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July
2021, 41(7): 3063-3092.
doi: 10.3934/dcds.2020398
Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions
| 1. | Department of Mathematics, TU Kaiserslautern, Paul-Ehrlich-Straße 31, 67663 Kaiserslautern, Germany |
| 2. | Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy |
| 3. | Departement of Mathematics, TU Darmstadt, Schlossgartenstr 7, 64289 Darmstadt, Germany |
The $ 3D $-primitive equations with only horizontal viscosity are considered on a cylindrical domain $ \Omega = (-h,h) \times G $, $ G\subset \mathbb{R}^2 $ smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering a vanishing vertical viscosity limit, we apply a direct approach which in particular avoids unnecessary boundary conditions on top and bottom. For the initial value problem, we obtain existence and uniqueness of local $ z $-weak solutions for initial data in $ H^1((-h,h),L^2(G)) $ and local strong solutions for initial data in $ H^1(\Omega) $. If $ v_0\in H^1((-h,h),L^2(G)) $, $ \partial_z v_0\in L^q(\Omega) $ for $ q>2 $, then the $ z $-weak solution regularizes instantaneously and thus extends to a global strong solution. This goes beyond the global well-posedness result by Cao, Li and Titi (J. Func. Anal. 272(11): 4606-4641, 2017) for initial data near $ H^1 $ in the periodic setting. For the time-periodic problem, existence and uniqueness of $ z $-weak and strong time periodic solutions is proven for small forces. Since this is a model with hyperbolic and parabolic features for which classical results are not directly applicable, such results for the time-periodic problem even for small forces are not self-evident.
Keywords:
Primitive equations,
horizontal viscosity,
initial value problem,
time-periodic solutions.
Mathematics Subject Classification:
Primary: 35Q35; Secondary: 35A01, 35K65, 35Q86, 35M10, 76D03, 86A05, 86A10.
Citation:
Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete and Continuous Dynamical Systems,
2021, 41
(7)
: 3063-3092.
doi: 10.3934/dcds.2020398
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Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482.
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Ch. Cao, J. Li and E. S. Titi,
Global well-posedness of the $3D$ primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., 69 (2016), 1492-1531.
doi: 10.1002/cpa.21576. |
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Ch. Cao, J. Li and E. S. Titi,
Strong solutions to the $3D$ primitive equations with only horizontal dissipation: Near $H^1$ initial data, J. Funct. Anal., 272 (2017), 4606-4641.
doi: 10.1016/j.jfa.2017.01.018. |
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Ch. Cao and E. S. Titi,
Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
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S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003. |
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T. Eiter and M. Kyed, Time-periodic linearized Navier-Stokes equations: An approach based on Fourier multipliers, In Particles in Flows, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, 2017, 77–137.
doi: 10.1007/978-3-319-60282-0_2. |
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P. Galdi, M. Hieber and T. Kashiwabara,
Strong time-periodic solutions to the 3D primitive equations subject to arbitrary large forces, Nonlinearity, 30 (2017), 3979-3992.
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M. Geissert, M. Hieber and T. H. Nguyen,
A general approach to time periodic incompressible viscous fluid flow problems, Arch. Ration. Mech. Anal., 220 (2016), 1095-1118.
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M. Hieber and T. Kashiwabara,
Global strong well-posedness of the three dimensional primitive equations in $L^p$-spaces, Arch. Rational Mech. Anal., 221 (2016), 1077-1115.
doi: 10.1007/s00205-016-0979-x. |
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Ch-H. Hsia and M. Ch. Shiue,
On the asymptotic stability analysis and the existence of time-periodic solutions of the primitive equations, Indiana Univ. Math. J., 62 (2013), 403-441.
doi: 10.1512/iumj.2013.62.4902. |
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Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Ration. Mech. Anal., 221 (2016), 1317-1344.
doi: 10.1007/s00205-016-0985-z. |
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N. Ju,
On $H^2$ solutions and $z$-weak solutions of the 3D primitive equations, Indiana Univ. Math. J., 66 (2017), 973-996.
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Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, J. Differential Equations, 250 (2011), 1719-1746.
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I. Kukavica and M. Ziane,
On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.
doi: 10.1088/0951-7715/20/12/001. |
| [20] |
M. Kyed and J. Sauer,
A method for obtaining time-periodic $L^p$ estimates, J. Differential Equations, 262 (2017), 633-652.
doi: 10.1016/j.jde.2016.09.037. |
| [21] |
J. Li and E. S. Titi, Recent advances concerning certain class of geophysical flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 933–971, Springer, Cham, 2018.
