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On the global wellposedness of free boundary problem for the Navier-Stokes system with surface tension.
- Source :
-
Journal of Differential Equations . Mar2024, Vol. 384, p1-92. 92p. - Publication Year :
- 2024
-
Abstract
- The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time t tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain Ω t ⊂ R 3 , t > 0 , to a problem in the lower half-space R − 3. We then establish some time-weighted estimate of solutions, in an L p -in-time and L q -in-space setting, for the linearized problem around the trivial steady state with the help of L r - L s time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R − 3 admits a global-in-time solution in the L p - L q setting and that the solution decays polynomially as time t tends to infinity under the assumption that p , q satisfy the conditions: 2 < p < ∞ , 3 < q < 16 / 5 , and (2 / p) + (3 / q) < 1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in R − 3 to prove our main results mentioned above for the original problem in Ω t. Here we want to emphasize that it is not allowed to take p = q in the above assumption about p , q , which means that the different exponents p , q of L p - L q setting play an essential role in our approach. [ABSTRACT FROM AUTHOR]
- Subjects :
- *SURFACE tension
*NAVIER-Stokes equations
*FREE surfaces
Subjects
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 384
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 174791435
- Full Text :
- https://doi.org/10.1016/j.jde.2023.11.020