We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be or singular. Taking 0 as the point where strong degeneracy occurs, we assume coefficient a = ( x ) in principal part of equation is such function → / L p (0,1) for some > 1. After establishing spectral estimates corresponding elliptic problem, prove null controllable energy spa...

Abstract. It is shown that there exists a weak solution to a degenerate and singular elliptic-parabolic partial integro-differential system of equations. These equations model two-phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium. The full model is of dual-porosity type, though the single porosity case is covered by sett...

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγ ρ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein t...

Nonlinear degenerate parabolic-hyperbolic equations are one of the most important classes of nonlinear partial differential equations. Nonlinearity and degeneracy are two main features of these equations and yield several striking phenomena that require new mathematical ideas, approaches, and theories. On the other hand, because of the importance of these equations in applications, there is a l...

In this paper, we prove a convergence theorem for singular perturbations problems class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to degenerate PDEs. Our approach mainly based on G-stochastic analysis argument. As byproduct, also establish averaging principle stochastic driven by G-Brow...

We study the existence theory for parabolic variational inequalities in weighted L spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L setting allows us to cover some singular cases...

We study the existence theory for parabolic variational inequalities in weighted L spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L setting allows us to cover some singular cases...