| Title: | Continuous dependence of 2D large scale primitive equations on the boundary conditions in oceanic dynamics (English) | 
| Author: | Li, Yuanfei | 
| Author: | Xiao, Shengzhong | 
| Language: | English | 
| Journal: | Applications of Mathematics | 
| ISSN: | 0862-7940 (print) | 
| ISSN: | 1572-9109 (online) | 
| Volume: | 67 | 
| Issue: | 1 | 
| Year: | 2022 | 
| Pages: | 103-124 | 
| Summary lang: | English | 
| . | 
| Category: | math | 
| . | 
| Summary: | In this paper, we consider an initial boundary value problem for the two-dimensional primitive equations of large scale oceanic dynamics. Assuming that the depth of the ocean is a positive constant, we establish rigorous a priori bounds of the solution to problem. With the aid of these a priori bounds, the continuous dependence of the solution on changes in the boundary terms is obtained. (English) | 
| Keyword: | a priori bounds | 
| Keyword: | primitive equation | 
| Keyword: | continuous dependence | 
| MSC: | 35B40 | 
| MSC: | 35Q30 | 
| MSC: | 76D05 | 
| MSC: | 76U60 | 
| idZBL: | Zbl 07478520 | 
| idMR: | MR4392408 | 
| DOI: | 10.21136/AM.2021.0076-20 | 
| . | 
| Date available: | 2022-02-08T10:50:46Z | 
| Last updated: | 2024-03-04 | 
| Stable URL: | http://hdl.handle.net/10338.dmlcz/149362 | 
| . | 
| Reference: | [1] Chen, W.: Cauchy problem for thermoelastic plate equations with different damping mechanisms.Commun. Math. Sci. 18 (2020), 429-457. Zbl 07327456, MR 4101316, 10.4310/CMS.2020.v18.n2.a7 | 
| Reference: | [2] Chen, W.: Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms.Nonlinear Anal., Theory Methods Appl., Ser. A 202 (2021), Article ID 112160, 23 pages. Zbl 1450.35172, MR 4156977, 10.1016/j.na.2020.112160 | 
| Reference: | [3] Chen, W., Dao, T. A.: On the Cauchy problem for semilinear regularity-loss-type $\sigma$-evolution models with memory term.Nonlinear Anal., Real World Appl. 59 (2021), Article ID 103265, 26 pages. Zbl 07347902, MR 4177989, 10.1016/j.nonrwa.2020.103265 | 
| Reference: | [4] Chiodaroli, E., Michálek, M.: Existence and non-uniqueness of global weak solutions to inviscid primitive and Boussinesq equations.Commun. Math. Phys. 353 (2017), 1201-1216. Zbl 1373.35238, MR 3652488, 10.1007/s00220-017-2846-5 | 
| Reference: | [5] Fang, D., Han, B.: Global well-posedness for the 3D primitive equations in anisotropic framework.J. Math. Anal. Appl. 484 (2020), Article ID 123714, 22 pages. Zbl 1433.35268, MR 4039433, 10.1016/j.jmaa.2019.123714 | 
| Reference: | [6] Hameed, A. A., Harfash, A. J.: Continuous dependence of double diffusive convection in a porous medium with temperature-dependent density.Basrah J. Sci. 37 (2019), 1-15. MR 4241308 | 
| Reference: | [7] Hardy, C. H., Littlewood, J. E., Pólya, G.: Inequalities.Cambridge University Press, Cambridge (1952). Zbl 0047.05302, MR 0046395 | 
| Reference: | [8] Hieber, M., Hussein, A., Kashiwabara, T.: Global strong $L^p$ well-posedness of the 3D primitive equations with heat and salinity diffusion.J. Differ. Equations 261 (2016), 6950-6981. Zbl 1351.35139, MR 3562316, 10.1016/j.jde.2016.09.010 | 
| Reference: | [9] Huang, D.-W., Guo, B.-L.: On two-dimensional large-scale primitive equations in oceanic dynamics. I.Appl. Math. Mech., Engl. Ed. 28 (2007), 581-592. Zbl 1231.35264, MR 2325168, 10.1007/s10483-007-0503-x | 
| Reference: | [10] Huang, D.-W., Guo, B.-L.: On two-dimensional large-scale primitive equations in oceanic dynamics. II.Appl. Math. Mech., Engl. Ed. 28 (2007), 593-600. Zbl 1231.35265, MR 2325169, 10.1007/s10483-007-0504-x | 
| Reference: | [11] Huang, D., Shen, T., Zheng, Y.: Ergodicity of two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise.Appl. Math. Lett. 102 (2020), Article ID 106146, 8 pages. Zbl 1441.60046, MR 4037702, 10.1016/j.aml.2019.106146 | 
