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Zbl 1034.35115
Xin, Zhouping; Zhang, Ping
On the uniqueness and large time behavior of the weak solutions to a shallow water equation.
(English)
[J] Commun. Partial Differ. Equations 27, No. 9-10, 1815-1844 (2002). ISSN 0360-5302; ISSN 1532-4133/e

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 $$\cases \partial_t u+u\partial_x u+\partial_xP=0,\ t>0,\ x\in\bbfR^1\\ P(t,x)=\tfrac 12 \int^\infty_{-\infty} e^{-\vert x-y\vert} \Bigl(u^2+\tfrac 12 (\partial_x u)^2\Bigr) (t,y)dy\\ u(0,x)=u_0(x)\in H^1(\bbfR^1),\endcases$$ 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 $t\to\infty$.
[Messoud A. Efendiev (Berlin)]
MSC 2000:
*35Q35 Other equations arising in fluid mechanics
35B40 Asymptotic behavior of solutions of PDE
76B15 Wave motions (fluid mechanics)
76B03 Existence, uniqueness, and regularity theory

Keywords: uniqueness; larger time behaviour; weak solutions; shallow water equation; Camass-Holm equation

Cited in: Zbl 1186.35003

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