Related papers: Spectral viscosity method with generalized Hermite…
We propose, analyze, and demonstrate a discontinuous Galerkin method for fractal conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear…
Recent work giving a classification of kinematic and vorticity conservation laws of compressible fluid flow for barotropic equations of state (where pressure is a function only of the fluid density) in $n>1$ spatial dimensions is extended…
In hyperspectral sparse unmixing, a successful approach employs spectral bundles to address the variability of the endmembers in the spatial domain. However, the regularization penalties usually employed aggregate substantial computational…
This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy…
In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $\Phi$,…
In this article we describe the applications of the relative entropy framework. In particular uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey…
A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant $\gamma\in [1,3)$. This formulation allows a family of invariant regions in the phase plane for…
Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic…
We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [8]. With the new definition, we prove the two important results till now…
This paper is devoted to a viscosity solution theory of the stochastic Hamilton-Jacobi-Bellman equation in the Wasserstein spaces for the mean-field type control problem which allows for random coefficients and may thus be non-Markovian.…
In this article we find the solution of the Burger equation with viscosity applying the boundary layer theory. In addition, we will observe that the solution of Burger equation with viscosity converge to the solution of Burger stationary…
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We…
In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function $f$ has on the entropy solution. More precisely, if the set $\{w:f''(w)\ne 0\}$ is dense,…
We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the…
We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on…
A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and…
We present a quantitative compensated compactness estimate for stochastic conservation laws, which generalises a previous result of Golse & Perthame (2013) for deterministic equations. With a stochastic modification of Kruzkov's…
Why reaction rate constants for enzymatic reactions are typically inversely proportional to fractional power exponents of solvent viscosity remains to be already a thirty years old puzzle. Available interpretations of the phenomenon invoke…
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…
The Van Leer approach for the approximation of nonlinear scalar conservation laws is studied in one space dimension. The problem can be reduced to a nonlinear interpolation and we propose a convexity property for the interpolated values. We…