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In questo lavoro studiamo il moto di un fluido viscoso e incomprimibile in una regione limitata tridimensionale Ω, con condizioni al contorno di superficie libera. Utilizzando un metodo, dovuto all’autore, che consiste nel considerare l’atmosfera o il vuoto come un secondo fluido, separato dal primo da un’interfaccia mobile Γ(t), dimostriamo l’esistenza di una sorta di soluzione debole, denominata soluzione quasi-debole.
Abstract
In this paper we are concerned with the flow of a viscous, incompressible fluid in a bounded, three-dimensional region Ω with free surface boundary conditions. Using a method introduced by the author, that consider a two-fluid system in which the atmosphere or the vacuum is considered as a second fluid, separated from the first one by a free interface Γ(t), we prove existence of a kind of weak solution that we call quasi-weak solution.
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References
P. G. Galdi,An introduction to the mathematical theory of Navier-Stokes equations, volume 1, 2, Springer Tracts in Natural Philosophy, vol.38 (1994).
H. Kawarada—K. Ohmory,A sharp interface capturing technique in the finite element approximation for two-fluid flows, Pitman Research Notes in Mathematics Series vol.,588, (ed. R. Salvi), pp. 310–321.
A. V. Kazhikov,Resolution of boundary value problems for nonhomogeneous viscous fluids, Dokl. Akad. Nauk.,216 (1974), pp. 1008–1010 (in Russian).
J. L. Lions,Quelques methodes de resolution des problemes aux limites nonlineares, Dunod, Paris, 1960.
P. L. Lions,Mathematical Topics in Fluid Mechanics, volume 1, Oxford Science Publication, 1996.
G. Nespoli—R. Salvi,On the existence of two-phase problem for incompressible flow, Quaderni di Matematica, volume 4 (ed. P. Maremonti), 1999.
R. Salvi,On the existence of weak solutions of boundary value problems in a diffusion model for an inhomogeneous liquid in regions with moving boundaries, Portugaliae Math.,43 (1985–86), pp. 213–233.
R. Salvi,On the Navier-Stokes equations in noncylindrical domains: On the existence and regularity, Math. Z.,170 (1988), pp. 153–170.
R. Salvi,The exterior nonostationary problem for the Navier-Stokes equations in regions with moving boundaries, J. Math. Japan,42 (1990), pp. 495–509.
R. Salvi—I. Straskraba,Global existence for viscous compressible fluids and their behavior as t→∞, j. Fac. Sci. Univ. Tokyo Sect. IA Math.,40 (1993), pp. 17–51.
J. Sokolowsky—J. P. Zolesio,Introduction to shape optimization, Springer Series in computational Mathematics,16 (1992).
V. A. Solonnikov,Solvability of a problem on the motion of a viscous incompressible fluid bounded by a free surface, Math. USSR Izv.,31 (1988), pp. 381–405.
N. Tanaka,Global existenceof two-phase nonhomogeneous viscous incompressible fluid flow, Comm. in P.D.E.,18 (1993), pp. 41–81.
R. Teman,Navier-Stokes Equations, Studies in Math. and Appl.,2, North-Holland, 1985.
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Work supported by Progetto Murst n. 9801262841.
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Salvi, R. On the existence of free surface problem for viscous incompressible flow. Ann. Univ. Ferrara 46, 251–266 (2000). https://doi.org/10.1007/BF02837301
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DOI: https://doi.org/10.1007/BF02837301