Related papers: On the loss of continuity for super-critical drift…
We consider a nonlocal nonlinear model with fractional diffusion motivated by studies of electroconvection phenomena in incompressible viscous fluids. We address the global well-posedness, global regularity and long time dynamics of the…
Transport phenomena plays an important role in science and technology. In the wide variety of applications both advection and diffusion may appear. Regarding diffusion, for long times, different type of decay rates are possible for…
In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in…
The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta)$ is investigated for the singular diffusion equation with critical gradient…
Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur showed that a certain class of weak solutions to the drift diffusion equation with initial data in $L^2$ gain H\"older continuity provided that the BMO norm of the drift…
We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is…
We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation…
Using the method of Krylov's estimates, we prove the existence of weak solutions of stochastic differential equations driven by purely discontinuous Levy processes satisfying an additional assumption. The diffusion coefficient is assumed to…
We establish well-posedness and regularity for parabolic diffusion equation in the case when the singularities of a general drift reach the critical magnitude. The latter dictates the need to work in an Orlicz space situated between all…
The aim of this paper is to study the generalized Fisher-KPP equation with nonlocal diffusion. In specific we prove the existence of a critical speed so that traveling front type solutions exist up to this critical speed and non-existence…
The paper studies a class of multidimensional optimal stopping problems with infinite horizon for linear switching diffusions. There are two main novelties in the optimal problems considered: the underlying stochastic process has…
We reveal that realistic fluids generate microscopic-level discontinuity constantly and the discontinuity spreads out with motion of particles rather rapidly and widely. These things cannot be treated by the standard kinetic equations, and…
We consider instabilities of a single mode with finite wavenumber in inversion symmetric spatially one dimensional systems, where the character of the bifurcation changes from sub- to supercritical behaviour. Starting from a general…
We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to…
We deal with the problem of separation of time-scales and filamentation in a linear drift-diffusion problem posed on the whole space $\mathbb{R}^2$. The passive scalar considered is stirred by an incompressible flow with radial symmetry. We…
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost…
Consider the Cauchy problem for the radial cubic wave equation in 1+3 dimensions with either the focusing or defocusing sign. This problem is critical in $\dot{H}^{\frac{1}{2}} \times \dot{H}^{-\frac{1}{2}}$ and subcritical with respect to…
It is known that classical solutions to the one-dimensional quasilinear Smoluchowski-Poisson system with nonlinear diffusion $a(u)=(1+u)^{-p}$ may blow up in finite time if $p>1$ and exist globally if $p<1$. The case $p=1$ thus appears to…
The global in-time semiclassical and relaxation limits of the bipolar quantum hydrodynamic model for semiconductors are investigated in $R^3$. We prove that the unique strong solution converges globally in time to the strong solution of…
We study the existence and infinite-speed propagation of solutions to models arising in porous media, when the mobility is highly degenerate (inverse power law). The approach is based on maximum principles for the fractional Laplacian, and…