Stability of Multidimensional Thermoelastic Contact Discontinuities

Abstract

We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface. Mathematically, this renders a nonlinear multidimensional hyperbolic problem with a characteristic free boundary. We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. The missing normal derivatives are recovered from the estimates of several quantities relating to physical involutions. In the estimate of tangential derivatives, there is a significant new difficulty, namely the presence of characteristic variables in the boundary conditions. To overcome this difficulty, we explore an intrinsic cancellation effect, which reduces the boundary terms to an instant integral. Then we can absorb the instant integral into the instant tangential energy by means of the interpolation argument and an explicit estimate for the traces on the hyperplane.

Appendices

Appendix A: Proof of Proposition 2.1

Assume that $[{ S }]=0$ on $\varGamma (t)$. Taking the scalar product of the last identity in (2.23) with N and utilizing (2.20e) yields that

$$\begin{aligned}&|N|^2\left( p(\rho ^+,S^+)-p(\rho ^-,S^+)\right) \\&\quad =|N|^2[p]= \rho ^+ F_{\ell N}^+[F_{\ell N}] =[\rho F_{\ell N}F_{\ell N}]=\sum _{j=1}^{d}(\rho ^+ F_{\ell N}^+)^2[ {\rho }^{-1}]. \end{aligned}$$

Then we infer from (2.11) and (2.22) that

$$\begin{aligned}{}[\rho ]=[p]=0, \end{aligned}$$

which, combined with (2.23), gives

$$\begin{aligned} F_{\ell N}^+[F_{\ell }]=0. \end{aligned}$$

(A.1)

Plug (2.20e) into (2.20f) to obtain

$$\begin{aligned} F_{kN}^{+}[F_{ij}]-F_{jN}^+[F_{ik}]=0 \qquad \,\,\text {for }i,j,k=1,\ldots ,d. \end{aligned}$$

(A.2)

For $d=2$, from (A.1)–(A.2), we have

$$\begin{aligned} (F_{1N}^{+})^2[F_{i2}]+(F_{2N}^{+})^2[F_{i2}] =F_{2N}^{+} \left( F_{1N}^{+}[F_{i1}]+F_{2N}^{+}[F_{i2}] \right) =0, \end{aligned}$$

which, along with (2.22), yields $[F_{i2}]=0$ for $i=1,2$. Then we utilize (A.2) again to obtain $[\varvec{F}]=0$ on $\varGamma (t)$.

For $d=3$, relations (A.2) are equivalent to

$$\begin{aligned} ( F_{1N}^+,\, F_{2N}^+,\, F_{3N}^+)^{{\mathsf {T}}}\times ([F_{i1}] ,\, [F_{i2}] ,\, [F_{i3}])^{{\mathsf {T}}}=0\qquad \,\text {for }i=1,2,3, \end{aligned}$$

which implies

$$\begin{aligned}{}[F_{ij}] =\omega _i F_{jN}^+ \end{aligned}$$

(A.3)

for some scalar functions $\omega _i$ and for all $i,j=1,2,3$. We plug (A.3) into (A.1) and utilize (2.22) to deduce that $\omega _i\equiv 0$ for all $i=1,\ldots ,d$. Then it follows from (A.3) that $[\varvec{F}]=0$ on $\varGamma (t)$.

In view of the second condition in (2.23), we find that $[U]=0$ on $\varGamma (t)$, i.e., the solution U is continuous across front $\varGamma (t)$. Therefore, there is no thermoelastic contact discontinuity for the case $[S]=0$. This completes the proof of Proposition 2.1.

Appendix B: Proof of Proposition 2.2

We omit indices ± in several places below to avoid overloaded expressions.

1: Proof of (2.35). In the original variables, we see from (2.15c) that

$$\begin{aligned} (\partial _t+v_{\ell } \partial _{\ell }) \det \varvec{F}&=\frac{\partial \det \varvec{F}}{\partial F_{ij}} (\partial _t+v_{\ell } \partial _{\ell }) F_{ij} =\det \varvec{F} (\varvec{F}^{-1})_{ji} F_{\ell j}\partial _{\ell } v_i \\&=\det \varvec{F} \delta _{\ell , i} \partial _{\ell } v_i=\det \varvec{F} \partial _{i} v_i, \end{aligned}$$

which, combined with the first equation in (2.5), yields

$$\begin{aligned} (\partial _t+v_{\ell } \partial _{\ell }) (\rho \det \varvec{F})=0. \end{aligned}$$

After transformation (2.29), we find

$$\begin{aligned} (\partial _t+w_{\ell } \partial _{\ell })(\rho \det \varvec{F})=0, \end{aligned}$$

where

$$\begin{aligned} w_1:=\frac{1}{\partial _1 \varPhi }\Big (v_1-\partial _t\varPhi -\sum _{j=2}^{d}v_j \partial _j\varPhi \Big ), \qquad w_i:=v_i \quad \text {for }i=2,\cdots ,d. \end{aligned}$$

Since $w_1|_{x_1=0}=0$ resulting from (2.32b), we can obtain identity (2.35) by the standard energy method.

