We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminarto-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate line...

A one parameter family of equations is considered which connects the well-known Perona-Malik equation to standard diffusion. The parameter acts as a regularization parameter which gradually modifies the ill-posed Perona-Malik equation, through a strongly locally well-posed equation, to a strongly globally well-posed one which exhibits a behavior akin to that of standard diffusion, which is itse...

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation ut = −(uuxxx)x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u0(x) ∈ H 1(IR), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L1-norm of the strong...

In this paper we consider the null controllability for a population model depending on time, space and age. Moreover, diffusion coefficient degenerate at boundary of domain. The novelty is that first time presence memory term, which makes computations more difficult. However, under suitable condition kernel deduce result original problem via new Carleman estimates adjoint associated to nonhomog...

We prove existence of the largest and smallest entropy solutions to Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish comparison principle in case when at least one initial functions is periodic. In function vanishes infinity (in sense strong average) long time decay an solution under exact nonlinearity-diffusivity condition.

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve result, we introduce new variant of micro-local defect functionals which are able ‘recognise’ changes type equation. As corollary, show existence weak solution for Cauchy problem non-linear degenerate parabolic flux. also strong traces a...

We consider the theory of erosion and investigate connections to the theory of optimal transport. The mathematical theory of erosion is based on two coupled nonlinear partial differential equations. The first one describing the water flow can be considered to be an averaged NavierStokes equation, and the second one describing the sediment flow is a very degenerate nonlinear parabolic equation. ...

These notes contain a short exposition of selected results about parabolic equations: Schauder estimates for linear parabolic equations with Hölder coefficients, some existence, uniqueness and regularity results for viscosity solutions of fully nonlinear parabolic equations (including degenerate ones), the Harnack inequality for fully nonlinear uniformly parabolic equations. MSC. 35K55, 35D40, ...

The null controllability of parabolic operators in bounded domains, with both boundary or locally distributed controls, is a well-established property, see, e.g., (Bensoussan et al., 1993) and (Fattorini, 1998). Such a property brakes down, however, for degenerate parabolic operators even when degeneracy occurs on ”small” subsets of the space domain, such as subsets of the boundary. This talk w...

Abstract. We propose a new notion of weak solutions (dissipative solutions) for nonisotropic, degenerate, second order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic,...