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

*(English)*Zbl 1090.35142We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis-Procesi equation

$${\partial}_{t}u-{\partial}_{txx}^{3}u+4u{\partial}_{x}u=3{\partial}_{x}u{\partial}_{xx}^{2}u+u{\partial}_{xxx}^{3}u\xb7\phantom{\rule{2.em}{0ex}}\left(\mathrm{DP}\right)$$

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 systems, integrability tests, bi-Hamiltonian structures, hierarchies |

35Q58 | Other completely integrable PDE (MSC2000) |

35L30 | Higher order hyperbolic equations, initial value problems |