On nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a symmetric cusp domain
Abstract
The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.
Keyword : stationary Navier–Stokes equations, nonhomogeneous boundary condition, cusp point singularity, multiply connected domain, nonzero fluxes
How to Cite
Kaulakytė, K., & Klovienė, N. (2021). On nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a symmetric cusp domain. Mathematical Modelling and Analysis, 26(1), 55-71. https://doi.org/10.3846/mma.2021.12173
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M. Chipot, K. Kaulakytė, K. Pileckas and W. Xue. On nonhomogeneous boundary value problems for the stationary Navier–Stokes equations in two-dimensional symmetric semi-infinite outlets. Analysis and Applications, 15(4):543–569, 2017. https://doi.org/10.1142/S0219530515500268
H. Fujita. On stationary solutions to Navier–Stokes equation in symmetric plane domain under general outflow condition. In Pitman research notes in mathematics, volume 388 of Proceeding of international conference on Navier-Stokes equations. Theory and numerical methods, pp. 16–30, Varenna, Italy, 1997.
K. Kaulakytė. On nonhomogeneous boundary value problem for the steady Navier–Stokes system in domain with paraboloidal and layer type outlets to infinity. Topol. Methods Nonlinear Anal., 46(2):835–865, 2015. https://doi.org/10.12775/TMNA.2015.070
K. Kaulakytė, N. Klovienė and K. Pileckas. Nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a domain with a cusp. Zeitschrift für angewandte Mathematik und Physik, 70(36), 2019. https://doi.org/10.1007/s00033-019-1075-5
K. Kaulakytė and K. Pileckas. On the nonhomogeneous boundary value problem for the Navier–Stokes system in a class of unbounded domains. Journal of Mathematical Fluid Mechanics, 14(4):693–716, 2012. https://doi.org/10.1007/s00021-011-0089-3
K. Kaulakytė and W. Xue. Nonhomogeneous boundary value problem for Navier–Stokes equations in 2D symmetric unbounded domains. Applicable Analysis, 96(11):1906–1927, 2017. https://doi.org/10.1080/00036811.2016.1198780
M.V. Korobkov, K. Pileckas, V.V. Pukhnachev and R. Russo. The flux problem for the Navier–Stokes equations. Russian Mathematical Surveys, 69(6):1065– 1122, 2014. https://doi.org/10.1070/RM2014v069n06ABEH004928
O.A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969.
O.A. Ladyzhenskaya and V.A. Solonnikov. Determination of the solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral. Journal of Soviet Mathematics, 21:728–761, 1983. https://doi.org/10.1007/BF01094437
H. Morimoto and H. Fujita. A remark on the existence of steady Navier–Stokes flows in 2D semi-infinite channel involving the general outflow condition. Mathematica Bohemica, 126(2):457–468, 2001. https://doi.org/10.21136/MB.2001.134017
H. Morimoto and H. Fujita. A remark on the existence of steady Navier–Stokes flows in a certain two-dimensional infinite channel. Tokyo Journal of Mathematics, 25(2):307–321, 2002. https://doi.org/10.3836/tjm/1244208856
J. Neustupa. On the steady Navier–Stokes boundary value problem in an unbounded 2D domain with arbitrary fluxes through the components of the boundary. Annali Dell Universita´ Di Ferrara´ , 55(2):353–365, 2009. https://doi.org/10.1007/s11565-009-0083-3
J. Neustupa. A new approach to the existence of weak solutions of the steady Navier–Stokes system with inhomoheneous boundary data in domains with noncompact boundaries. Archive for Rational Mechanics and Analysis, 198(1):331– 348, 2010. https://doi.org/10.1007/s00205-010-0297-7
V.A. Solonnikov. On the solvability of boundary and initial-boundary value problems for the Navier–Stokes system in domains with noncompact boundaries. Pacific J. Math., 93(2):443–458, 1981. https://doi.org/10.2140/pjm.1981.93.443
V.A. Solonnikov and K. Pileckas. Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier–Stokes system of equations in domains with noncompact boundaries. Journal of Soviet Mathematics, 34:2101– 2111, 1986. https://doi.org/10.1007/BF01741584
E.M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1970.
