zbMATH — the first resource for mathematics

On the Neumann problem for some linear hyperbolic systems of 2nd order with coefficients in Sobolev spaces. (English) Zbl 0706.35082
The paper is concerned with unique existence of solutions to the following equations (N): \(\partial^ 2_ t\vec u-\partial_ i(A^{i0}\partial_ t\vec u+A^{ij}\partial_ j\vec u)=\vec f_{\Omega}\text{ in } (0,T)\times \Omega,\nu_ iA^{ij}\partial_ j\vec u+B^ j\partial_ j\vec u+B^ 0\partial_ t\vec u=\vec f_{\Gamma}\text{ on } (0,T)\times \Gamma,\vec u(0,x)=\vec u_ 0(x)\text{ and } \partial_ t\vec u(0,x)=\vec u_ 1(x)\text{ in } \Omega,\)where \(\vec u=^ t(u_ 1,...,u_ m)\) \((=\) m row vector), \(\Omega\) is a domain in \({\mathbb{R}}^ n\) with boundary \(\Gamma\) which is a \(C^{\infty}\) and compact hypersurface, \(\partial_ t=\partial /\partial t\), \(\partial_ j=\partial /\partial x\), t is a time variable, \(x=(x_ 1,...,x_ n)\in {\mathbb{R}}^ n\), \(\nu =(\nu_ 1,...,\nu_ n)\) is a unit outer normal to \(\Gamma\) and the summation convention is understood. The \(A^{i\ell}=A^{i\ell}(t,x)\) and \(B^{\ell}=B^{\ell}(t,x)\) are \(m\times m\) matrices of functions satisfying the following five assumptions:
(A.1) \(A^{i\ell}=A_{\infty}^{i\ell}+A_ S^{i\ell}\) where \(A_{\infty}^{i\ell}\in {\mathcal B}^ K\) and \(A_ S^{i\ell}\in Y^{K-1,1}([0,T],\Omega)\) and \(B^{\ell}\in Y^{K- 1,1/2}([0,T],\Gamma)\), where \[ Y^{\ell +1,r}(J,G)=\{u\in X^{\ell,r}(J,G)| \quad \partial^ j_ tu\in L^{\infty}(J,H^{\ell +1+r-j}(G))\cap Lip(J,H^{\ell +r-j}(G))\text{ for } 0\leq j\leq \ell \}, \] \(X^{\ell,r}(J,G)=C^ 0(J,H^{\ell +r}(G))\cap...\cap C^{\ell}(J,H^ r(G))\), \(H^ s(G)\) is a usual Sobolev space of order s over G.
(A.2) \({}^ tA^{i0}=A^{i0}\), \({}^ tA^{ij}=A^{ji}\), \({}^ tB^ 0=B^ 0\), \({}^ tB^ i+B^ i=0\) \((i,j=1,...,n).\)
(A.3) There exist \(\delta_ 1,\delta_ 2>0\) such that \[ \int_{\Omega}A^{ij}\partial_ j\vec v\cdot \partial_ i\vec v dx+\int_{\Gamma}B^ j\partial_ j\vec v\cdot \vec v d\Gamma \geq \delta_ 1\| \vec v\|^ 2_ 1-\delta_ 2\| \vec v\|^ 2_ 0,\quad \vec v\in H^ 2(\Omega), \] where \(\| \|_ s\) is a norm of \(H^ s(\Omega).\)
(A.4) \(\nu_ i(x)B^ i(t,x)=0\) on [0,T]\(\times \Gamma.\)
(A.5) \((-\nu_ i(x)A^{i0}(t,x)+2B^ 0(t,x))\eta \cdot \eta \leq 0\) for \(\eta \in {\mathbb{R}}^ m\) on [0,T]\(\times \Gamma.\)
Then, we have: Theorem. Let K be an integer \(>n/2+1\) and \(2\leq L\leq K\). If \(\vec u_ 0\in H^ L(\Omega)\), \(\vec u_ 1\in H^{L-1}(\Omega)\), \(\vec f_{\Omega}\in X^{L-2,0}([0,T),\Omega)\), \(\partial_ t^{L- 2}\vec f_{\Omega}\in Lip([0,T],L^ 2(\Omega))\), \(\vec f_{\Gamma}\in X^{L-2,1/2}([0,T],\Gamma)\), \(\partial_ t^{L-2}\vec f_{\Gamma}\in Lip([0,T],H^{1/2}(\Gamma))\) and the L-2th order compatibility condition is satisfied, then (N) admits a unique solution \(\vec u\in X^{L,0}([0,T],\Omega)\). Moreover, we get the suitable energy estimates.
The purpose is to improve the local existence theorem for the quasilinear hyperbolic system with Neumann conditions to minimal Sobolev order.
Reviewer: Y.Shibata

35L55 Higher-order hyperbolic systems
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI