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In this paper, we consider the global existence, uniqueness and estimates of weak solutions to quasilinear parabolic equation of -Laplacian type in with zero Dirichlet boundary condition in . Further, we obtain the estimate of the solution and for with the initial data
We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem.
We prove the local in time existence of solutions for an aggregation equation in Besov spaces. The Fourier localization technique and Littlewood-Paley theory are the main tools used in the proof.
We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data , and for and any . The initial regularity of the micro-rotational velocity is weaker than velocity of the fluid .
We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation
corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation
We then give conditions for the convergence, as , of the solution of the evolution equation to its stationary state.
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