318 research outputs found
Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics
We use De Giorgi techniques to prove H\"older continuity of weak solutions to
a class of drift-diffusion equations, with initial data and divergence
free drift velocity that lies in . We apply this
result to prove global regularity for a family of active scalar equations which
includes the advection-diffusion equation that has been proposed by Moffatt in
the context of magnetostrophic turbulence in the Earth's fluid core.Comment: To appear in Annales de l'Institut Henri Poincare - Analyse non
lineair
A minimum problem with free boundary in Orlicz spaces
We consider the optimization problem of minimizing in the class of functions with
, for a given and bounded.
is the class of weakly differentiable functions with
. The conditions on the function G allow
for a different behavior at 0 and at . We prove that every solution u
is locally Lipschitz continuous, that they are solution to a free boundary
problem and that the free boundary, , is a regular
surface. Also, we introduce the notion of weak solution to the free boundary
problem solved by the minimizers and prove the Lipschitz regularity of the weak
solutions and the regularity of their free boundaries near
``flat'' free boundary points
Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient
In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray-Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis
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