convergence of minimizers of a Ginzburg-Landau functional.
Let be a cylinder in and . It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator in the Morrey spaces , , , supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.
Let be a linear partial differential operator with holomorphic coefficients, whereandWe consider Cauchy problem with holomorphic dataWe can easily get a formal solution , bu in general it diverges. We show under some conditions that for any sector with the opening less that a constant determined by , there is a function holomorphic except on such that and as in .
The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation with the initial boundary value conditions or with the initial boundary value conditions are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.
As a model for elliptic boundary value problems, we consider the Dirichlet problem for an elliptic operator. Solutions have singular expansions near the conical points of the domain. We give formulas for the coefficients in these expansions.
We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac-like inequalities. As part of our methods, we give a different approach to a priori estimates of Foiaş, Guillopé and Temam.
In the context of suitable weak solutions to the Navier-Stokes equations we present local conditions of Prodi-Serrin’s type on velocity and pressure under which is a regular point of . The conditions are imposed exclusively on the outside of a sufficiently narrow space-time paraboloid with the vertex and the axis parallel with the -axis.