Related papers: On gradient stability in nonlinear PDE models and …
We study the existence and the rate of equilibration of weak solutions to a two-component system of non-linear diffusion-aggregation equations, with small cross diffusion effects. The aggregation term is assumed to be purely attractive, and…
Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to…
We develop a new finite element method for solving planar elasticity problems involving of heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the `broken'…
In the previous paper \cite{J1}, we established pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves $\bar u$ of a system of reaction diffusion equations, and also obtained…
We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron…
We present an exact solution to an interface model representing the dynamics of a domain wall in a two-phase Ising system. The model is microscopically motivated, yet we find that in the scaling regime our results are consistent with those…
This paper is devoted to the problem of approximating non-linear Stochastic Partial Differential Equations (SPDEs) via interacting particle systems. In particular, we consider the Stochastic McKean-Vlasov equation, which is the…
Inspired by the gradient flow viewpoint of the Landau equation and the corresponding dynamic formulation of the Landau metric in [arXiv:2007.08591], we develop a novel implicit particle method for the Landau equation in the framework of the…
In this paper we focus on the pathwise stability of mild solutions for a class of stochastic partial differential equations which are driven by switching-diffusion processes with jumps. In comparison to the existing literature, we show…
For inhomogeneous systems with interfaces, the inclusion of long-range dispersion interactions is necessary to achieve consistency between molecular simulation calculations and experimental results. For accurate and efficient incorporation…
We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction-diffusion equations with periodic source terms, in one spatial dimension. We formulate an \textit{a priori} condition for the…
In this paper, we study the polynomial stability of analytical solution and convergence of the semi-implicit Euler method for non-linear stochastic pantograph differential equations. Firstly, the sufficient conditions for solutions to grow…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
We consider kinetic systems and prove their stability working in weighted spaces in which the systems are symmetric. We prove stability for various explicit and implicit semi-discrete and fully discrete schemes. The applications include…
We present a statistical learning framework for robust identification of partial differential equations from noisy spatiotemporal data. Extending previous sparse regression approaches for inferring PDE models from simulated data, we address…
The quasi-neutral hybrid model with kinetic ions and fluid electrons is a promising approach for bridging the inherent multi-scale nature of many problems in space and laboratory plasmas. Here, a novel, implicit, particle-in-cell based…
We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the…
We deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolution form. We describe the case of a convex (or…
We consider a lattice gas with general short range interactions and a Kac potential $J_\gamma({r})$ of range $\gamma^{-1}$, $\gamma>0$, evolving via particles hopping to nearest neighbor empty sites with rates which satisfy detailed balance…
In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully…