Related papers: Uniqueness issues for evolution equations with den…
Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution…
We consider variational inequality solutions with prescribed gradient constraints for first order linear boundary value problems. For operators with coefficients only in $L^2$, we show the existence and uniqueness of the solution by using a…
We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. The existence and uniqueness of the solution are shown. We prove that as the competition rate goes to infinity…
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…
We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the…
We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first…
We consider a path-dependent Hamilton--Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test…
We are interested in the uniqueness of solutions of a nonlinear, pseudomonotone, stochastic diffusion evolution problem with homogeneous Dirichlet boundary conditions with reflection, where the noise term is additive and given by a…
We develop deterministic particle schemes to solve non-local scalar conservation laws with congestion. We show that the discrete approximations converge to the unique entropy solution with an explicit rate of convergence under more general…
We study the problem of coupling Einstein's equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a…
We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential…
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions…
We prove an exponential deviation inequality for the convex hull of a finite sample of i.i.d. random points with a density supported on an arbitrary convex body in $\R^d$, $d\geq 2$. When the density is uniform, our result yields rate…
Much effort has been spent in recent years on restoring uniqueness of McKean-Vlasov SDEs with non-smooth coefficients. As a typical instance, the velocity field is assumed to be bounded and measurable in its space variable and…
We study the large population limit of the Moran process, assuming weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral…
We study a McKendrick-von Foerster type equation with renewal. This model is represented by a single equation which describes one species which produces young individuals. The renewal condition is linear, but takes into account some history…
We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist…
We analyze a coupled Cahn-Hilliard-Forchheimer system featuring concentration-dependent mobility, mass source and convective transport. The velocity field is governed by a generalized quasi-incompressible Forchheimer equation with…
We prove the existence of nonnegative weak solutions to a class of second and fourth order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space $\mathbb{R}^d$, for…
We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighbourhood. In the first-order model the location of each…