Existence and regularity results for terminal value problem for nonlinear fractional wave equations

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods are showed by applying the results to two interesting models: Time fractional Ginzburg-Landau equation, and Time fractional Burgers equation, where time and spatial regularity estimates are obtained

Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation

This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the $L_2$ and ${L_\infty }$ error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method

Burgers Turbulence

The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure

Freely floating objects on a fluid governed by the Boussinesq equations

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear non dispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has an interest of its own and for which we consequently provide a general analysis.Comment: 63 pages, 2 figure

Classical and Quantum Mechanical Models of Many-Particle Systems

This workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed

Multiscale Computation with Interpolating Wavelets

Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution representation of a function from its sample values on a finite set of points in space. We present a detailed study of the application of wavelet concepts to physical problems expressed in such bases. The manuscript describes algorithms for the associated transforms which, for properly constructed grids of variable resolution, compute correctly without having to introduce extra grid points. We demonstrate that for the application of local homogeneous operators in such bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtain less stringent conditions on the grids, we generalize the non-standard multiply so that communication may proceed between non-adjacent levels. The manuscript concludes with timing comparisons against naive algorithms and an illustration of the scale-independence of the convergence rate of the conjugate gradient solution of Poisson's equation using a simple preconditioning, suggesting that this approach leads to an O(n) solution of this equation.Comment: 33 pages, figures available at http://laisla.mit.edu/muchomas/Papers/nonstand-figs.ps . Updated: (1) figures file (figs.ps) now appear with the posting on the server; (2) references got lost in the last submissio

Nonlinear Waves and Dispersive Equations

Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics and general relativity. The conference did focus on the analytic aspects and PDE aspects

Regularity results for some models in geophysical fluid dynamics

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 12-04-2019This thesis centers on the study of two di erent problems of partial di erential equations arising from geophysics and uid mechanics: the surface quasi-geostrophic equation and the so called, Incompressible Slice Model. The surface quasi-geostrophic equation is a two dimensional nonlo- cal partial di erential equation of geophysical importance, describing the evolution of a surface buoyancy in a rapidly rotating, strati ed potential vorticity uid. In the rst part of the talk, we will present some global regularity results for its dissipative analogue in the critical regime for the two dimensional sphere. After that, we will introduce the Incompressible Slice Model deal- ing with oceanic and atmospheric uid motions taking place in a ver- tical slice domain R2, with smooth boundary. The ISM can be understood as a toy model for the full 3D Euler-Boussinesq equa- tions. We will study the solution properties of the Incompressible Slice Model: characterizing a class of equilibrium solutions, establishing the local existence of solutions and providing a blow-up criterion.This thesis has been funded by a Severo Ochoa FPI scholarship for Centres of Excellence in R&D (SEV-2015-0554) and by the grant MTM2017-83496-P from the Spanish Ministry of Economy and Competitiveness