Sharp energy estimates for nonlinear fractional diffusion equations

Sharp Energy Estimates for Nonlinear Fractional Diffusion Equations

We study the nonlinear fractional equation (−∆)u = f(u) in R, for all fractions 0 < s < 1 and all nonlinearities f . For every fractional power s ∈ (0, 1), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional ...

Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Lévy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinea...

Nonlinear Elliptic Equations with Fractional Diffusion

Quantitative Local and Global A Priori Estimates for Fractional Nonlinear Diffusion Equations

We establish quantitative estimates for solutions u(t, x) to the fractional nonlinear diffusion equation, ∂tu + (−∆)(u) = 0 in the whole range of exponents m > 0, 0 < s < 1. The equation is posed in the whole space x ∈ R. We first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data. In the core of the paper we obtain quantitative po...

SHARP Lp ESTIMATES FOR SINGULAR TRANSPORT EQUATIONS

We prove that L estimates for a singular transport equation are sharp by building what we call a cascading solution; the equation we consider studies the combined effect of multiplying by a bounded function and application of the Hilbert transform. Along the way we prove an invariance result for the Hilbert transform which could be of independent interest. Finally, we give an example of a bound...