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This paper is devoted to a model of magnetohydrodynamics described by a parabolic system of partial differential equations coupling the nonhomogeneous incompressible Navier--Stokes equations and Maxwell's equations. In the case of two-dimensional flows, we prove global regularity results under the assumption that the fluids' viscosities are close enough to their average. On the other hand, a more detailed description of the interface and of the regularity of the third component of the magnetic field is given when the fluids have the same viscosity.
We show that smooth solutions of porous medium equations satisfy a simple $L^2$ gradient estimate on sets where the solution itself is small. Along with known continuity estimates for solutions and an estimate on second derivatives of smooth solutions, this estimate allows us to show that approximating smooth solutions of a porous medium equation converge strongly to the weak solution.
We prove the existence of weak solutions that belong to some local Sobolev space of a quasilinear elliptic problem in an exterior domain. The free-boundary problem considered here arises from the discontinuous behavior of the nonlinearity involved. The method of upper and lower solutions, extremality results for quasilinear elliptic problems in bounded domains, gradient estimates and abstract fixed point principles in partially ordered sets are the main tools used in the proof of our main result. An application to a superlinear discontinuous elliptic problem with the p-Laplacian is given.
A threshold result on the global dynamics of the scalar asymptotically periodic Kolmogorov equation is proved and then applied to models of single-species growth and $n$-species competition in a periodically operated chemostat. The operating parameters and the species-specific response functions can be periodic functions of time. Species-specific removal rates are also permitted. Sufficient conditions ensure uniform persistence of all of the species and guarantee that the full system admits at least one positive, periodic solution. In the special case when the species-specific removal rates are all equal to the dilution rate, the single-species growth model has a threshold between global extinction and uniform persistence, in the form of a positive, periodic coexistence state. Improved results in the case of 3-species competition are also given, including an example illustrating competition-mediated coexistence of three species.
We propose a variational approach to the method of moving planes which easily applies to quasilinear equations of type (1-1) with $f$ locally Lipschitz continuous. To do this we use a characterization of Lipschitz continuous functions which allows us to get symmetry results without writing an equation for the difference between the solution and its reflection.
We provide a method for the count and the variational characterization of periodic solutions to forced pendulum-type equation. This is achieved by reducing the problem to the study of a real function of one variable. This paper generalizes some results obtained by G. Tarantello.