On singular solutions of the stationary Navier-Stokes system in power cusp domains
Abstract
The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.
Keyword : stationary Navier-Stokes problem, power cusp domain, singular solutions, asymptotic expansion
How to Cite
Pileckas, K., & Raciene, A. (2021). On singular solutions of the stationary Navier-Stokes system in power cusp domains. Mathematical Modelling and Analysis, 26(4), 651-668. https://doi.org/10.3846/mma.2021.13836
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R.A. Adams and J. Fournier. Sobolev Spaces. Academic Press (Elsevier), 2003.
G. Cardone, S.A. Nazarov and J. Sokolowski. Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary. Asymptotic Analysis, 62(1-2):41–88, 2009. https://doi.org/10.3233/ASY-2008-0915
A. Eismontaite and K. Pileckas. On singular solutions of time-periodic and steady Stokes problems in a power cusp domain. Applicable Analysis, 97(3):415–437, 2018. https://doi.org/10.1080/00036811.2016.1269321
A. Eismontaite and K. Pileckas. On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain. Applicable Analysis, 98(13):2400–2422, 2019. https://doi.org/10.1080/00036811.2018.1460815
I.V. Kamotski and V.G. Maz’ya. On the third boundary value problem in domains with cusps. Journal of Mathematical Sciences, 173(5):609–631, 2011. https://doi.org/10.1007/s10958-011-0262-5
L.V. Kapitanskii and K. Pileckas. Certain problems of vector analysis. J. Sov. Math., 32(5):469–483, 1986. https://doi.org/10.1007/BF01372197
K. Kaulakyte and N. Kloviene. On nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a symmetric cusp domain. Mathematical Modelling and Analysis, 26(1):55–71, 2021. https://doi.org/10.3846/mma.2021.12173
K. Kaulakyte, N. Kloviene and K. Pileckas. Nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a domain with a cusp. Z. Angew. Math. Phys., 70(36), 2019. https://doi.org/10.1007/s00033-019-1075-5
K. Kaulakyte and K. Pileckas. Nonhomogeneous boundary value problem for the time periodic linearized Navier-Stokes system in a domain with outlet to infinity. J. of Mathematical Analysis and Applications, 489(1), 2020. https://doi.org/10.1016/j.jmaa.2020.124126
H. Kim and H. Kozono. A removable isolated singularity theorem for the stationary Navier-Stokes equations. J. Diff. Equations, 220(1):68–84, 2006. https://doi.org/10.1016/j.jde.2005.02.002
M.B. Korobkov, K. Pileckas, V.V. Pukhnachev and R. Russo. The flux problem for the Navier-Stokes equations. Russian Math. Surveys, 69(6):1065–1122, 2014. https://doi.org/10.1070/RM2014v069n06ABEH004928
V. Kozlov and J. Rossmann. On the nonstationary Stokes system in a cone. J. of Diff. Equations, 260(12):8277–8315, 2016. https://doi.org/10.1016/j.jde.2016.02.024
V. Kozlov and J. Rossmann. On the nonstationary Stokes system in a cone: Asymptotics of solutions at infinity. J. of Mathematical Analysis and Applications, 486(10), 2020. https://doi.org/10.1016/j.jmaa.2019.123821
V. Kozlov and J. Rossmann. On the nonstationary Stokes system in a cone (Lp theory). J. of Mathematical Fluid Mechanics, 22(42), 2020. https://doi.org/10.1007/s00021-020-00502-w
O.A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. Gordon and Breach, 1969.
V.G. Maz’ya, S.A. Nazarov and B.A. Plamenevskii. Asymptotic theory of elliptic boundary value problems in singularly perturbed domain, Vol. 2. BirkhauserVerlag, 2000. https://doi.org/10.1007/978-3-0348-8432-7
V.G. Maz’ya and B.A. Plamenevskii. Estimates in Lp and Ho¨lder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr., 81:25– 82, 1978.
A. Movchan and S.A. Nazarov. Asymptotics of the stressed-deformed state near a spatial peak-type inclusion. Mech. Composite Materials, 5:792–800, 1985.
A. Movchan and S.A. Nazarov. The stressed-deformed state near a vertex of a three-dimensional absolutely rigid peak imbedded in an elastic body. Prikl. Mech., 25(12):10–19, 1989.
S.A. Nazarov. The structure of solutions of elliptic boundary value problems in thin domain. Vestnik Leningrad. Univ., Ser. Mat. Mekh. Astr., 7(2):65–68, 1982.
