Strong solutions to a modified Michelson-Sivashinsky equation. (English) Zbl 1479.35225

Summary: We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space \(W^{1,\infty} (\mathbb{R}^d)\) 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 by Kiselev, Nazarov, Volberg and Shterenberg 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. This work provides an example where regularity is shown to persist even when a priori bounds are not available.


35D35 Strong solutions to PDEs
35B50 Maximum principles in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
35R11 Fractional partial differential equations
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