doi: 10.1007/978-3-319-10151-4_22-1. |
| [22] |
J. L. Lions, R. Temam and Sh. H. Wang,
New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
doi: 10.1088/0951-7715/5/2/001. |
| [23] |
J. L. Lions, R. Temam and Sh. H. Wang,
On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
doi: 10.1088/0951-7715/5/5/002. |
| [24] |
J. L. Lions, R. Temam and Sh. H. Wang, Models for the coupled atmosphere and ocean. (CAO I, II), Comput. Mech. Adv., 1 (1993), 120pp. |
| [25] |
G. Łukaszewicz, E. E. Ortega-Torres and M. A. Rojas-Medar,
Strong periodic solutions for a class of abstract evolution equations, Nonlinear Anal., 54 (2003), 1045-1056.
doi: 10.1016/S0362-546X(03)00125-1. |
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A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, (Courant Lecture Notes in Mathematics vol 9), Providence, RI: American Mathematical Society, 2003.
doi: 10.1090/cln/009. |
| [27] |
H. Morimoto,
Survey on time periodic problem for fluid flow under inhomogeneous boundary condition, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 631-639.
doi: 10.3934/dcdss.2012.5.631. |
| [28] |
J. Pedlosky, Geophysical Fluid Dynamics, Second Edition, Springer, New York, 1987.
doi: 10.1007/978-1-4612-4650-3. |
| [29] |
J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhäuser/Springer, [Cham], 2016.
doi: 10.1007/978-3-319-27698-4. |
| [30] |
M. Saal, Primitive Equations with half horizontal viscosity, Advances in Differential Equations, 25 (2020), 651–685, https://arXiv.org/abs/1807.05045 arXiv: 1807.05045. |
| [31] |
J. Serrin,
A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122.
doi: 10.1007/BF00284169. |
| [32] |
Th. Tachim Medjo,
Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean, Discrete Contin. Dyn. Syst., 26 (2010), 1491-1508.
doi: 10.3934/dcds.2010.26.1491. |
| [33] |
H. Triebel, Theory of Function Spaces, (Reprint of 1983 edition) Springer AG, Basel, 2010.
doi: 10.1007/978-3-0346-0416-1. |
| [34] |
G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Second Edition, Cambridge Univ. Press, 2006.
|
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W. M. Washington and C. L. Parkinson, An Introduction to Three Dimensional Climate Modeling, Second Edition, University Science Books, 2005. |
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T. K. Wong, Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143 (2015), 1119–1125.
doi: 10.1090/S0002-9939-2014-12243-X. |
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M. Wrona, Liquid Crystals and the Primitive Equations: An Approach by Maximal Regularity, Dissertation. Technische Universität Darmstadt, 2020.
doi: 10.25534/tuprints-00011551. |
show all references
References:
| [1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
| [2] |
D. Bresch, F-Guillén-González, N. Masmoudi and M. A. Rodríguez-Bellido, On the uniqueness of weak solutions of the two-dimensional primitive equations, Differential Integral Equations, 16 (2003), 77–94, https://projecteuclid.org/euclid.die/1356060697. |
| [3] |
A. Celik and M. Kyed,
Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.
doi: 10.1002/zamm.201600280. |
| [4] |
Ch. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi,
Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482.
doi: 10.1007/s00220-015-2365-1. |
| [5] |
Ch. Cao, J. Li and E. S. Titi,
Global well-posedness of the $3D$ primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., 69 (2016), 1492-1531.
doi: 10.1002/cpa.21576. |
| [6] |
Ch. Cao, J. Li and E. S. Titi,
Strong solutions to the $3D$ primitive equations with only horizontal dissipation: Near $H^1$ initial data, J. Funct. Anal., 272 (2017), 4606-4641.
doi: 10.1016/j.jfa.2017.01.018. |
| [7] |
Ch. Cao and E. S. Titi,
Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math. (2), 166 (2007), 245-267.
doi: 10.4007/annals.2007.166.245. |
| [8] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003. |
| [9] |
T. Eiter and M. Kyed, Time-periodic linearized Navier-Stokes equations: An approach based on Fourier multipliers, In Particles in Flows, Adv. Math. Fluid Mech., Birkhäuser/Springer, Cham, 2017, 77–137.
doi: 10.1007/978-3-319-60282-0_2. |
| [10] |
P. Galdi, M. Hieber and T. Kashiwabara,
Strong time-periodic solutions to the 3D primitive equations subject to arbitrary large forces, Nonlinearity, 30 (2017), 3979-3992.
doi: 10.1088/1361-6544/aa8166. |
| [11] |
M. Geissert, M. Hieber and T. H. Nguyen,
A general approach to time periodic incompressible viscous fluid flow problems, Arch. Ration. Mech. Anal., 220 (2016), 1095-1118.
