Fractal first order partial differential equations
نویسندگان
چکیده
The present paper is concerned with semilinear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local HamiltonJacobi equations. The idea is to combine an integral representation of the operator and Duhamel’s formula to prove, on the one side, the key a priori estimates for the scalar conservation law and the Hamilton-Jacobi equation and, on the other side, the smoothing effect of the operator. As far as Hamilton-Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law. Mathematical subject classifications: 35B45, 35B65, 35A35, 35S30
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