Preventing blow up by convective terms in dissipative PDEs

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

On the Normal Semilinear Parabolic Equations Corresponding to 3D Navier-Stokes System

Part 5: Flow ControlInternational audienceThe semilinear normal parabolic equations corresponding to 3D Navier-Stokes system have been derived. The explicit formula for solution of normal parabolic equations with periodic boundary conditions has been obtained. It was shown that phase space of corresponding dynamical system consists of the set of stability (where solutions tends to zero as time t → ∞), the set of explosions (where solutions blow up during finite time) and intermediate set. Exact description of these sets has been given