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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

Amru Hussein 1,, , Martin Saal 2, and  Marc Wrona 3,

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

* Corresponding author: Amru Hussein

Received  September 2019 Revised  October 2020 Published  July 2021 Early access  December 2020

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 & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Second Edition, Springer, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

H. Triebel, Theory of Function Spaces, (Reprint of 1983 edition) Springer AG, Basel, 2010. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[34] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Second Edition, Cambridge Univ. Press, 2006.   Google Scholar
[35]

W. M. Washington and C. L. Parkinson, An Introduction to Three Dimensional Climate Modeling, Second Edition, University Science Books, 2005. Google Scholar

[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.  Google Scholar

[37]

M. Wrona, Liquid Crystals and the Primitive Equations: An Approach by Maximal Regularity, Dissertation. Technische Universität Darmstadt, 2020. doi: 10.25534/tuprints-00011551.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Second Edition, Springer, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

H. Triebel, Theory of Function Spaces, (Reprint of 1983 edition) Springer AG, Basel, 2010. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[34] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Second Edition, Cambridge Univ. Press, 2006.   Google Scholar
[35]

W. M. Washington and C. L. Parkinson, An Introduction to Three Dimensional Climate Modeling, Second Edition, University Science Books, 2005. Google Scholar

[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.  Google Scholar

[37]

M. Wrona, Liquid Crystals and the Primitive Equations: An Approach by Maximal Regularity, Dissertation. Technische Universität Darmstadt, 2020. doi: 10.25534/tuprints-00011551.  Google Scholar

Figure 1.  From $ z $-weak to global solution
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