[L'existence d'une solution en 3D pour la dynamique de l'océan à grande échelle]
L'auteur considère le système 3D d'équations décrivant la dynamique de l'océan à grande échelle en coordonnées cartésiennes. Il démontre, pour tout coefficient de viscosité et toute donnée initiale, l'existence et l'unicité d'une solution sur un intervalle de temps
For the 3D system of equations describing large-scale ocean dynamics in the Cartesian coordinate system existence and uniqueness of a solution on an arbitrary time interval
Accepté le :
Publié le :
Georgij M. Kobelkov 1
@article{CRMATH_2006__343_4_283_0, author = {Georgij M. Kobelkov}, title = {Existence of a solution {\textquoteleft}in the large{\textquoteright} for the {3D} large-scale ocean dynamics equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {283--286}, publisher = {Elsevier}, volume = {343}, number = {4}, year = {2006}, doi = {10.1016/j.crma.2006.04.020}, language = {en}, }
Georgij M. Kobelkov. Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations. Comptes Rendus. Mathématique, Volume 343 (2006) no. 4, pp. 283-286. doi : 10.1016/j.crma.2006.04.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.020/
[1] Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics (16 Nov 2005) | arXiv
[2] Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, Volume 14 (2001) no. 1, pp. 1381-1408
[3] The primitive equations on the large scale ocean under the small depth hypothesis, Discrete and Continuous Dynamical Systems, Volume 9 (January 2003) no. 1
[4] Analyse Mathématique et Océnographie, Masson, Paris, 1997
[5] On the equations of the large-scale ocean, Nonlinearity, Volume 5 (1992), pp. 1007-1053
[6] New formulations of the primitive equations of the atmosphere and applications, Nonlinearity, Volume 5 (1992), pp. 237-288
[7] Mathematical Models of Ocean Circulation (G.I. Marchuk; A.S. Sarkisyan, eds.), Nauka, Novosibirsk, 1980 (in Russian)
[8] Some mathematical problems in geophysical fluid dynamics (S. Frielander; D. Serr, eds.), Handbook of Mathematical Fluid Dynamics, vol. 3, Elsevier, 2004, pp. 535-658
- Global well-posedness of strong solutions to the primitive equations without vertical diffusivity, Applied Mathematics Letters, Volume 150 (2024), p. 108943 | DOI:10.1016/j.aml.2023.108943
- Nonuniqueness of Generalised Weak Solutions to the Primitive and Prandtl Equations, Journal of Nonlinear Science, Volume 34 (2024) no. 4 | DOI:10.1007/s00332-024-10032-8
- The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain, Nonlinearity, Volume 37 (2024) no. 7, p. 075024 | DOI:10.1088/1361-6544/ad5131
- Global existence and large-time behavior for primitive equations with the free boundary, Science China Mathematics, Volume 67 (2024) no. 10, p. 2303 | DOI:10.1007/s11425-022-2191-4
- On the Rigorous Mathematical Derivation for the Viscous Primitive Equations with Density Stratification, Acta Mathematica Scientia, Volume 43 (2023) no. 3, p. 1081 | DOI:10.1007/s10473-023-0306-1
- Blow-up criterion of solutions of the horizontal viscous primitive equations with horizontal eddy diffusivity, Applied Mathematics Letters, Volume 145 (2023), p. 108743 | DOI:10.1016/j.aml.2023.108743
- Rigorous derivation of the full primitive equations by the scaled Boussinesq equations with rotation, Bulletin of the Malaysian Mathematical Sciences Society, Volume 46 (2023) no. 3 | DOI:10.1007/s40840-023-01482-6
- On energy conservation for the hydrostatic Euler equations: an Onsager conjecture, Calculus of Variations and Partial Differential Equations, Volume 62 (2023) no. 8 | DOI:10.1007/s00526-023-02558-8
