Related papers: A viscosity approach to degenerate complex Monge-A…
This paper is devoted to the global existence of weak solutions to the three-dimensional compressible Navier-Stokes equations with heat-conducting effects in a bounded domain. The viscosity and the heat conductivity coefficients are assumed…
We consider a scalar, possibly degenerate parabolic equation with a source term, in several space dimensions. For initial data with bounded variation we prove the existence of solutions to the initial-value problem. Then we show that these…
We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity…
Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds,…
We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations.
We show a very general existence theorem to the complex Monge-Amp\`ere type equation on hyperconvex domains.
This paper is devoted to study the following degenerate Monge-Amp\`ere equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=\Lambda_q (-u)^q \quad \text{in}\quad \Omega,\\ u=0 \quad\text{on}\quad \partial\Omega \end{cases}…
We show several results on convergence of the Monte Carlo method applied to consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our method is based on combination of several new concepts. We…
In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic…
In this paper, we considered the isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosities in $\mathbb{R}^3$. These systems come from the Boltzmann equations through the Chapman-Enskog expansion to the…
Here, we address a uniqueness structure of viscosity solutions for ergodic problems of weakly coupled Hamilton-Jacobi systems. In particular, we study comparison principle with respect to generalized Mather measures as a generalization of…
We show removability of half-line singularities for viscosity solutions of fully nonlinear elliptic PDEs which have classical density and a Jacobi inequality. An example of such a PDE is the Monge-Amp\`ere equation, and the original proof…
By using the global deformation of almost complex structures which are compatible with a symplectic form off a Lebesgue measure zero subset, we construct a (measurable) Lipschitz Kahler metric such that the one-form type Calabi-Yau equation…
This paper is devoted to a complete classification on the existence and nonexistence results of viscosity solutions to the general Dirichlet problem for a class of eigenvalue type equations. With the distance function included in the…
For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme…
In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids with viscosities depending on density in a power law and allowing initial vacuum. We introduce the notion of regular solutions and…
In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous…
This paper is dedicated to the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows…
We establish the existence and compactness of global martingale entropy solutions with finite relative-energy for the stochastically forced system of isentropic Euler equations governed by a general pressure law. To achieve these, a…
We study the parabolic complex Monge-Amp\`ere type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a…