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Register a new userConvergence analysis of upwind type schemes for the aggregation equation with pointy potential
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Franc{c}ois Delarue
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A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order 1/2 in Wasserstein distance. This result is illustrated by numerical simulations, which show the optimality of the order of convergence.
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This work is devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one dimensional space. The aggregation equation is by now widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigate an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigate the numerical discretization of this relaxation limit by uniformly accurate schemes.
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In this paper, we show the scattering of the solution for the focusing inhomogenous nonlinear Schrodinger equation with a potential begin{align*} ipartial_t u+Delta u- Vu=-|x|^{-b}|u|^{p-1}u end{align*} in the energy space $H^1(mathbb R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory.
The Special Lagrangian Potential Equation for a function $u$ on a domain $Omegasubset {bf R}^n$ is given by ${rm tr}{arctan(D^2 ,u) } = theta$ for a contant $theta in (-n {piover 2}, n {piover 2})$. For $C^2$ solutions the graph of $Du$ in $Omegatimes {bf R}^n$ is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem for this equation, but the existence result relies on explicitly computing the associated boundary conditions (or, otherwise said, computing the pseudo-convexity for the associated potential theory). This is done in this paper, and the answer is interesting. The result carries over to many related equations -- for example, those obtained by taking $sum_k arctan, lambda_k^{{mathfrak g}} = theta$ where ${{mathfrak g}} : {rm Sym}^2({bf R}^n)to {bf R}$ is a Garding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformed Hermitian-Yang-Mills equation which appears in mirror symmetry. Another example is $sum_j arctan kappa_j = theta$ where $kappa_1, ... , kappa_n$ are the principal curvatures of the graph of $u$ in $Omegatimes {bf R}$. We also discuss the inhomogeneous Dirichlet Problem ${rm tr}{arctan(D^2_x ,u)} = psi(x)$ where $psi : overline{Omega}to (-n {piover 2}, n {piover 2})$. This equation has the feature that the pull-back of $psi$ to the Lagrangian submanifold $Lequiv {rm graph}(Du)$ is the phase function $theta$ of the tangent spaces of $L$. On $L$ it satisfies the equation $ abla psi = -JH$ where $H$ is the mean curvature vector field of $L$.
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