Initial-boundary value problem for a class of linear relaxation systems in arbitrary space dimensions
The author studies the behaviour of solutions of the following initial-boundary value problem in the half-space \(x_1\geq 0\), \(x'=(x_2,\dots,x_n)\in \mathbb{R}^{n-1}\) \[ \begin{cases} \frac{\partial u^{\varepsilon}}{\partial t}+ \sum_{k=1}^{n}\frac{\partial v_k^{\varepsilon}}{\partial x_k} =0,\\ \frac{\partial v_j^{\varepsilon}}{\partial t} +a_j\frac{\partial u^{\varepsilon}}{\partial x_j}= -\frac{1}{\varepsilon}(v^{\varepsilon}_j-\lambda_ju^{\varepsilon}), \quad j=1,\dots,n, \end{cases} \tag{1} \] \[ u^{\varepsilon}(x,0)=u_0(x),\quad v^{\varepsilon}_j(x,0)=v_{j0}(x), \quad j=1,\dots,n, \tag{2} \] \[ B_0u^{\varepsilon}(0,x',t)+B_1v_1^{\varepsilon}(0,x',t)=b(x',t) \tag{3} \] as \(\varepsilon \to 0\), where \(n\geq 2\), \(u^{\varepsilon}, v^{\varepsilon}, \lambda_j\in \mathbb{R}^n\), \(a_j>0\) and \(\varepsilon\) is a small positive parameter. The problem (1), (2), (3) will be called stiffly well posed if for any \(\alpha >0\), there exists a constant \(K=K(\alpha)>0\) independent on \(\varepsilon\), \(U_0\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\) and \(b\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\) such that \[ \begin{multlined} \int_0^{\infty}\int_{\mathbb{R}^n_+}|U^{\varepsilon}(x,t)|^2e^{-2\alpha t} dx dt+ \int_0^{\infty}\int_{\mathbb{R}^{n-1}_{+}}|U^{\varepsilon}(0,x',t)|^2e^{-2\alpha t} dx' dt+\\ \leq K\Big(\int_0^{\infty}\int_{\mathbb{R}^{n-1}}|b(x',t)|^2e^{-2\alpha t} dx' dt+ \int_{\mathbb{R}^n_+}|U_0(x)|^2 dx\Big),\end{multlined} \] where \(U^{\varepsilon}=\text{col}(u^{\varepsilon},v^{\varepsilon}_1, \dots,v^{\varepsilon}_n)\). Assuming that \( \sum_{j=1}^n\lambda_j^2/a_j\leq 1 \) and \(U_0(x)\equiv 0\) the author proves that the problem (1),(2),(3) is stiffly well posed if and only if the boundary condition (3) satisfies the Uniform Kreiss Condition (UKC): \(B_0B_1>0\). If (UKC) holds and \(b\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\), then there exists a unique solution \(U= \text{col}(u,v_1,\dots,v_n)\) of the problem \[ \frac{\partial u}{\partial t} +\sum_{j=1}^n\lambda _l \frac{\partial u}{\partial x_j} ,\quad v_j=\lambda _ju, \quad j=1,\dots,n \] such that \[ \int_0^{\infty}\int_{\mathbb{R}^n_+}|U^{\varepsilon}(x,t)-U(x,t)|^2e^{-2\alpha t} dx dt \to 0 \quad\text{as}\quad \varepsilon \to 0, \] for any \(\alpha >0\). If in addition \(b\in L^2(\mathbb{R}^+\times \mathbb{R}^{n-1})\) and \(b(x',0)=\frac{\partial b(x',0)}{\partial t}=0\), \(x'\in \mathbb{R}^{n-1}\), then \[ \int_ 0^{\infty}\int_{\mathbb{R}^n_+}|U^{\varepsilon}(x,t)-U(x,t)|^2 e^{-2\alpha t} dx dt\leq \begin{cases} O(1)\varepsilon^2\|b\|^2_{H^2}, \quad &\text{if }\lambda_1>0,\\ O(1)\varepsilon\|b\|^2_{L^2}, \quad &\text{if } \lambda_1<0,\\ O(1)\varepsilon^{1/2}\|b\|^2_{L^2}, \quad &\text{if }\lambda_1=0.\end{cases} \] {} In the cases \(\lambda_1<0\) and \(\lambda _1=0\) there exists a boundary layer \[ U^{bl}= \begin{cases} U^{bl}(x_1/\varepsilon,x',t),\quad &\text{if }\lambda_1<0,\\ U^{bl}(x_1/\sqrt{\varepsilon},x',t),\quad &\text{if }\lambda_1=0, \end{cases} \] such that \[ \int_ 0^{\infty}\int_{R^n_+}|U^{\varepsilon}(x,t)-U(x,t)- U^{bl}(x,\varepsilon,t)|^2e^{-2\alpha t} dx dt\leq \begin{cases} O(1)\varepsilon^2\|b\|^2_{H^2}, \quad &\text{if } \lambda_1<0,\\ O(1)\varepsilon^{3/2}\|b\|^2_{H^2}, \quad &\text{if }\lambda_1=0.\end{cases} \] {}.
scientific article; zbMATH DE number 1836803