| Reference: | [12] Jiu, Q., Li, M., Wang, F.: Uniqueness of the global weak solutions to 2D compressible primitive equations.J. Math. Anal. Appl. 461 (2018), 1653-1671. Zbl 1406.35281, MR 3765508, 10.1016/j.jmaa.2017.12.035 | 
| Reference: | [13] Li, Y.: Continuous dependence on boundary parameters for three-dimensional viscous primitive equation of large-scale ocean atmospheric dynamics.J. Jilin Univ., Sci. 57 (2019), 1053-1059 Chinese. Zbl 1449.35036, MR 3970526, 10.13413/j.cnki.jdxblxb.2019038 | 
| Reference: | [14] Li, Y.: Continuous dependence on viscosity coefficient for the primitive equations.J. Shandong Univ., Nat. Sci. 54 (2019), 12-23 Chinese. Zbl 1449.35037, MR 4114791, 10.6040/j.issn.1671-9352.0.2019.539 | 
| Reference: | [15] Li, Y., Lin, C.: Continuous dependence for the nonhomogeneous Brinkman-Forchheimer equations in a semi-infinite pipe.Appl. Math. Comput. 244 (2014), 201-208. Zbl 1335.35198, MR 3250570, 10.1016/j.amc.2014.06.082 | 
| Reference: | [16] Lions, J. L., Temam, R., Wang, S.: New formulations of the primitive equations of atmosphere and applications.Nonlinearity 5 (1992), 237-288. Zbl 0746.76019, MR 1158375, 10.1088/0951-7715/5/2/001 | 
| Reference: | [17] Lions, J. L., Temam, R., Wang, S.: On the equations of the large-scale ocean.Nonlinearity 5 (1992), 1007-1053. Zbl 0766.35039, MR 1187737, 10.1088/0951-7715/5/5/002 | 
| Reference: | [18] Lions, J. L., Temam, R., Wang, S.: Models of the coupled atmosphere and ocean (CAO I).Comput. Mech. Adv. 1 (1993), 5-54. Zbl 0805.76011, MR 1252502 | 
| Reference: | [19] Lions, J. L., Temam, R., Wang, S.: Mathematical theory for the coupled atmosphere-ocean models (CAO III).J. Math. Pures Appl., IX. Sér. 74 (1995), 105-163. Zbl 0866.76025, MR 1325825 | 
| Reference: | [20] Liu, Y.: Continuous dependence for a thermal convection model with temperature-dependent solubitity.Appl. Math. Comput. 308 (2017), 18-30. Zbl 1411.35228, MR 3638153, 10.1016/j.amc.2017.03.004 | 
| Reference: | [21] Liu, Y., Xiao, S., Lin, Y.: Continuous dependence for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded domain.Math. Comput. Simul. 150 (2018), 66-82. Zbl 07316235, MR 3783079, 10.1016/j.matcom.2018.02.009 | 
| Reference: | [22] Mitrinović, D. S.: Analytical Inequalities.Die Grundlehren der mathematischen Wissenschaften 165. Springer, Berlin (1970). Zbl 199.38101, MR 0274686, 10.1007/978-3-642-99970-3 | 
| Reference: | [23] Petcu, M., Temam, R., Wirosoetisno, D.: Existence and regularity results for the primitive equations in the two dimensions.Commun. Pure Appl. Anal. 3 (2004), 115-131. Zbl 1060.35033, MR 2033944, 10.3934/cpaa.2004.3.115 | 
| Reference: | [24] Scott, N. L., Straughan, B.: Continuous dependence on the reaction terms in porous convection with surface reactions.Q. Appl. Math. 71 (2013), 501-508. Zbl 1275.35022, MR 3112825, 10.1090/S0033-569X-2013-01289-X | 
| Reference: | [25] Sun, J., Cui, S.: Sharp well-posedness and ill-posedness of the three-dimensional primitive equations of geophysics in Fourier-Besov spaces.Nonlinear Anal., Real World Appl. 48 (2019), 445-465. Zbl 1453.35152, MR 3915400, 10.1016/j.nonrwa.2019.02.003 | 
| Reference: | [26] Sun, J., Yang, M.: Global well-posedness for the viscous primitive equations of geophysics.Bound. Value Probl. 2016 (2016), Article ID 21, 16 pages. Zbl 1330.35349, MR 3451597, 10.1186/s13661-016-0526-6 | 
| Reference: | [27] You, B., Li, F.: Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics.Z. Angew. Math. Phys. 69 (2018), Article ID 114, 13 pages. Zbl 1400.35037, MR 3846299, 10.1007/s00033-018-1007-9 | 
| . |