Kernel-adapted Schauder estimates in critical Hölder/Besov spaces yield local and global well-posedness for the Muskat equation with surface tension and Peskin problems with nonlinear elastic tension.
Ignatova, Construction of solutions of the critical SQG equation in bo unded domains, Advances in Mathematics, 351 (2019), 1000–1023
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Schauder-type Estimates and Well-posedness for Nonlocal Quasilinear Evolution Equations in Fluid Dynamics
Kernel-adapted Schauder estimates in critical Hölder/Besov spaces yield local and global well-posedness for the Muskat equation with surface tension and Peskin problems with nonlinear elastic tension.