }-\varvec{\varphi }_n||_{L^2(\Omega )} \rightarrow 0, \quad \text {as }n\rightarrow \infty . \end{aligned}$$

Now, we can give meaning to $\int _\Omega {\textbf{u}}_t(t)\cdot \varvec{\varphi }$ for all t by constructing a sequence of $t_n$ such that $\int _\Omega {\textbf{u}}_t(t_n)\cdot \varvec{\varphi }$ is finite for all n and $t_n\rightarrow t$ as $n\rightarrow \infty$. From the above inequality, $\int _\Omega {\textbf{u}}_t(t_n)\cdot \varvec{\varphi }$ forms a Cauchy sequence which converges to a unique limit $a_{\varvec{\varphi }}\in {\mathbb {R}}$ for any $\varvec{\varphi }$, independent of the choice of the sequence $\{t_n\}_{n\in {\mathbb {N}}}$. Now, since $\left| \int _\Omega {\textbf{u}}_t(t_n)\cdot \varvec{\varphi }\right| \le C||\varphi ||_{L^2(\Omega )}$, one has that $|a_{\varvec{\varphi }}|\le C||\varphi ||_{L^2(\Omega )}$. This combined with linearity of $\varvec{\varphi }\mapsto a_{\varvec{\varphi }}$ allows us to identify $a_{\varvec{\varphi }}$ with a function ${\textbf{u}}_t(t)\in L^2(\Omega )$, by the Riesz theorem. The proof is now finished. $\square$