- Global existence and asymptotic stability of the free boundary problem of the primitive equations with heat insulation, Journal of Differential Equations, Volume 371 (2023), p. 549 | DOI:10.1016/j.jde.2023.06.042
- Global well‐posedness of the 3D primitive equations with only horizontal eddy diffusivity and delays in both convective and heat source terms, Mathematical Methods in the Applied Sciences, Volume 46 (2023) no. 14, p. 14895 | DOI:10.1002/mma.9353
- Higher-order error estimates for physics-informed neural networks approximating the primitive equations, Partial Differential Equations and Applications, Volume 4 (2023) no. 4 | DOI:10.1007/s42985-023-00254-y
- The exponential behavior and stability of the stochastic three-dimensional primitive equations driven by Lévy noise, Stochastics and Dynamics, Volume 23 (2023) no. 01 | DOI:10.1142/s0219493723500077
- Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations, Stochastics and Partial Differential Equations: Analysis and Computations, Volume 11 (2023) no. 4, p. 1470 | DOI:10.1007/s40072-022-00266-6
- On the Effect of Rotation on the Life-Span of Analytic Solutions to the 3D Inviscid Primitive Equations, Archive for Rational Mechanics and Analysis, Volume 243 (2022) no. 2, p. 747 | DOI:10.1007/s00205-021-01748-y
- Global well-posedness of the three-dimensional viscous primitive equations with bounded delays, Discrete and Continuous Dynamical Systems - B, Volume 27 (2022) no. 11, p. 6771 | DOI:10.3934/dcdsb.2022019
- Martingale Solutions of the Stochastic 2D Primitive Equations with Anisotropic Viscosity, ESAIM: Probability and Statistics, Volume 26 (2022), p. 243 | DOI:10.1051/ps/2022006
- The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations, Journal of Differential Equations, Volume 306 (2022), p. 492 | DOI:10.1016/j.jde.2021.10.048
- Uniform stability and convergence of the iterative solutions of the 3D steady viscous primitive equations of the ocean under the small depth assumption, Journal of Differential Equations, Volume 335 (2022), p. 549 | DOI:10.1016/j.jde.2022.07.011
- On the Effect of Fast Rotation and Vertical Viscosity on the Lifespan of the 3D Primitive Equations, Journal of Mathematical Fluid Mechanics, Volume 24 (2022) no. 3 | DOI:10.1007/s00021-022-00705-3
- Global well-posedness of z-weak solutions to the primitive equations without vertical diffusivity, Journal of Mathematical Physics, Volume 63 (2022) no. 2 | DOI:10.1063/5.0065114
- Local Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations, Archive for Rational Mechanics and Analysis, Volume 241 (2021) no. 2, p. 729 | DOI:10.1007/s00205-021-01662-3
- Finite-time blowup and ill-posedness in Sobolev spaces of the inviscid primitive equations with rotation, Journal of Differential Equations, Volume 286 (2021), p. 557 | DOI:10.1016/j.jde.2021.03.037
- Global well-posedness of strong solutions to the 2D nonhomogeneous incompressible primitive equations with vacuum, Journal of Differential Equations, Volume 286 (2021), p. 624 | DOI:10.1016/j.jde.2021.03.042
- Global smooth solution of 2D temperature-dependent tropical climate model, Nonlinearity, Volume 34 (2021) no. 8, p. 5662 | DOI:10.1088/1361-6544/ac0d44
- Global well–posedness for the 3D primitive equations in anisotropic framework, Journal of Mathematical Analysis and Applications, Volume 484 (2020) no. 2, p. 123714 | DOI:10.1016/j.jmaa.2019.123714
- On the solutions of the 3D steady and unsteady primitive equations of the ocean, Journal of Mathematical Analysis and Applications, Volume 491 (2020) no. 1, p. 124243 | DOI:10.1016/j.jmaa.2020.124243
