Noticias en español
Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities.
Abstract: The problem of the asymptotic stability of kinks in classical nonlinear scalar field equations on the real line leads to the study of the decay of small solutions to 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities. I will discuss the occurrence of a novel modified scattering behavior of such solutions that involves a logarithmic slow-down of the decay rate along certain rays. It is caused by a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. This talk is based...
Read MoreGevrey regularity for the Vlasov-Poisson system.
Abstract: The importance of understanding high regularity solutions for the Vlasov-Poisson system has been underscored by the work on Landau damping of Mouhot-Villani and several other follow ups. These works on Landau damping make use of the propagation in time of high order regularity in a perturbative regime around homogeneous stationary solutions of the Vlasov-Poisson system. In this talk, we prove a general result on propagation of Gevrey regularity for the Vlasov-Poisson system on $\T^d\times \R^d$ using a Fourier space method in analogy to results proved for the 2D-Euler system by...
Read MoreModified scattering for the nonlinear Schrödinger equation in an external field.
Abstract: In this talk, we consider the cubic nonlinear Schrödinger equation with an external potential. We prove the existence of modified scattering for this model, that is, linear scattering modulated by a phase. Our approach is based on the spectral theorem for the perturbed linear Schrödinger operator and a factorization technique, which allows us to control the resonant nonlinear term. This approach requires a detailed and subtle study of the low-energy properties of the scattering data. Therefore, we study the cases of generic and exceptional potentials separately. The exceptional...
Read MoreSimulating Microenvironmental effects in tumor progression.
Abstract. In this presentation, we will see the preliminary numerical exploration of a mathematical approach that captures and explores a wide range of mechanisms and biological variabilities in tumor progression is presented. Precisely, the mathematical model captures cell-cell interactions including micro-environment effects in solid tumor progression. The biological principle consists in to assume that the tumor cells interact with Tumor-Associated Macrophages, and simultaneously with the extracellular matrix. Such two different mechanisms are modeled coupling the parabolic-elliptic...
Read MoreA regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem.
Abstract: We will study a bilinear optimal control problem associated to a 3D chemo-repulsion model with linear production. We prove the existence of weak solutions, and establish a regularity criterion to get global-in-time strong solutions. As a consequence, we deduce the existence of a global optimal solution with bilinear control, and, using a Lagrange multipliers theorem in Banach spaces, we derive first-order necessary optimality conditions for each local optimal solution.
Read MoreLong time asymptotics and soliton stability for the sine-Gordon equation.
Abstract: We compute the long time asymptotics of the sine-Gordon equation whose initial condition supports finitely many solitons(kinks/breathes). Our approach is the nonlinear steepest descent for Riemann-Hilbert problems. Through certain type of nonlinear Fourier transform, we characterize various quantities (radiation, phase shift, error estimates) by the regularity of the initial condition. We then extend our results to lower regularity spaces through a refined approximation argument. In particular, we characterize the initial condition that rules out wobbling kinks.
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