Decay of almost periodic solutions of conservation laws
The mathematical apparatus of some mechanical problems consists in dealing with systems of conservation laws described by the equation \[ \partial_t u+\sum^d_{k=1} \partial_{x_k}f^k(u)= \sum_{kl}\partial^2_{x_kx_l} a_{kl}(u), \quad x\in\mathbb{R}^d,\;t>0, \] where \(u(x,t)\) takes values in an open set \(U\subseteq \mathbb{R}^n\), while \(f^k\), \(a_{kl}\) are smooth functions from \(U\) into \(\mathbb{R}^n\). An initial condition of the form \(u(x,0)=u_0(x)\) is prescribed. The concept of an entropy solution is defined and its decay at infinity is investigated. Many conditions utilized are too technical to be reproduced here, but for getting a flavor of the nature of the results we will briefly mention the following: for a solution \(u(x,t)\) satisfying the initial condition, let us denote \(u^T(x,t)=u(Tx,Tt)\); let \(\overline u\) be defined by \(\overline u=\lim_{L \to\infty} (2L)^{-d} \int_{|x|\leq L}u_0(x)dx\); then \(u^T(x,t) \to \overline u\) in a precised sense as \(T\to\infty\); moreover, \[ \lim\left(T^{-1} \int^T_0M_x \biggl(\bigl|u(x,t)- \overline u\bigr|\biggr) dt\right)=0 \] as \(T \to\infty\). In the inviscid case one can obtain the stronger result \(M_x (|u(x,t)- \overline u|)=0\). Various applications are provided. The concept of almost periodicity used by the author is that due to Stepanoff.
scientific article; zbMATH DE number 1735404