- On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation, Journal of Mathematical Fluid Mechanics, Volume 22 (2020) no. 3 | DOI:10.1007/s00021-020-00495-6
- Numerical Simulations of the Two-Dimensional Inviscid Hydrostatic Primitive Equations with Humidity and Saturation, Journal of Scientific Computing, Volume 83 (2020) no. 2 | DOI:10.1007/s10915-020-01215-y
- Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity, Physica D: Nonlinear Phenomena, Volume 412 (2020), p. 132606 | DOI:10.1016/j.physd.2020.132606
- Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems, Computers Mathematics with Applications, Volume 77 (2019) no. 9, p. 2291 | DOI:10.1016/j.camwa.2018.12.020
- The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: Rigorous justification of the hydrostatic approximation, Journal de Mathématiques Pures et Appliquées, Volume 124 (2019), p. 30 | DOI:10.1016/j.matpur.2018.04.006
- The Barotropic Quasi-Geostrophic Equation under a Free Surface, SIAM Journal on Mathematical Analysis, Volume 51 (2019) no. 3, p. 1836 | DOI:10.1137/17m1146816
- Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities, SIAM Journal on Mathematical Analysis, Volume 51 (2019) no. 3, p. 1913 | DOI:10.1137/18m1211994
- Recent Advances Concerning Certain Class of Geophysical Flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2018), p. 933 | DOI:10.1007/978-3-319-13344-7_22
- The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality, Nonlinearity, Volume 31 (2018) no. 10, p. 4692 | DOI:10.1088/1361-6544/aad525
- Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics, Zeitschrift für angewandte Mathematik und Physik, Volume 69 (2018) no. 5 | DOI:10.1007/s00033-018-1007-9
- Time discrete approximation of weak solutions to stochastic equations of geophysical fluid dynamics and applications, Chinese Annals of Mathematics, Series B, Volume 38 (2017) no. 2, p. 425 | DOI:10.1007/s11401-017-1077-6
- Strong solutions to the 3D primitive equations with only horizontal dissipation: Near H1 initial data, Journal of Functional Analysis, Volume 272 (2017) no. 11, p. 4606 | DOI:10.1016/j.jfa.2017.01.018
- Strong time-periodic solutions to the 3D primitive equations subject to arbitrary large forces, Nonlinearity, Volume 30 (2017) no. 10, p. 3979 | DOI:10.1088/1361-6544/aa8166
- Existence and Uniqueness of Weak Solutions to Viscous Primitive Equations for a Certain Class of Discontinuous Initial Data, SIAM Journal on Mathematical Analysis, Volume 49 (2017) no. 1, p. 1 | DOI:10.1137/15m1050513
- Existence and Regularity Results for the Inviscid Primitive Equations with Lateral Periodicity, Applied Mathematics Optimization, Volume 73 (2016) no. 3, p. 501 | DOI:10.1007/s00245-016-9345-5
- Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion, Communications on Pure and Applied Mathematics, Volume 69 (2016) no. 8, p. 1492 | DOI:10.1002/cpa.21576
- Averaging method applied to the three-dimensional primitive equations, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 10, p. 5681 | DOI:10.3934/dcds.2016049
- The finite dimensional global attractor for the 3D viscous Primitive Equations, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 12, p. 7001 | DOI:10.3934/dcds.2016104
- Global well-posedness of strong solutions to a tropical climate model, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 8, p. 4495 | DOI:10.3934/dcds.2016.36.4495
- Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete and Continuous Dynamical Systems - Series B, Volume 21 (2016) no. 9, p. 3053 | DOI:10.3934/dcdsb.2016087
