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Existence and Analyticity of Solutions of Nonlinear Parabolic Model Equations with Singular Data
Journal article   Peer reviewed

Existence and Analyticity of Solutions of Nonlinear Parabolic Model Equations with Singular Data

David M Ambrose, Milton C Lopes Filho and Helena J Nussenzveig Lopes
IMA journal of applied mathematics, Forthcoming
07 May 2026
DOI: https://doi.org/10.1093/imamat/hxag011

Abstract

Kuramoto-Sivashinsky flame fronts Constantin-Lax-Majda vortex stretching radius of analyticity low-regularity data
We explore two approaches to proving existence and analyticity of solutions to nonlinear parabolic differential equations. One of these methods works well for more general nonlinearities, while the second method gives stronger results when the nonlinearity is simpler. The first approach uses the exponentially weighted Wiener algebra, and is related to prior work of Duchon and Robert for vortex sheets. The second approach uses two norms, one with a supremum in time and one with an integral in time, with the integral norm representing the parabolic gain of regularity. As an example of the first approach we prove analyticity of small solutions of a class of generalized one-dimensional Kuramoto-Sivashinsky equations, which model the motion of flame fronts and other phenomena. To illustrate the second approach, we prove existence and analyticity of solutions of the dissipative Constantin-Lax-Majda equation (which models vortex stretching), with and without added advection, with two classes of rough data. The classes of data treated include both data in the Wiener algebra with negative-power weights, as well as data in pseudomeasure spaces with negative-power weights.

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