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Free Boundary Problems for incompressible and inhomogeneous Navier-Stokes equations
- Speaker(s)
- Tomasz Piasecki
- Affiliation
- MIM UW
- Language of the talk
- English
- Date
- May 7, 2026, 12:30 p.m.
- Room
- room 5070
- Seminar
- Seminar of Mathematical Physics Equations Group
The talk will be based on my recent joint work with Piotr B. Mucha and Yoshihiro Shibata. We investigate free boundary problems for incompressible and inhomogeneous (with variable density and divergence-free velocity) Navier-Stokes equations, where the spatial domain is either a half-space or its perturbation.
We prove the global existence of regular solutions in the maximal regularity setting, in space dimension greater than or equal to 2, under certain smallness assumptions on the initial velocity. In the case of an inhomogeneous system, we assume additionally that the initial density is close to constant. The proof relies on reformulation of the problem in Lagrangian coordinates and already known maximal regularity estimates for the Stokes system, which we extend to cover the case of the perturbed half-space. Part of the nonlinearities is estimated in quite a direct way by applying the Sobolev embedding. The essential difficulty lies in the treatment of the nonlinearities coming from the transformation of the boundary conditions. To overcome this issue, we prove a general interpolation result which can be of independent interest, as it applies to a class of operators that typically arise in Lagrangian transformation, and it can be easily modified and extended.
We prove the global existence of regular solutions in the maximal regularity setting, in space dimension greater than or equal to 2, under certain smallness assumptions on the initial velocity. In the case of an inhomogeneous system, we assume additionally that the initial density is close to constant. The proof relies on reformulation of the problem in Lagrangian coordinates and already known maximal regularity estimates for the Stokes system, which we extend to cover the case of the perturbed half-space. Part of the nonlinearities is estimated in quite a direct way by applying the Sobolev embedding. The essential difficulty lies in the treatment of the nonlinearities coming from the transformation of the boundary conditions. To overcome this issue, we prove a general interpolation result which can be of independent interest, as it applies to a class of operators that typically arise in Lagrangian transformation, and it can be easily modified and extended.
