Related papers: Local boundedness and higher integrability for the…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the…
In this work we consider weak solutions of the incompressible 2-D porous media equation. By using the approach of De Lellis-Sz\'ekelyhidi we prove non-uniqueness for solutions in $L^\infty$ in space and time.
We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.
Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[…
In this paper, we show that the minimal solution of a backward stochastic differential equation gives a probabilistic representation of the minimal viscosity solution of an integro-partial differential equation both with a singular terminal…
In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having $p$-structure depending only on the symmetric part of the gradient.
We aim at providing a characterization of the ability to maintain a stochastic coupled system with porous media components in a prescribed set of constraints by using internal controls. This property is proven via a quasi-tangency…
Using the generalized variational framework, the strong/weak existence and uniqueness of solutions are derived for a class of distribution dependent stochastic porous media equations on general measure spaces, which also extends the…
A system of quasilinear elliptic equations on an unbounded domain is considered. The existence of a sequence of radially symmetric weak solutions is proved via variational methods.
The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied. To this end, a rescaled zero-range…
We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type…
We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for…
A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space $L^2(H,\nu)$ where $\nu$ is an infinitesimally excessive measure. Then a weak solution is…
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the…
We prove that the standard conditions that provide unique solvability of a mixed stochastic differential equations also guarantee that its solution possesses finite moments. We also present conditions supplying existence of exponential…
We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This…
We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.
We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term.…
One proves existence and uniqueness of strong solutions to stochastic porous media equations under minimal monotonicity conditions on the nonlinearity. In particular, we do not assume continuity of the drift or any growth condition at…