Smooth Solutions for a Class of Nonlinear Parabolic Evolution Equations

2000 ◽  
Vol 61 (1) ◽  
pp. 216-244 ◽  
Author(s):  
Markus Poppenberg
Keyword(s):  
Evolution Equations ◽  
Smooth Solutions ◽  
Nonlinear Parabolic ◽  
Parabolic Evolution Equations

scholarly journals Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

2020 ◽  
pp. 1-31
Author(s):  
Bogdan–Vasile Matioc ◽  
Georg Prokert
Keyword(s):  
Stokes Flow ◽  
Evolution Equations ◽  
Single Layer ◽  
Two Phase ◽  
Layer Potential ◽  
Two Fluids ◽  
Flat Interface ◽  
Nonlinear Parabolic ◽  
Parabolic Evolution Equations ◽  
2D Space

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

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