S.A. Nazarov. Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid. Siberian Math. Journal, 31(2):296–307, 1990. https://doi.org/10.1007/BF00970660
S.A. Nazarov. On the flow of water under a still stone. Math. Sbornik, 186(11):1621–1658, 1995. https://doi.org/10.1070/SM1995v186n11ABEH000086
S.A. Nazarov. On the essential spectrum of boundary value problems for systems of differential equations in a bounded domain with a peak. Funkt. Anal. i Prilozhen., 43(1):55–67, 2009.
S.A. Nazarov and K. Pileckas. The Reynolds flow of a fluid in a thin three dimensional channele structure of solutions of elliptic boundary value problems in thin domain. Litovskii Matem. Sb., 30:772–783, 1990.
S.A. Nazarov and K. Pileckas. Asymptotics of solutions to Stokes and NavierStokes equations in domains with paraboloidal outlets to infinity. Rend. Sem. Math. Univ. Padova, 99:1–43, 1998.
S.A. Nazarov and O.R. Polyakova. Asymptotic behaviour of the stressstrain state near a spatial singularity of the boundary of the beak tip type. J. Appl. Math. Mech., 57(5):887–902, 1993. https://doi.org/10.1016/0021-8928(93)90155-F
S.A. Nazarov, J. Sokolowski and J. Taskinen. Neumann Laplacian on a domain with tangential components in the boundary. Ann. Acad. Sci. Fenn. Math., 34:131–143, 2009.
K. Pileckas and A. Raciene. Non-stationary Navier-Stokes equation in 2D power cusp domain. ii. existence of the solution. Advances in Nonlinear Analysis, 10:1011–1038, 2020. https://doi.org/10.1515/anona-2020-0165
K. Pileckas and A. Raciene. Non-stationary Navier-Stokes equation in 2D power cusp domain. i. Construction of the formal asymptotic decomposition. Advances in Nonlinear Analysis, 10:982–1010, 2021. https://doi.org/10.1515/anona-2020-0164
V.V. Pukhnachev. Singular solutions of Navier-Stokes equations. AAMS Transl. Series 2., 232:193–218, 2014. https://doi.org/10.1090/trans2/232/11
A. Russo and A. Tartaglione. On the existence of singular solutions of the stationary Navier-Stokes problem. Lithuanian Math. J., 53(4):423–437, 2013. https://doi.org/10.1007/s10986-013-9219-3
G. Cardone, S.A. Nazarov and J. Sokolowski. Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary. Asymptotic Analysis, 62(1-2):41–88, 2009. https://doi.org/10.3233/ASY-2008-0915
A. Eismontaite and K. Pileckas. On singular solutions of time-periodic and steady Stokes problems in a power cusp domain. Applicable Analysis, 97(3):415–437, 2018. https://doi.org/10.1080/00036811.2016.1269321
A. Eismontaite and K. Pileckas. On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain. Applicable Analysis, 98(13):2400–2422, 2019. https://doi.org/10.1080/00036811.2018.1460815
I.V. Kamotski and V.G. Maz’ya. On the third boundary value problem in domains with cusps. Journal of Mathematical Sciences, 173(5):609–631, 2011. https://doi.org/10.1007/s10958-011-0262-5
L.V. Kapitanskii and K. Pileckas. Certain problems of vector analysis. J. Sov. Math., 32(5):469–483, 1986. https://doi.org/10.1007/BF01372197
K. Kaulakyte and N. Kloviene. On nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a symmetric cusp domain. Mathematical Modelling and Analysis, 26(1):55–71, 2021. https://doi.org/10.3846/mma.2021.12173
K. Kaulakyte, N. Kloviene and K. Pileckas. Nonhomogeneous boundary value problem for the stationary Navier-Stokes equations in a domain with a cusp. Z. Angew. Math. Phys., 70(36), 2019. https://doi.org/10.1007/s00033-019-1075-5
K. Kaulakyte and K. Pileckas. Nonhomogeneous boundary value problem for the time periodic linearized Navier-Stokes system in a domain with outlet to infinity. J. of Mathematical Analysis and Applications, 489(1), 2020. https://doi.org/10.1016/j.jmaa.2020.124126
H. Kim and H. Kozono. A removable isolated singularity theorem for the stationary Navier-Stokes equations. J. Diff. Equations, 220(1):68–84, 2006. https://doi.org/10.1016/j.jde.2005.02.002
M.B. Korobkov, K. Pileckas, V.V. Pukhnachev and R. Russo. The flux problem for the Navier-Stokes equations. Russian Math. Surveys, 69(6):1065–1122, 2014. https://doi.org/10.1070/RM2014v069n06ABEH004928
V. Kozlov and J. Rossmann. On the nonstationary Stokes system in a cone. J. of Diff. Equations, 260(12):8277–8315, 2016. https://doi.org/10.1016/j.jde.2016.02.024
V. Kozlov and J. Rossmann. On the nonstationary Stokes system in a cone: Asymptotics of solutions at infinity. J. of Mathematical Analysis and Applications, 486(10), 2020. https://doi.org/10.1016/j.jmaa.2019.123821
V. Kozlov and J. Rossmann. On the nonstationary Stokes system in a cone (Lp theory). J. of Mathematical Fluid Mechanics, 22(42), 2020. https://doi.org/10.1007/s00021-020-00502-w
O.A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. Gordon and Breach, 1969.
