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Strong ergodic properties of a first-order partial differential equation. (English) Zbl 0673.35012

Consider the initial value problem of the first order nonlinear PDE \[ (1)\quad u_ t+C(x)u_ x=f(x,u),\quad (x,t)\in [0,\infty)\times [0,1],\quad u(0,x)=v(x),\quad x\in [0,1]. \] Under some conditions, (1) generates a semiflow \(\{S_ t\}_{t\geq 0}\) on C[0,1] defined by \(S_ tv(x)=u(t,x)\), where u is the solution of (1). This paper shows one existence of an exact invariant measure \(\mu\) having some additional properties. From that follow additional features of \(\{S_ t\}:\) chaos and the existence of turbulent trajectories.
Reviewer: J.H.Tian

MSC:

35F25 Initial value problems for nonlinear first-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
37A99 Ergodic theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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