Preprint A37/2001
Decay of Almost Periodic Solutions of Conservation Laws
Hermano Frid
Keywords: Conservation laws | parabolic equations | almost periodic functions
We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H.\ Bohr. We prove that if $u(x,t)$ is such a solution whose inclusion intervals at time $t$, with respect to $\ve>0$, satisfy $l_\ve(t)/t\to0$ as $t\to\infty$, and so that the scaling sequence $u^T(x,t)=u(T x,T t)$ is pre-compact as $T\to\infty$ in $L_{\loc}^1(\R^{d+1}_+)$ then $u(x,t)$ decays to its mean value $\bar u$, which is independent of $t$, as $t\to\infty$. The decay considered here is in $L^1_{\loc}$ of the variable $\xi=x/t$, which implies, as we show, that $M_x(|u(x,t)-\bar u|)\to0$, as $t\to\infty$, where $M_x$ denotes taking the mean value with respect to $x$. In many cases we show that the solutions are almost periodic in the generalized sense if the initial data are. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals $l_\ve(t)$ with $t$, as $t\to\infty$, for fixed $\ve>0$, to a condition on the growth of $l_\ve(0)$ with $\ve$, as $\ve\to0$, which amounts to impose restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as $\ve\to0$. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and many viscous $2\times2$ systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the $L^\infty$ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.