Well-posedness and asymptotics for initial boundary value problems of linear relaxation systems in one space variable
This paper deals with the linear relaxation system \[ \begin{aligned} \partial_tu^{\varepsilon}(x,t)+ \partial_xv^{\varepsilon}(x,t)&= q_1(x,t),\\ \partial_tv^{\varepsilon}(x,t)+ a\partial_xu^{\varepsilon}(x,t)&= q_2(x,t)-\tfrac{1}{\varepsilon} (v^{\varepsilon}-f(u^{\varepsilon}))\end{aligned} \tag{1} \] in \(\{x>0,t>0\}\), associated with the initial conditions \(u_o(x),v_o(x)\) and the boundary condition \(B_uu^{\varepsilon}(0,t)+B_vv^{\varepsilon}(0,t)=b(t)\). Most of the paper is devoted to the scalar case i.e. \(u^{\varepsilon},v^{\varepsilon}\) are \(\mathbb R\)-valued. Under the assumptions: \(f(u)=\lambda u,\lambda\in\mathbb R,a\geq\lambda^2,\) the stiff Kreiss condition \(\{B_v=0\) or \(\frac{B_u}{B_v}\notin [-\sqrt{a}-\frac{\lambda+| \lambda| }{2}]\},\) (1) is \(L^2\)-stiffly well-posed [cf. \textit{Z. Xin} and \textit{W-Q. Xu}, J. Differ. Equations 167, 388--437 (2000; Zbl 0964.35092)]. Set \(U^{\varepsilon}= (u^{\varepsilon},v^{\varepsilon}),\) the solution to (1). Let \[ \partial_tu+\partial_xf(u)= q_1(x,t), \quad v=f(u) \tag{2} \] be the corresponding equilibrium system. Under suitable regularity assumptions on the data, it is proved that there exists a unique solution \(U=(u,v)\) to (2) such that \[ I_{\varepsilon}=\int_0^{\infty}\int_0^{\infty} | U^{\varepsilon}-U| ^2 e^{-2\alpha t}\,dx\,dt \to 0, \] as \(\varepsilon\to0,\) for any \(\alpha>0\). Sharp estimates of \(I_{\varepsilon}\) are obtained, initial layers and boundary layers are investigated. Previous results by Z. Xin and W-Q. Xu (loc. cit.), in the homogeneous case of system (1), are used in the proofs. A possible extension of the above results in case \(u^{\varepsilon},v^{\varepsilon}\) are \(\mathbb R^n\)-valued, \(n>1\), is briefly discussed.
scientific article; zbMATH DE number 5152717