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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

t u+u x u+ x P=0,t>0,x 1 P(t,x)=1 2 - e -|x-y| u 2 + 1 2 ( x u) 2 (t,y)dyu(0,x)=u 0 (x)H 1 ( 1 ),

which is formally equivalent to the Camass-Holm equation

t u- x 2 t u+3u x u=2 x u x 2 u+u x 3 u·

Moreover, the authors show that the admissible weak solutions (under some additional condition on the solutions) tend to 0 pointwisely as t.


MSC:
35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03Existence, uniqueness, and regularity theory (fluid mechanics)