doi: 10.1007/s00205-015-0949-8. |
| [12] |
Y. Giga, M. Gries, A. Hussein, M. Hieber and T. Kashiwabara, The Primitive Equations in the scaling invariant space $L^{\infty}(L^1)$, Preprint, https://arXiv.org/abs/1710.04434 arXiv: 1710.04434, 2017. |
| [13] |
M. Hieber and A. Hussein, An approach to the primitive equations for ocean and atmospheric dynamics by evolution equations, In: Fluids Under Pressure, T. Bodnar et al. (eds.), Advances in Math. Fluid Mech., Birkhäuser, 2020. |
| [14] |
M. Hieber and T. Kashiwabara,
Global strong well-posedness of the three dimensional primitive equations in $L^p$-spaces, Arch. Rational Mech. Anal., 221 (2016), 1077-1115.
doi: 10.1007/s00205-016-0979-x. |
| [15] |
Ch-H. Hsia and M. Ch. Shiue,
On the asymptotic stability analysis and the existence of time-periodic solutions of the primitive equations, Indiana Univ. Math. J., 62 (2013), 403-441.
doi: 10.1512/iumj.2013.62.4902. |
| [16] |
D. Han-Kwan and T. T. Nguyen,
Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Ration. Mech. Anal., 221 (2016), 1317-1344.
doi: 10.1007/s00205-016-0985-z. |
| [17] |
N. Ju,
On $H^2$ solutions and $z$-weak solutions of the 3D primitive equations, Indiana Univ. Math. J., 66 (2017), 973-996.
doi: 10.1512/iumj.2017.66.6065. |
| [18] |
I. Kukavica, R. Temam, V. Vicol and M. Ziane,
Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, J. Differential Equations, 250 (2011), 1719-1746.
doi: 10.1016/j.jde.2010.07.032. |
| [19] |
I. Kukavica and M. Ziane,
On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.
doi: 10.1088/0951-7715/20/12/001. |
| [20] |
M. Kyed and J. Sauer,
A method for obtaining time-periodic $L^p$ estimates, J. Differential Equations, 262 (2017), 633-652.
doi: 10.1016/j.jde.2016.09.037. |
| [21] |
J. Li and E. S. Titi, Recent advances concerning certain class of geophysical flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 933–971, Springer, Cham, 2018.
doi: 10.1007/978-3-319-10151-4_22-1. |
| [22] |
J. L. Lions, R. Temam and Sh. H. Wang,
New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
doi: 10.1088/0951-7715/5/2/001. |
| [23] |
J. L. Lions, R. Temam and Sh. H. Wang,
On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.
doi: 10.1088/0951-7715/5/5/002. |
| [24] |
J. L. Lions, R. Temam and Sh. H. Wang, Models for the coupled atmosphere and ocean. (CAO I, II), Comput. Mech. Adv., 1 (1993), 120pp. |
| [25] |
G. Łukaszewicz, E. E. Ortega-Torres and M. A. Rojas-Medar,
Strong periodic solutions for a class of abstract evolution equations, Nonlinear Anal., 54 (2003), 1045-1056.
doi: 10.1016/S0362-546X(03)00125-1. |
| [26] |
A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, (Courant Lecture Notes in Mathematics vol 9), Providence, RI: American Mathematical Society, 2003.
doi: 10.1090/cln/009. |
| [27] |
H. Morimoto,
Survey on time periodic problem for fluid flow under inhomogeneous boundary condition, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 631-639.
doi: 10.3934/dcdss.2012.5.631. |
| [28] |
J. Pedlosky, Geophysical Fluid Dynamics, Second Edition, Springer, New York, 1987.
doi: 10.1007/978-1-4612-4650-3. |
| [29] |
J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhäuser/Springer, [Cham], 2016.
doi: 10.1007/978-3-319-27698-4. |
| [30] |
M. Saal, Primitive Equations with half horizontal viscosity, Advances in Differential Equations, 25 (2020), 651–685, https://arXiv.org/abs/1807.05045 arXiv: 1807.05045. |
| [31] |
J. Serrin,
A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122.
doi: 10.1007/BF00284169. |
| [32] |
Th. Tachim Medjo,
Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean, Discrete Contin. Dyn. Syst., 26 (2010), 1491-1508.
doi: 10.3934/dcds.2010.26.1491. |
| [33] |
H. Triebel, Theory of Function Spaces, (Reprint of 1983 edition) Springer AG, Basel, 2010.
doi: 10.1007/978-3-0346-0416-1. |
| [34] |
G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Second Edition, Cambridge Univ. Press, 2006.
|
| [35] |
W. M. Washington and C. L. Parkinson, An Introduction to Three Dimensional Climate Modeling, Second Edition, University Science Books, 2005. |
| [36] |
T. K. Wong, Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143 (2015), 1119–1125.
doi: 10.1090/S0002-9939-2014-12243-X. |
| [37] |
M. Wrona, Liquid Crystals and the Primitive Equations: An Approach by Maximal Regularity, Dissertation. Technische Universität Darmstadt, 2020.
doi: 10.25534/tuprints-00011551. |
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