10 research outputs found
Local and global behaviour of nonlinear equations with natural growth terms
This paper concerns a study of the pointwise behaviour of positive solutions
to certain quasi-linear elliptic equations with natural growth terms, under
minimal regularity assumptions on the underlying coefficients. Our primary
results consist of optimal pointwise estimates for positive solutions of such
equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto
Riesz potentials and nonlinear parabolic equations
The spatial gradient of solutions to nonlinear degenerate parabolic equations
can be pointwise estimated by the caloric Riesz potential of the right hand
side datum, exactly as in the case of the heat equation. Heat kernels type
estimates persist in the nonlinear cas
Blow-up of critical Besov norms at a potential Navier-Stokes singularity
We show that the spatial norm of any strong Navier-Stokes solution in the space X must become unbounded near a singularity, where X may be any critical homogeneous Besov space in which local existence of strong solutions to the 3-d Navier-Stokes system is known. In particular, the regularity of these spaces can be arbitrarily close to -1, which is the lowest regularity of any Navier-Stokes critical space. This extends a well-known result of Escauriaza-Seregin-Sverak (2003) concerning the Lebesgue space , a critical space with regularity 0 which is continuously embedded into the spaces we consider. We follow the "critical element" reductio ad absurdum method of Kenig-Merle based on profile decompositions, but due to the low regularity of the spaces considered we rely on an iterative algorithm to improve low-regularity bounds on solutions to bounds on a part of the solution in spaces with positive regularity
Recent advances in nonlinear potential theory
Recent developments in Nonlinear Potential Theory show the possibility of estimating solutions to nonlinear degenerate equations via potentials. We give a brief description of these results
Nonlinear Measure Data Problems
We describe some basic results from regularity theory for solutions to elliptic quasilinear equations involving an assigned measure datum and we include some new integrability and differentiability results for sublinear problem