Abstract
Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density 6 to the L2-adjoints of these operators evaluated at the density 6' are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the L2-adjoints of these operators. An ...
Abstract
Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density 6 to the L2-adjoints of these operators evaluated at the density 6' are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the L2-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces Wps (R) with p E (1,co) and s E (1 + 1/p, 2). (c) 2023 Elsevier Inc. All rights reserved.
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