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On the well-posedness of the Degasperis-Procesi equation. (English) Zbl 1090.35142
We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation \[ \partial_tu-\partial^3_{txx}u+4u\partial_xu=3\partial_xu\partial^2_{xx}u+u\partial^3_{xxx}u.\tag{DP} \] This equation can be regarded as a model for shallow water dynamics and its asymptotic accuracy is the same as for the Camassa-Holm equation (one order more accurate than the Korteweg-de Vries equation). We prove existence and \(L^1\) stability (uniqueness) results for entropy weak solutions belonging to the class \(L^1\cap BV\), while existence of at least one weak solution, satisfying a restricted set of entropy inequalities, is proved in the class \(L^2\cap L^4\). Finally, we extend our results to a class of generalized Degasperis-Procesi equations.

MSC:
35Q35 PDEs in connection with fluid mechanics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
35L30 Initial value problems for higher-order hyperbolic equations
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