Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations

Abstract

The global regularity for the two-and three-dimensional Kuramoto-Sivashinsky equations is one of the major open questions in nonlinear analysis. Inspired by this question, we introduce in this paper a family of hyper-viscous Hamilton-Jacobi-like equations parametrized by the exponent in the nonlinear term, p, where in the case of the usual Hamilton-Jacobi nonlinearity, p = 2. Under certain conditions on the exponent p we prove the short-time existence of weak and strong solutions to this family of equations. We also show the uniqueness of strong solutions. Moreover, we prove the blow-up in finite time of certain solutions to this family of equations when the exponent p > 2. Furthermore, we discuss the difference in the formation and structure of the singularity between the viscous and hyper-viscous versions of this type of equation.

Preventing Blow up by Convective Terms in Dissipative PDE’s

“…blow up in finite time, see e.g., [24] and references therein, see also [1,2,3,11,14,17,18,19,22] for the analogous results for more complicated equations.…”

“…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.blow up in finite time, see e.g., [24] and references therein, see also [1,2,3,11,14,17,18,19,22] for the analogous results for more complicated equations.…”

Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

“…It arises as a model in hydrodynamics (a thin film flow down an inclined plane in the presence of an electric field), in combustion theory (propagation of flame fronts), phase turbulence and plasma physics, as well as a model for spatio-temporal chaos; c.f. [2,14] for a short review of applications with key references.…”

Maximum principle for optimal boundary control of the Kuramoto–Sivashinsky equation

“…For example, by the KS equation, Matar et al [24] study the nonlinear stability and dynamic behavior of falling fluid films; Annunziato et al [2] consider the unstable flame front propagation in uniform mixtures; and Ramaswamy et al [27] analyze the interfacial instabilities in thin-film flows. For other interesting results, refer to [4,9,28,31,33], name just a few. Nonetheless, when the KS equation is involved in control issues, the situation turns to be completely different and there are few known researches yet.…”