We prove some maximum and gradient estimates for classical solutions to a wide class of quasilinear degenerate parabolic equations, including first order ones. The proof is elementary and exploits the smallness of the domain in the time direction.

This paper studies the rst order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial di erential equations with pointwise state constraints. Pontryagin type maximum principle is obtained.

We develop further in this work the high order paracontrolled calculus setting to deal with analytic part of study quasilinear singular PDEs. Continuity results for a number operators are proved that purpose. Unlike regularity structures approach subject by Gerencser & Hairer and Otto, Sauer, Smith Weber, or Furlan Gubinelli’ two dimensional parabolic Anderson model equation, we do not use para...

In this paper, we study the system of equations a boundary layer for nonlinearly viscous, electrically conductive liquid described by rheological law proposed O. A. Ladyzhenskaya incompressible media. The boundary-layer model were first obtained from Prandtl’s axioms. By Mises transform, can be reduced to single quasilinear equation. main method used in paper is Crocco which turns into degenera...

The existence of solutions for quasilinear parabolic equation with boundary conditions and initial conditions can be discussed by maximal regularity, and more and more works on this field show that the maximal regularity method is efficient. Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to t...

be linear and quasilinear evolution equations of parabolic type in a Banach space X respectively. By "parabolic type" we mean that A(t) and A(t,u) are all the infinitesimal generators of analytic linear semigroups on X we do not necessarily assume that the domains of the operators A(t) and A("t,u) are dense subspaces of X, so the semigroups generated by them may not be of class c0 J. The domain...

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-sca...

We prove two different types of comparison results between semicontinuous viscosity suband supersolutions of the generalized Dirichlet problem (in the sense of viscosity solutions theory) for quasilinear parabolic equations: the first one is an extension of the Strong Comparison Result obtained previously by the second author for annular domains, to domains with a more complicated geometry. The...