On the backward behavior of some dissipative evolution equations

abstract

• 2015 Elsevier B.V. Abstract We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the periodic 2D Navier-Stokes equations studied by Constantin et al. (1997), but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica and Malcok (2005). Also we study the backward behavior of solutions to the damped driven nonlinear Schrdinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, we provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, we discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by the investigation of the Bardos-Tartar conjecture on the Navier-Stokes equations stated in Bardos and Tartar (1973).

authors

• Titi, Edriss

published proceedings

• PHYSICA D-NONLINEAR PHENOMENA

altmetric score

• 0.5

author list (cited authors)

• Guo, Y., & Titi, E. S.

citation count

• 0

complete list of authors

• Guo, Yanqiu||Titi, Edriss S

publication date

• June 2015

publisher

• Elsevier  Publisher

published in

• Physica D : Non-linear phenomena  Journal

keywords

• Backward Behaviors
• Bardos-tartar Conjecture
• Burgers' Equation
• Korteweg-de Vries Equation
• Kuramoto-sivashinsky Equation
• Turbulence

Digital Object Identifier (DOI)

• 10.1016/j.physd.2015.05.011

start page

• 34

end page

• 47

volume

• 306

URL

• http%3A%2F%2Fdx.doi.org%2F10.1016%2Fj.physd.2015.05.011