Abstract.
We consider fluid in a smooth rigid container whose lateral boundary is a piece of vertical cylinder, bounded above by a free upper surface. As basic flow we consider the non homogeneous rest state in the presence of gravity, and of a surface tension. Under these assumptions, we study the existence of a steady free boundary Γ and a steady motion in Ω of an isothermal viscous gas, resulting as perturbation to the rest state in correspondence of small non potential perturbations to the (large potential) gravitational force. We linearize the problem by prescribing the unknown domain Ω, then we make use of the iterative scheme introduced by Heywood and Padula. Our method is based on an iteration between the Neumann problem for a non homogeneous Stokes system for the velocity, the Neumann problem for an elliptic problem on Γ for height, and a steady transport equation for the perturbation to the density. The difference of boundary condition between lateral boundary and free upper surfaces causes a singularity at the intersection (contact line). To avoid singularities on the contact line, we adopt weighted Sobolev spaces.
Similar content being viewed by others
References
Bause, M., Heywood, J.G., Novotny, A., Padula, M.: Numerical approximation schemes for the Poisson-Stokes equations of steady compressible viscous flow 1. Math. Fluid Mech., Birkhauser-Verlag ed., 2001
Bause, M., Heywood, J.G., Novotny, A., Padula, M.: Numerical approximation schemes for the Poisson-Stokes equations of steady compressible viscous flow 2. Math. Fluid Mech., Birkhauser-Verlag ed., 2001
Beale, J.T.: Large time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84(4), 307–352 (1984)
Beiraó da Veiga, H.: Existence results in Sobolev spaces for a stationary transport equations. Ricerche Mat. Vol. in honor of C. Miranda, 1987
Beiraó da Veiga, H.: An L p-theory for the n-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions. Commun. Math. Phys. 109, 229–248 (1987)
Borchers, W., Pileckas, K.: Existence, uniqueness and asymptotics of steady jets. Arch. Rat. Mech. Anal. 120, 1–49 (1992)
Choe, H.J., Jin, B.J.: Existence of solutions of stationary compressible Navier-Stokes equations with large force. Journal of Functional Analysis, 2000
Finn, R.: Equilibrium capillary surfaces. Grundelehren der mathematischen Wissenschaften 284, Springer-Verlag, New-York, Berlin, Heidelberg, Tokyo, 1986
Finn, R.: Equations of capillarity. JMFM 3, 139–151 (2001)
Farwig, R.: Stationary solutions of the Navier-Stokes equations with slip boundary conditions. Comm. Partial Diff. Eqs. 14, 1579–1606 (1989)
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer-Verlag New York, 1994
Gilbarg and Trudinger: Elliptic partial differential equations of second order. Springer-Verlag, Berlin and New York, 1983
Grisvard, P.: Elliptic problems in nonsmooth domains. Boston, London, Melbourne, Pitman Pub. Co., 1985
Kondratev, V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moscow Math. Soc. 16, 227–313 (1967)
Massari, U., Miranda, M.: Minimal surfaces of codimension one. Math. Studies 91, ed. L. Nachbin North Holland, Amsterdam, New York, Oxford, 1984
Jin, B.J., Padula, M.: On the existence of compressible viscous flow in a horizontal layer with free upper surface. To appear in Comm. Pure and Applied Analysis
Heywood, J.G., Padula, M.: On the Existence and Uniqueness Theory for steady compressible Viscous flow, Fundamental directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel-Boston-Berlin, 1999
Matsumura, A., Nishida, T.: Initial boundary value problems for the equations of motions of compressible viscous and heat-conductive fluids. Comm. Math. Phys. 89, 445–464 (1983)
Matsumura, A., Padula, M.: Stability of the stationary solutions of compressible Viscous fluids with large external potential forces. Stab. Appl. Anal. Cont. Media 2, 183–202 (1992)
Nazarov, S.A., Novotny, A., Pileckas, K.: On steady compressible Navier-Stokes equations in plane domain with corners. Math. Ann. 302, 121–150 (1996)
Nazarov, S.A., Plamenevsky, B.A.: Elliptic problems in domains with piecewise smooth boundaries. De Gruyter expositions in Mathematics 13, Berlin, New-York, 1994