- About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows, Discrete and Continuous Dynamical Systems - Series S, Volume 9 (2016) no. 5, p. 1565 | DOI:10.3934/dcdss.2016063
- Recent Advances Concerning Certain Class of Geophysical Flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2016), p. 1 | DOI:10.1007/978-3-319-10151-4_22-1
- Local well-posedness for the tropical climate model with fractional velocity diffusion, Kinetic and Related Models, Volume 9 (2016) no. 3, p. 551 | DOI:10.3934/krm.2016006
- A tropical atmosphere model with moisture: global well-posedness and relaxation limit, Nonlinearity, Volume 29 (2016) no. 9, p. 2674 | DOI:10.1088/0951-7715/29/9/2674
- Finite Dimensions of the Global Attractor for 3D Primitive Equations with Viscosity, Journal of Nonlinear Science, Volume 25 (2015) no. 1, p. 131 | DOI:10.1007/s00332-014-9223-8
- The primitive equations of the atmosphere in presence of vapour saturation, Nonlinearity, Volume 28 (2015) no. 3, p. 625 | DOI:10.1088/0951-7715/28/3/625
- Local and Global Well-Posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity, Archive for Rational Mechanics and Analysis, Volume 214 (2014) no. 1, p. 35 | DOI:10.1007/s00205-014-0752-y
- Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, Journal of Differential Equations, Volume 257 (2014) no. 11, p. 4108 | DOI:10.1016/j.jde.2014.08.003
- Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations, Journal of Mathematical Physics, Volume 55 (2014) no. 5 | DOI:10.1063/1.4875104
- Global Attractors for the Three-Dimensional Viscous Primitive Equations of Large-Scale Atmosphere in Log-Pressure Coordinate, Abstract and Applied Analysis, Volume 2013 (2013), p. 1 | DOI:10.1155/2013/758730
- Averaging of a 3D primitive equations with oscillating external forces, Applicable Analysis, Volume 92 (2013) no. 5, p. 869 | DOI:10.1080/00036811.2011.640628
- Asymptotic analysis for the 3D primitive equations in a channel, Discrete Continuous Dynamical Systems - S, Volume 6 (2013) no. 2, p. 401 | DOI:10.3934/dcdss.2013.6.401
- Well-posedness for the stochastic 2D primitive equations with Lévy noise, Science China Mathematics, Volume 56 (2013) no. 8, p. 1629 | DOI:10.1007/s11425-013-4590-4
- Non-autonomous 3D primitive equations with oscillating external force and its global attractor, Discrete Continuous Dynamical Systems - A, Volume 32 (2012) no. 1, p. 265 | DOI:10.3934/dcds.2012.32.265
- Numerical approximation of the inviscid 3D primitive equations in a limited domain, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 46 (2012) no. 3, p. 619 | DOI:10.1051/m2an/2011058
- Local existence of solutions to the free boundary value problem for the primitive equations of the ocean, Journal of Mathematical Physics, Volume 53 (2012) no. 10 | DOI:10.1063/1.4753991
- Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, Volume 25 (2012) no. 7, p. 2093 | DOI:10.1088/0951-7715/25/7/2093
- On the convergence of difference schemes for the equations of ocean dynamics, Sbornik: Mathematics, Volume 203 (2012) no. 8, p. 1091 | DOI:10.1070/sm2012v203n08abeh004256
- О сходимости разностных схем для уравнений динамики океана, Математический сборник, Volume 203 (2012) no. 8, p. 17 | DOI:10.4213/sm7896
- Cauchy convergence schemes for some nonlinear partial differential equations, Applicable Analysis, Volume 90 (2011) no. 1, p. 85 | DOI:10.1080/00036811003735956
- Pathwise Solutions of the 2-D Stochastic Primitive Equations, Applied Mathematics Optimization, Volume 63 (2011) no. 3, p. 401 | DOI:10.1007/s00245-010-9126-5
- Convergence of difference schemes for the large-scale ocean dynamics equations, Doklady Mathematics, Volume 84 (2011) no. 2, p. 702 | DOI:10.1134/s1064562411060342