V.G. Maz’ya, S.A. Nazarov and B.A. Plamenevskii. Asymptotic theory of elliptic boundary value problems in singularly perturbed domain, Vol. 2. BirkhauserVerlag, 2000. https://doi.org/10.1007/978-3-0348-8432-7
V.G. Maz’ya and B.A. Plamenevskii. Estimates in Lp and Ho¨lder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr., 81:25– 82, 1978.
A. Movchan and S.A. Nazarov. Asymptotics of the stressed-deformed state near a spatial peak-type inclusion. Mech. Composite Materials, 5:792–800, 1985.
A. Movchan and S.A. Nazarov. The stressed-deformed state near a vertex of a three-dimensional absolutely rigid peak imbedded in an elastic body. Prikl. Mech., 25(12):10–19, 1989.
S.A. Nazarov. The structure of solutions of elliptic boundary value problems in thin domain. Vestnik Leningrad. Univ., Ser. Mat. Mekh. Astr., 7(2):65–68, 1982.
S.A. Nazarov. Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid. Siberian Math. Journal, 31(2):296–307, 1990. https://doi.org/10.1007/BF00970660
S.A. Nazarov. On the flow of water under a still stone. Math. Sbornik, 186(11):1621–1658, 1995. https://doi.org/10.1070/SM1995v186n11ABEH000086
S.A. Nazarov. On the essential spectrum of boundary value problems for systems of differential equations in a bounded domain with a peak. Funkt. Anal. i Prilozhen., 43(1):55–67, 2009.
S.A. Nazarov and K. Pileckas. The Reynolds flow of a fluid in a thin three dimensional channele structure of solutions of elliptic boundary value problems in thin domain. Litovskii Matem. Sb., 30:772–783, 1990.
S.A. Nazarov and K. Pileckas. Asymptotics of solutions to Stokes and NavierStokes equations in domains with paraboloidal outlets to infinity. Rend. Sem. Math. Univ. Padova, 99:1–43, 1998.
S.A. Nazarov and O.R. Polyakova. Asymptotic behaviour of the stressstrain state near a spatial singularity of the boundary of the beak tip type. J. Appl. Math. Mech., 57(5):887–902, 1993. https://doi.org/10.1016/0021-8928(93)90155-F
S.A. Nazarov, J. Sokolowski and J. Taskinen. Neumann Laplacian on a domain with tangential components in the boundary. Ann. Acad. Sci. Fenn. Math., 34:131–143, 2009.
K. Pileckas and A. Raciene. Non-stationary Navier-Stokes equation in 2D power cusp domain. ii. existence of the solution. Advances in Nonlinear Analysis, 10:1011–1038, 2020. https://doi.org/10.1515/anona-2020-0165
K. Pileckas and A. Raciene. Non-stationary Navier-Stokes equation in 2D power cusp domain. i. Construction of the formal asymptotic decomposition. Advances in Nonlinear Analysis, 10:982–1010, 2021. https://doi.org/10.1515/anona-2020-0164
V.V. Pukhnachev. Singular solutions of Navier-Stokes equations. AAMS Transl. Series 2., 232:193–218, 2014. https://doi.org/10.1090/trans2/232/11
A. Russo and A. Tartaglione. On the existence of singular solutions of the stationary Navier-Stokes problem. Lithuanian Math. J., 53(4):423–437, 2013. https://doi.org/10.1007/s10986-013-9219-3