We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation $$\pt u-\ptxx u+4u\px u =3\px u\pxx u +u\pxxx u. \tag{DP}$$ 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 KdV 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.

### On the well-posedness of the Degasperis-Procesi equation

#### Abstract

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation $$\pt u-\ptxx u+4u\px u =3\px u\pxx u +u\pxxx u. \tag{DP}$$ 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 KdV 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.
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2006
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/12916
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