Novotny, A.: About the steady transport equations. L p approach in domains with sufficiently smooth boundaries. Comment. Math. Univ. Carolin., 1996
Novotny, A., Padula, M.: Existence and uniqueness of stationary solutions for viscous compressible heat conductive fluid with large potential and small nonpotential forces. Siberian Math. J. 34, 120–146 (1993)
Novotny, A., Pileckas, K.: Steady compressible Navier-Stokes equations with large potential forces via a method of decomposition. Math. Meth. Appl. Sci. 21, 665–684 (1998)
Padula, M.: Existence and uniqueness for viscous steady compressible motions. Arch. Rat. Mech. Anal. 97, 89–102 (1987)
Padula, M., Solonnikov, V.A.: On Reyleigh-Taylor stability. Annali dell’Universita’ di Ferrara, vol. 45, 2000
Padula, M.: On the uniqueness of viscous compressible steady flows. Trends in Appl. of Pure Math. to Mech. Vol. IV, ed. Brilla, Pitman, London, 1981, pp. 186–196
Padula, M.: Steady flows of barotropic viscous fluids. Quaderni di Matematica 1, 253–345 (1997)
Pileckas, K., Zajaczkowski, W.M.: On the free boundary problem for a stationary compressible Navier-Stokes equations. Commun. Math. Phys. 129, 169–204 (1990)
Sattinger, D.H.: On the free surface of a viscous fluid motion. Proc. R. Soc: London, A 349, 183–204 (1976)
Scadilov, Solonnikov, V.A.: Solvability of the initial boundary value problem for the equations of a viscous compressible fluid. J. Sov. Math 14, (1980)
Solonnikov, V.A.: Solvability of a problem on the plane motion of a heavy viscous incompressible capillary liquid partially filling a container. Isv. Ahad. Nauk SSSR, ser. Mat. 43, (1979); Math. USSR Izvestija 14, 193–221 (1980)
Solonnikov, V.A.: On the Stokes equation in domains with nonsmooth boundaries and on a viscous incompressible flow with a free surface. Nonlinear Part. Diff. Equations~3, College de France Seminar, 1980/1981, pp. 340–423
Solonnikov, V.A.: Solvability of two stationary free boundary problems for the Navier-Stokes equations. Bollettino U.M.I. (8) 1-B, 283–342 (1998)
Solonnikov, V.A.: Solvability of the initial boundary value problem for the equations of a viscous compressible fluid. J. Sov. Math 14 (1980)
Solonnikov, V.A.: Free boundary problems for the Navier-Stokes equations with moving contact points. Proceedings of the Conference on Nonlinear Evolution Equations and Infinite-dimensional Dynamics Systems, 1995, pp. 180–189
Solonnikov, V.A.: Free boundary problems for the Navier-Stokes equations with moving contact points. Free boundary problems: theory and applications, (Toledo, 1993). Pitman Res. Notes Math. Ser., 323. Longman Sci. Tech, Harlow, 1995, pp. 203–214
Solonnikov, V.A.: Solvability of two-dimensional free boundary problem for the Navier-Stokes equations for limiting values of contact angle. Recent developments in partial differential equations, Quad. Mat,, 2. Arance, Rome, 1998, pp. 163–210
Solonnikov, V.A.: On some free boundary problems for the Navier-Stokes equations with moving contact points and lines. Math. Ann. 302(4), 743–772 (1995)
Solonnikov, V.A., Tani, A.: Evolution free boundary problem for equations of motion of viscous compressible barotropic liquids. The Navier-Stokes equation II: Theory and numerical methods, Proceedings Oberwolfach, Lecture notes in Math., 1530, 1991, pp. 30–58
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Univ. press, 1970
Tani, A.: On the free boundary value problem for compressible viscous fluid motion. J. Math. Kyoto. Univ. 839(859), 21–4 (1981)
Tani, A.: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. RIMS. Kyoto Univ 13, 193–253 (1977)
Valli, A.: Periodic and stationary solutions for compressible Navier-Stokes equations by stability method. Ann. Sc. Norm. Super. pisa 607–647 (1984)
Valli, A.: On the existence of stationary solutions for compressible Navier-Stokes equation. Ann. Inst. H. Poincaré 4, 99–113 (1987)
Maxwell, J.C.: Capillary attraction. Encyclopedia Britannica, 9th ed., vol. 5, S.L. Hall, New York, 1878
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ja Jin, B., Padula, M. Steady flows of compressible fluids in a rigid container with upper free boundary. Math. Ann. 329, 723–770 (2004). https://doi.org/10.1007/s00208-004-0535-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-004-0535-0