2019/11/08 by Francisco Gancedo, Gancedo, Francisco, Rafael Granero-Belinchón +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Lattice Boltzmann Simulation Studies #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1911.03331
openalex publication_date 2019/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies the dynamics of an incompressible fluid driven by gravity\nand capillarity forces in a porous medium. The main interest is the\nstabilization of the fluid in Rayleigh-Taylor unstable situations where the\nfluid lays on top of a dry region. An important feature considered here is that\nthe layer of fluid is under an impervious wall. This physical situation have\nbeen widely study by mean of thin film approximations in the case of small\ncharacteristic high of the fluid considering its strong interaction with the\nfixed boundary. Here, instead of considering any simplification leading to\nasymptotic models, we deal with the complete free boundary problem. We prove\nthat, if the fluid interface is smaller than an explicit constant, the solution\nis global in time and it becomes instantly analytic. In particular, the fluid\ndoes not form drops in finite time. Our results are stated in terms of Wiener\nspaces for the interface together with some non-standard Wiener-Sobolev\nanisotropic spaces required to describe the regularity of the fluid pressure\nand velocity. These Wiener-Sobolev spaces are of independent interest as they\ncan be useful in other problems. Finally, let us remark that our techniques do\nnot rely on the irrotational character of the fluid in the bulk and they can be\napplied to other free boundary problems.\n