Preventing blow up by convective terms in dissipative PDEs

TL;DR

This paper investigates how strong convective terms can prevent finite time blow-up in various dissipative PDEs, establishing global well-posedness through weighted energy estimates rather than maximum principles.

Contribution

It introduces a novel approach using weighted energy estimates to show convective terms prevent blow-up in dissipative PDEs, extending beyond Burger's equations.

Findings

Strong convective terms prevent blow-up in dissipative PDEs.

Weak convective terms may still allow finite time blow-up.

Weighted energy estimates replace maximum principle in analysis.

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 Kuramoto-Sivashinsky 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.