相关论文: Pluripotential solutions versus viscosity solution…
In this paper, we consider the Monge-Amp\`{e}re type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence…
We prove that if the modulus of continuity of a plurisubharmonic subsolution satisfies a Dini type condition then the Dirichlet problem for the complex Monge-Amp\`ere equation has the continuous solution. The modulus of continuity of the…
We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is…
A new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is…
We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of…
In this paper, we consider a version of parabolic complex Monge-Ampere equations, and use a PDE approach similar to Phong et al to establish $L^{\infty}$ and H\"older estimates. We also generalize the $L^{\infty}$ estimates to parabolic…
The aim of this paper is to further develop the theory of the degenerate complex Hessian equations on compact Hermitian manifolds. Building upon the generalization of the Bedford-Taylor pluripotential theory to complex Hessian equations by…
Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent…
Several theories for weakly damped free-surface flows have been formulated. In this paper we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to…
This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Amp\`ere equation $\det(D^2u^0)=f$ based on the vanishing moment method which was proposed recently by…
We show the existence and uniqueness of bounded solutions to the degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds. We also study the asymptotics of these solutions. As applications, we give partial answers to…
In this paper, we prove a uniform estimate for the modulus of continuity of solutions to degenerate complex Monge--Amp\`ere equation in big cohomology classes. This improves the previous results of Di Nezza--Lu and of the first author.
We deal with the equations of a planar magnetohydrodynamic compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity proportional to a positive power of the temperature. Under the same…
We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDEs given as a sup--envelope function. This is done through a dynamic programming principle derived from Denis, Hu, Peng (2010). The…
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some…
Let $\mu$ be a non-negative measure defined on bounded $\mathcal F$-hyperconvex domain $\Omega$. We are interested in giving sufficient conditions on $\mu$ such that we can find a plurifinely plurisubharmonic function satisfying $NP (dd^c…
Tropical toric varieties are partial compactifications of finite dimensional real vector spaces associated with rational polyhedral fans. We introduce plurisubharmonic functions and a Bedford--Taylor product for Lagerberg currents on open…
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…
In this work we introduce a viscosity-based notion of solution for general approximation schemes associated with partial differential equations, such as dynamic programming principles~(DPPs). A key feature of our approach is that it…
Multiscale mixed methods based on non-overlapping domain decompositions can efficiently handle the solution of significant subsurface flow problems in very heterogeneous formations of interest to the industry, especially when implemented on…