Preventing Blow up by Convective Terms in Dissipative PDE’s

Abstract

Abstract. We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger's type equations, convective Cahn-Hilliard equation, generalized KuramotoSivashinsky equation and KdV type equations, we establish the following common scenario: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similarly to the case when the equation does not involve convective term.This kind of result has been previously known for the case of Burger's type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.

Attractors. Then and now

“…The structure of the Lorenz attractor is well understood nowadays, but this theory is far beyond the scope of this survey, so we restrict ourselves to mentioning a few interesting properties of this object. Further details can be found in [1], [87], [91], [92], [103], [118], [15], and [189].…”

“…We recall that the possible non-uniqueness of solutions of (4.7) is formally caused by the absence of the local Lipschitz continuity of the nonlinearity f . However, in contrast to ODEs, adding this natural assumption would not change the situation drastically due to a possible blow up of smooth solutions, which may occur despite the dissipative energy estimate (4.11); see [21] for the case of the 3D complex Ginzburg-Landau equation, and see [98], [15], and [186] for different classes of reaction-diffusion systems.…”