Notes on some recent methods in bifurcation theory
Nouvelles estimations pour les solutions d'équations aux dérivées partielles non linéaires
Novel criteria on global robust exponential stability to a class of reaction-diffusion neural networks with delays.
Novel method for generalized stability analysis of nonlinear impulsive evolution equations
In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value...
Null form estimates for symbols and local existence for a quasilinear dirichlet-wave equation
Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system
We prove almost optimal local well-posedness for the coupled Dirac–Klein–Gordon (DKG) system of equations in dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman–Machedon type. It has been known for some time that the Klein–Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.
Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations
We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing...
Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations
We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with...
Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations
We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector (with...
Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions
Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers’ equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this...
Numerical blow-up for the -Laplacian equation with a nonlinear source
Numerical blow-up solutions for some semilinear heat equations.
Numerical blow-up time for a semilinear parabolic equation with nonlinear boundary conditions.
Numerical homogenization: survey, new results, and perspectives
These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the...
Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium
The paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach...
Numerical solutions of a fractional predator-prey system.
Numerical study on the blow-up rate to a quasilinear parabolic equation
In this paper, we consider the blow-up solutions for a quasilinear parabolic partial differential equation . We numerically investigate the blow-up rates of these solutions by using a numerical method which is recently proposed by the authors [3].