- The exponential behavior of the stochastic three-dimensional primitive equations with multiplicative noise, Nonlinear Analysis: Real World Applications, Volume 12 (2011) no. 2, p. 799 | DOI:10.1016/j.nonrwa.2010.08.007
- Extreme events in solutions of hydrostatic and non-hydrostatic climate models, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 369 (2011) no. 1939, p. 1156 | DOI:10.1098/rsta.2010.0244
- Local martingale and pathwise solutions for an abstract fluids model, Physica D: Nonlinear Phenomena, Volume 240 (2011) no. 14-15, p. 1123 | DOI:10.1016/j.physd.2011.03.009
- Existence 'in the large' of a solution to the system of equations of large-scale ocean dynamics on a manifold, Sbornik: Mathematics, Volume 202 (2011) no. 10, p. 1463 | DOI:10.1070/sm2011v202n10abeh004195
- Существование “в целом” решения системы уравнений крупномасштабной динамики океана на многообразии, Математический сборник, Volume 202 (2011) no. 10, p. 55 | DOI:10.4213/sm7769
- On the uniqueness of z-weak solutions of the three-dimensional primitive equations of the ocean, Nonlinear Analysis: Real World Applications, Volume 11 (2010) no. 3, p. 1413 | DOI:10.1016/j.nonrwa.2009.02.031
- Pullback attractors for closed cocycles, Nonlinear Analysis: Theory, Methods Applications, Volume 73 (2010) no. 8, p. 2737 | DOI:10.1016/j.na.2010.06.057
- Stability of the Slow Manifold in the Primitive Equations, SIAM Journal on Mathematical Analysis, Volume 42 (2010) no. 1, p. 427 | DOI:10.1137/080733358
- Splitting schemes for the ocean dynamics equations, Moscow University Mathematics Bulletin, Volume 64 (2009) no. 1, p. 29 | DOI:10.3103/s0027132209010070
- The primitive equations of the ocean with delays, Nonlinear Analysis: Real World Applications, Volume 10 (2009) no. 2, p. 779 | DOI:10.1016/j.nonrwa.2007.11.003
- Boundary Value Problems for the Inviscid Primitive Equations in Limited Domains, Special Volume: Computational Methods for the Atmosphere and the Oceans, Volume 14 (2009), p. 481 | DOI:10.1016/s1570-8659(08)00211-1
- Some Mathematical Problems in Geophysical Fluid Dynamics, Special Volume: Computational Methods for the Atmosphere and the Oceans, Volume 14 (2009), p. 577 | DOI:10.1016/s1570-8659(08)00212-3
- RANDOM ATTRACTOR FOR THE 3D VISCOUS STOCHASTIC PRIMITIVE EQUATIONS WITH ADDITIVE NOISE, Stochastics and Dynamics, Volume 09 (2009) no. 02, p. 293 | DOI:10.1142/s0219493709002683
- Simulations of the 2.5D inviscid primitive equations in a limited domain, Journal of Computational Physics, Volume 227 (2008) no. 23, p. 9865 | DOI:10.1016/j.jcp.2008.08.005
- A Small Eddy Correction Algorithm for the Primitive Equations of the Ocean, Mathematical Modeling, Simulation, Visualization and e-Learning (2008), p. 107 | DOI:10.1007/978-3-540-74339-2_8
- STOCHASTIC SOLUTIONS OF THE TWO-DIMENSIONAL PRIMITIVE EQUATIONS OF THE OCEAN AND ATMOSPHERE WITH AN ADDITIVE NOISE, Analysis and Applications, Volume 05 (2007) no. 02, p. 183 | DOI:10.1142/s0219530507000948
- A 2.5D MODEL FOR THE EQUATIONS OF THE OCEAN AND THE ATMOSPHERE, Analysis and Applications, Volume 05 (2007) no. 03, p. 199 | DOI:10.1142/s021953050700095x
- On the regularity of the primitive equations of the ocean, Nonlinearity, Volume 20 (2007) no. 12, p. 2739 | DOI:10.1088/0951-7715/20/12/001
- Finite Dimensional Behaviors of the Primitive Equations Under Small Depth Assumption, Numerical Functional Analysis and Optimization, Volume 28 (2007) no. 7-8, p. 853 | DOI:10.1080/01630560701493248
Cité par 87 documents. Sources : Crossref
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier