Related papers: Degenerate diffusion with a drift potential: a vis…
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of…
The earlier paper [LK] (arXiv:0910.3432) contains a lower bound of the solution in terms of its $L^1$ norm, which is incorrect. In this note we explain the mistake and present a correction to it under the restriction that the permeability…
In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.
We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of…
In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the…
We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection…
We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are…
The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is…
This paper is concerned with a strongly degenerate convection-diffusion equation in one space dimension whose convective flux involves a non-linear function of the total mass to one side of the given position. This equation can be…
This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve uniform H\"{o}lder regularity in terms of $L^p$-bound on the…
This is the sequel of the recent work (Du, Huang, Pu, Commun. Math. Phys, 2023, doi: 10.1007/s00220-023-04651-7) on axially symmetric gravity water waves with general vorticities, which has investigated the singular wave profile of the free…
We establish H\"older regularity for the weak solution to a degenerate diffusion equation in the presence of a local (drift) potential and nonlocal (interaction) term, posed in a bounded domain with no-flux boundary conditions. The…
We study the long-time dynamics of the nonlinear processes modeled by diffusion-transport partial differential equations in non-divergence form with drifts. The solutions are subject to some inhomogeneous Dirichlet boundary condition.…
In this paper a strongly degenerate parabolic equation derived from a density dependent particle flow model is studied. Furthermore, a free boundary problem and its connection to the strongly degenerate parabolic equation is investigated.…
In this work we study the degenerate diffusion equation $\partial_{t}=x^{\alpha}a\left(x\right)\partial_{x}^{2}+b\left(x\right)\partial_{x}$ for $\left(x,t\right)\in\left(0,\infty\right)^{2}$, equipped with a Cauchy initial data and the…
In this work, we study convection-diffusion equations in the cases of bounded drifts and drifts induced by the gradient of a potential. We define a new notion of solution and prove its existence and uniqueness. Furthermore, we show the…
The theory of degenerate parabolic equations of the forms \[ u_t=(\Phi(u_x))_{x} \quad {\rm and} \quad v_{t}=(\Phi(v))_{xx} \] is used to analyze the process of contour enhancement in image processing, based on the evolution model of…
We consider a spatially homogeneous advection-diffusion equation in which the diffusion tensor and drift velocity are time-independent, but otherwise general. We derive asymptotic expressions, valid at large distances from a steady point…
In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity…
We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in \cite{MonsaingonNovikovRoquejoffre}. We set up a finite difference scheme…