Some nonlocal operators in porous medium equationthe extension problem and regularity theory

Résumé

In this PhD Thesis we shall deal with problems related to fractional nonlocal operators, namely to the fractional Laplacian and to some other types of fractional derivatives and their usefulness in the study of nonlinear diffusion phenomena such gas, fluid in porous medium. First of all, we shall make an extensive introduction to the fractional derivative, present some related contemporary research results, and we will add some original material. We shall study the regularity theory for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel, passing through existence of weak solutions. Also, to point out that the (nonlocal) character of the fractional Laplacian and fractional derivative gives rise to some surprising nonlocal effects, we will study the existence, uniqueness and regularity properties for the time fractional porous medium equation. Finally we shall also develop a theory of existence, uniqueness and regularity for the time fractional porous medium equation with fractional diffusion using the so-called Caffarelli-Silvestre extension.In this PhD Thesis we shall deal with problems related to fractional nonlocal operators, namely to the fractional Laplacian and to some other types of fractional derivatives and their usefulness in the study of nonlinear diffusion phenomena such gas, fluid in porous medium. First of all, we shall make an extensive introduction to the fractional derivative, present some related contemporary research results, and we will add some original material. We shall study the regularity theory for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel, passing through existence of weak solutions. Also, to point out that the (nonlocal) character of the fractional Laplacian and fractional derivative gives rise to some surprising nonlocal effects, we will study the existence, uniqueness and regularity properties for the time fractional porous medium equation. Finally we shall also develop a theory of existence, uniqueness and regularity for the time fractional porous medium equation with fractional diffusion using the so-called Caffarelli-Silvestre extension.