2: Proof of (2.36). A straightforward calculation shows that solutions of (2.18) satisfy (see, e.g., the proof of Qian–Zhang [24, Proposition 1])

$$\begin{aligned} (\partial _t+v_{\ell } \partial _{\ell })( {F}_{\ell k}\partial _{\ell }{F}_{ij}-{F}_{\ell j}\partial _{\ell }{F}_{i k}) =\partial _{m }v_i({F}_{\ell k}\partial _{\ell }{F}_{m j}-{F}_{\ell j}\partial _{\ell }{F}_{m k}). \end{aligned}$$

After transformation (2.29), we have

$$\begin{aligned} (\partial _t+w_{\ell } \partial _{\ell })M_{k,i,j}=\partial _{m }^{\varPhi }v_i M_{k,m, j} \end{aligned}$$

with $M_{k,i,j}:={F}_{\ell k}\partial _{\ell }^{\varPhi }{F}_{ij}-{F}_{\ell j}\partial _{\ell }^{\varPhi }{F}_{ik}$. Here we recall the differentials with respect to (2.29) from definition (2.40). Similar to the proof of Hu–Wang [19, Lemma A.2], we can use integration by parts and $w_1|_{x_1=0}=0$ to obtain (2.36).

3: Proof of (2.37) and (2.39). In the original variables, system (2.15) gives

$$\begin{aligned}&(\partial _t+v_{\ell } \partial _{\ell })(\rho {F}_{ij})+\rho {F}_{ij} {\partial _{\ell } v_{\ell }}-\rho {F}_{\ell j}\partial _{\ell } v_i=0. \end{aligned}$$

(B.1)

After transformation (2.29), equation (B.1) becomes

$$\begin{aligned} (\partial _t+w_{\ell } \partial _{\ell } )(\rho {F}_{ij})+\rho {F}_{ij} \partial _{\ell }^{\varPhi } v_{\ell }-\rho {F}_{\ell j} \partial _{\ell }^{\varPhi } v_i=0. \end{aligned}$$

(B.2)

By virtue of (2.32b), we have

$$\begin{aligned} (\partial _t+w_{\ell } \partial _{\ell })\partial _i\varphi =\partial _i v\cdot N \qquad \,\, \text {on }\partial \varOmega ,\text { for }i=2,\ldots ,d. \end{aligned}$$

Then it follows from the restriction of (B.2) on $\partial \varOmega$ that

$$\begin{aligned} (\partial _t+w_{\ell } \partial _{\ell } )(\rho {F}_{jN})+\rho {F}_{jN} \sum _{\ell =2}^d \partial _{\ell } v_{\ell }=0\qquad \, \text {on }\partial \varOmega . \end{aligned}$$

(B.3)

Since $w_1|_{x_1=0}=0$ and $[v]=0$, we can derive (2.37) and (2.39) by employing the method of characteristics.

4: Proof of (2.38). It follows from (B.3) that

$$\begin{aligned}&(\partial _t+w_{\ell } \partial _{\ell } )(\rho {F}_{kN} F_{ij} -\rho F_{jN} F_{ik}) -\rho {F}_{kN} (\partial _t+w_{\ell } \partial _{\ell } ) F_{ij}\\&\quad +\rho {F}_{jN} (\partial _t+w_{\ell } \partial _{\ell } ) F_{ik}+ \sum _{\ell =2}^d \partial _{\ell } v_{\ell } (\rho {F}_{kN}F_{ij} -\rho F_{jN}F_{ik})=0 \qquad \text {on }\partial \varOmega . \end{aligned}$$

Since

$$\begin{aligned} (\partial _t+w_{\ell } \partial _{\ell } ) F_{ij}=F_{\ell j}\partial _{\ell }^{\varPhi }v_i= \frac{\partial _1 v_i}{\partial _1 \varPhi } F_{jN}+\sum _{\ell =2}^d F_{\ell j}\partial _{\ell } v_i, \end{aligned}$$

we have

$$\begin{aligned} (\partial _t+w_{\ell } ^+\partial _{\ell } ) [I_{k,i,j}]+\sum _{\ell =2}^d \partial _{\ell } v_i^+ [ I_{j,\ell ,k}] + \sum _{\ell =2}^d \partial _{\ell } v_{\ell }^+ [I_{k,i,j}]=0 \qquad \text {on }\partial \varOmega , \end{aligned}$$

for $I_{k,i,j}:=\rho {F}_{kN} F_{ij} -\rho F_{jN} F_{ik}$. Since (2.38) holds at the initial time, i.e., $[I_{k,i,j}]=0$ at $t=0$ for $i,j,k=1,\ldots ,d$, we employ the standard argument of the energy method to derive that (2.38) is satisfied for all $t\in [0,T]$.

5: Proof of (2.41). It suffices to prove (2.12) in the original variables. We note that (2.6)–(2.7) hold in virtue of (2.35)–(2.36) so that

$$\begin{aligned} \partial _{\ell }(\rho F_{\ell k})&= \partial _{\ell }((\det \varvec{F})^{-1} F_{\ell k})\\&=(\det \varvec{F})^{-1} \partial _{\ell } F_{\ell k}- (\det \varvec{F})^{-2} F_{\ell k}\frac{\partial \det \varvec{F}}{\partial F_{i j}} \partial _{\ell } F_{i j}\\&=(\det \varvec{F})^{-1} \left( \partial _{\ell } F_{\ell k}- (\varvec{F}^{-1})_{j i} F_{\ell k} \partial _{\ell } F_{i j} \right) \\&=(\det \varvec{F})^{-1} \left( \partial _{\ell } F_{\ell k}- (\varvec{F}^{-1})_{ j i} F_{\ell j} \partial _{\ell } F_{i k} \right) \\&=(\det \varvec{F})^{-1} \left( \partial _{\ell } F_{\ell k}- \delta _{\ell ,i} \partial _{\ell } F_{i k} \right) =0. \end{aligned}$$

This completes the proof of Proposition 2.2.