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On the uniqueness and large time behavior of the weak solutions to a shallow water equation. (English) Zbl 1034.35115

The authors deal with the uniqueness and larger time behaviour of weak solutions to the Cauchy problems for the following one-dimensional shallow water equation \[ \begin{cases} \partial_t u+u\partial_x u+\partial_xP=0,\;t>0,\;x\in\mathbb{R}^1\\ P(t,x)=\tfrac 12 \int^\infty_{-\infty} e^{-| x-y|} \Bigl(u^2+\tfrac 12 (\partial_x u)^2\Bigr) (t,y)dy\\ u(0,x)=u_0(x)\in H^1(\mathbb{R}^1),\end{cases} \] which is formally equivalent to the Camass-Holm equation \[ \partial_tu-\partial_x^2 \partial_tu+ 3u\partial_x u=2\partial_x u\partial_x^2u+u \partial_x^3u. \] Moreover, the authors show that the admissible weak solutions (under some additional condition on the solutions) tend to 0 pointwisely as \(t\to\infty\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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