Fractal First-Order Partial Differential Equations

Abstract

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 Hamilton-Jacobi 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, 35S30Key words. Lévy operator -fractal conservation laws -maximum principle -non-local Hamilton-Jacobi equations -smoothing effect -a priori estimates -global-in-time existence -rate of convergence.

Fractal Hamilton-Jacobi-KPZ equations

“…✷ Remark 2.2. After this paper was completed we received a preprint of [11] which studied a nonlinear-nonlocal viscous Hamilton-Jacobi equation of the form…”

“…Here, the theory of the viscosity solutions provides a good framework to study these equations. We refer the reader to the works of Jakobsen and Karlsen [15,16], and Droniou and Imbert [13,11] for more detailed information and references. Fractional conservation laws, including the fractional Burgers equation, were studied in [5,17,18,22] via probabilistic techniques such as nonlinear McKean processes and interacting diffusing particle systems.…”

Decay of mass for fractional evolution equation with memory term

“…More details are given in a recent paper of Roberts and Olmstead [23]. Furthermore, nonlinear evolution problems involving fractional Laplacian describing the anomalous diffusion (or β-stable Lévy diffusion) have been extensively studied in the mathematical and physical literature [2,6,13]. One of the possible ways to understand the interaction between the anomalous diffusion operator (given by (−∆) β/2 or, more generally, by the Lévy diffusion operator) and the nonlinearity in the equation (1.1) is the study of the large time asymptotics of solutions to such equations.…”

Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

“…The fractional dissipation operator severs to model many physical phenomena (see [17]) in hydrodynamics and molecular biology such as anomalous diffusion in semiconductor growth (see [36]). We remark the convention that by α = 0 we mean that there is no dissipation in (1.1) 1 , and similarly β = 0 represents that there is no dissipation in (1.1) 2 .…”