On equations of a~nonlinear compressible fluid with a~discontinuous constitutive law (Q1307153)

A~system describing the motion of a~barotropic fluid \(\rho_t+(\rho u)_x=0\), \(\rho(u_t+uu_x)=-(p(\rho))_x+(\sigma(u_x))_x\) is considered in the cylinder \(Q_T=\{(x,t):0<x<1\), \(0<t<T<+\infty\}\). Here \(u(x,t)\) is the velocity, \(\rho(x,t)\) is the density, and \(\sigma\) is the deviatoric part of the stress. The authors study the initial-boundary value problem with conditions \(u=0\) for \(x=0\), \(x=1\), \(u=u_0(x)\), \(\rho=\rho_0(x)\) for \(t=0\). It is proven that this problem has a~unique solution for \(u\in L_\infty(0,T;W_2^1(0,1)) \cap L_2(0,T;W_2^2(0,1))\), \(\rho\in L_\infty(0,T;W_2^1(0,1))\), \(\rho>0\), \(\rho_t\in L_\infty(0,T;L_2(0,1))\), \(u_t,{\partial\over\partial x}\sigma\in L_2(Q_T)\), \(F(u_x)\in L_\infty(0,T;L_1(0,1))\), where \(F(s)= \int_0^s \sigma(\xi)d\xi\). Moreover, the function \(\sigma\) may have discontinuities (for instance, \(\sigma\) may have the form \(\sigma(s)=ks+\sigma_0 \text{ sign } s\), \(k>0\), \(\sigma_0>0\)).