Nonlocal Burgers Equation Research Articles

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

A nonlocal model describing tumor angiogenesis

In this paper, we derive and study a new mathematical model that describes the onset of angiogenesis. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. For a particular value of the parameters, the equation reduces to ∂tp−12(−Δ)(α−1)/2H∂tp=−12(−Δ)α/2p+p∂xp−∂xp, where H denotes the Hilbert transform. In addition to the derivation of the new model, the main novelty of the present paper is that we also prove a number of well-posedness results. Finally, some preliminary numerical results are shown. These numerical results suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time.

A nonlocal Burgers equation in atmospheric dynamical system and its exact solutions**Project supported by the National Natural Science Foundation of China (Grant Nos. 11405110, 11275129, and 11472177) and the Natural Science Foundationof Zhejiang Province of China (Grant No. LY18A050001).

From a two-vortex interaction model in atmospheric and oceanic systems, a nonlocal counterpart with shifted parity and delayed time reversal is derived by a simple AB reduction. To obtain some approximate analytic solutions of this nonlocal system, the multi-scale expansion method is applied to get an AB-Burgers system. Various exact solutions of the AB-Burgers equation, including elliptic periodic waves, kink waves and solitary waves, are obtained and shown graphically. To show the applications of these solutions in describing correlated events, a simple approximate solution for the two-vortex interaction model is given to show two correlated dipole blocking events at two different places. Furthermore, symmetry reduction solutions of the nonlocal AB-Burgers equation are also given by using the standard Lie symmetry method.

Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation

We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in [6].

Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on a one dimensional bounded domain. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain.

Global Well-Posedness of a Non-local Burgers Equation: the periodic case

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads u_t − u|∇|u + |∇|(u^2) = 0. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in L ∞. We show that any weak solution is instantaneously regularized into C ∞. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.

Existence of traveling waves for the nonlocal Burgers equation

We study the equation u t + u u x + u − K ∗ u = 0 in the case of an arbitrary K ≥ 0 , which is a generalization of a model for radiating gas, in which K ( y ) = 1 2 e − | y | . Using a monotone iteration scheme argument we establish the existence of traveling waves, which gives a solution to an open question raised by D. Serre [ L 1 -stability of nonlinear waves in scalar conservation laws, in: Evolutionary Equations, in: Handb. Differ. Equ., vol. I, North-Holland, Amsterdam, 2004, pp. 473–553].

A CONTRIBUTION TO NONAUTONOMOUS INERTIAL MANIFOLDS

The goal of the paper is to construct nonautonomous inertial manifolds via outflowing invariant manifolds. As an example, a nonautonomous, nonlocal Burgers equation is considered.