Strong solutions to a modified Michelson-Sivashinsky equation

Abstract

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension. Local in time well-posedness (and regularity) in the space W 1,∞ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced to handle the critically dissipative surface quasi-geostrophic equation and the critically dissipative fractional Burgers equation. Namely, the global regularity result is achieved by constructing a time-dependent modulus of continuity that must be obeyed by the solution of the initial-value problem for all time, preventing blowup of the gradient of the solution.

Lipschitz continuity of solutions to drift-diffusion equations in the presence of nonlocal terms

“…By tracking the evolution of Lipschitz moduli of continuity (see Definition 1.2, below), they were able to prevent a gradient blowup scenario, from which a global regularity result follows. As a followup to a recent work of ours [14], we attempt to extend such ideas from their current domain of scalar equations to advectiondiffusion systems in the presence of nonlocal forcing terms, which include the incompressible Navier-Stokes equations, in any spatial dimensions d ≥ 3 and in the absence of physical boundaries. A key ingredient used in analyzing the pressure term is a subtle observation regarding its regularity made by Silvestre in an unpublished work [28], as well as Constantin [6], Isett [15] (see also Isett and Oh [16]) and De Lellis and Székelyhidi Jr. [11].…”

“…In particular, time dependent moduli of continuity were introduced in [17] mainly in order to obtain certain eventual regularity results for a class of active scalar evolution equations. We recently extended this technique in [14] to a modified Michelson-Sivashinsky equation (see discussion below), where we were able to obtain a global regularity result. To our knowledge, this program has never been tried for the NS system.…”

“…Tracking the evolution of moduli of continuity is in particular very useful for analyzing certain nonlinear parabolic equations in the presence of nonlocal terms, as it provides a "medium" capable of comparing stabilizing (dissipative) terms against destabilizing (certain nonlinear and/or nonlocal) terms at the same level. To more concretely demonstrate the last point, we consider the following problem analyzed in [14], which essentially is what motivated this paper:…”

“…Nevertheless, the singular integral operator (−∆) α is known to map C 0,β Hölder continuous functions to C 0,β−2α , provided 0 < 2α < β ≤ 1 [27]. Similarly, one can show that while it doesn't quite preserve abstract moduli of continuity, it doesn't distort them too much either [14]. Therefore one can successfully employ the above mentioned strategy and construct a certain modulus of continuity that must be obeyed by the solution for all time.…”

“…The MS model is a refined combustion model based on the Darrieus-Landau flame stability analysis [30]. The reason for restricting α ∈ (0, 1/2) in [14] is because of the lack of any "obvious" continuity (or even L ∞ ) estimate for the square root of the Laplacian of a function in terms of those known of function itself. To see this, recall the representation…”