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

* Corresponding author: Amru Hussein

Received  September 2019 Revised  October 2020 Published  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.

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 - A, 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. CaoS. IbrahimK. 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. CaoJ. 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. CaoJ. 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. GaldiM. 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. GeissertM. 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. KukavicaR. TemamV. 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. LionsR. 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. LionsR. 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. ŁukaszewiczE. 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. CaoS. IbrahimK. 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. CaoJ. 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. CaoJ. 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. GaldiM. 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. GeissertM. 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. KukavicaR. TemamV. 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. LionsR. 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. LionsR. 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. ŁukaszewiczE. 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
[1]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[2]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[3]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[4]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[5]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[6]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[7]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[8]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[9]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[10]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075

[11]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[12]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[13]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[14]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[15]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[16]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

[17]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[18]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[19]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[20]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (18)
  • HTML views (108)
  • Cited by (0)

Other articles
by authors

[Back to Top]