A. Takeshita. A remark on Leray’s inequality. Pacific Journal of Mathematics, 157(1):151–158, 1993. https://doi.org/10.2140/pjm.1993.157.151
H. Fujita. On stationary solutions to Navier–Stokes equation in symmetric plane domain under general outflow condition. In Pitman research notes in mathematics, volume 388 of Proceeding of international conference on Navier-Stokes equations. Theory and numerical methods, pp. 16–30, Varenna, Italy, 1997.
K. Kaulakytė. On nonhomogeneous boundary value problem for the steady Navier–Stokes system in domain with paraboloidal and layer type outlets to infinity. Topol. Methods Nonlinear Anal., 46(2):835–865, 2015. https://doi.org/10.12775/TMNA.2015.070
K. Kaulakytė, N. Klovienė and K. Pileckas. Nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a domain with a cusp. Zeitschrift für angewandte Mathematik und Physik, 70(36), 2019. https://doi.org/10.1007/s00033-019-1075-5
K. Kaulakytė and K. Pileckas. On the nonhomogeneous boundary value problem for the Navier–Stokes system in a class of unbounded domains. Journal of Mathematical Fluid Mechanics, 14(4):693–716, 2012. https://doi.org/10.1007/s00021-011-0089-3
K. Kaulakytė and W. Xue. Nonhomogeneous boundary value problem for Navier–Stokes equations in 2D symmetric unbounded domains. Applicable Analysis, 96(11):1906–1927, 2017. https://doi.org/10.1080/00036811.2016.1198780
M.V. Korobkov, K. Pileckas, V.V. Pukhnachev and R. Russo. The flux problem for the Navier–Stokes equations. Russian Mathematical Surveys, 69(6):1065– 1122, 2014. https://doi.org/10.1070/RM2014v069n06ABEH004928
O.A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969.
O.A. Ladyzhenskaya and V.A. Solonnikov. Determination of the solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral. Journal of Soviet Mathematics, 21:728–761, 1983. https://doi.org/10.1007/BF01094437
H. Morimoto and H. Fujita. A remark on the existence of steady Navier–Stokes flows in 2D semi-infinite channel involving the general outflow condition. Mathematica Bohemica, 126(2):457–468, 2001. https://doi.org/10.21136/MB.2001.134017
H. Morimoto and H. Fujita. A remark on the existence of steady Navier–Stokes flows in a certain two-dimensional infinite channel. Tokyo Journal of Mathematics, 25(2):307–321, 2002. https://doi.org/10.3836/tjm/1244208856
J. Neustupa. On the steady Navier–Stokes boundary value problem in an unbounded 2D domain with arbitrary fluxes through the components of the boundary. Annali Dell Universita´ Di Ferrara´ , 55(2):353–365, 2009. https://doi.org/10.1007/s11565-009-0083-3
J. Neustupa. A new approach to the existence of weak solutions of the steady Navier–Stokes system with inhomoheneous boundary data in domains with noncompact boundaries. Archive for Rational Mechanics and Analysis, 198(1):331– 348, 2010. https://doi.org/10.1007/s00205-010-0297-7
V.A. Solonnikov. On the solvability of boundary and initial-boundary value problems for the Navier–Stokes system in domains with noncompact boundaries. Pacific J. Math., 93(2):443–458, 1981. https://doi.org/10.2140/pjm.1981.93.443
V.A. Solonnikov and K. Pileckas. Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier–Stokes system of equations in domains with noncompact boundaries. Journal of Soviet Mathematics, 34:2101– 2111, 1986. https://doi.org/10.1007/BF01741584
E.M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1970.
A. Takeshita. A remark on Leray’s inequality. Pacific Journal of Mathematics, 157(1):151–158, 1993. https://doi.org/10.2140/pjm.1993.157.151