Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type — Oak Academic Publishing
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Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type
University of Shanghai for Science and Technology, Shanghai, China
1 University of Shanghai for Science and Technology, Shanghai, China
In this paper, we study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum. It is proved that there exists a unique strong solution for the two-dimensional Cauchy problem established that the initial density and the initial temperature decay not extremely slow. Particularly, it is allowed to be arbitrarily large for the initial data and vacuum states for the initial density, even including the compact support. Moreover, when the density depends on the Korteweg term with the viscosity coefficient and capillary coefficient, we obtain a consistent priority estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, when the pressure and external force are not affected, we deform the fluid models of Korteweg type, we can obtain the large time decay rates of the gradients of velocity, temperature and pressure.
KeywordsIncompressible Boussinesq EquationKorteweg TypeGlobal Strong SolutionsLarge Time BehaviorVacuum
Cannon, J. and DiBenedetto, E. (1980) The Initial Value Problem for the Boussinesq Equations with Data in L p . In: Rautmann, R., Ed., Approximation Methods for Navier-Stokes Problems. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086903
Hou, T.Y. and Li, C. (2005) Global Well-Posedness of the Viscous Boussinesq Equations. Discrete & Continuous Dynamical Systems, 12, 1-12. https://doi.org/10.3934/dcds.2005.12.1
Chae, D. (2006) Global Regularity for the 2D Boussinesq Equations with Partial Viscosity Terms. Advances in Mathematics, 203, 497-513. https://doi.org/10.1016/j.aim.2005.05.001
Lorca, S.A. and Boldrini, J.L. (1996) The Initial Value Problem for a Generalized Boussinesq Model: Regularity and Global Existence of Strong Solutions. Matemática Contemporanea, 11, 71-94.
Lorca, S.A. and Boldrini, J.L. (1999) The Initial Value Problem for a Generalized Boussinesq Model. Nonlinear Analysis, 36, 457-480. https://doi.org/10.1016/S0362-546X(97)00635-4
Qiu, H. and Yao, Z. (2017) Well-Posedness for Density-Dependent Boussinesq Equations without Dissipation Terms in Besov Spaces. Computers & Mathematics with Applications, 73, 1920-1931. https://doi.org/10.1016/j.camwa.2017.02.041
Wang, J. and Xie, F. (2015) Zero Dissipation Limit and Stability of Boundary Layers for the Heat Conductive Boussinesq Equations in a Bounded Domain. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145, 611-637. https://doi.org/10.1017/S0308210513000875
Haspot, B. (2009) Existence of Strong Solutions for Nonisothermal Korteweg System. Annales Mathématiques Blaise Pascal, 16, 431-481. https://doi.org/10.5802/ambp.274
Haspot, B. (2016) Existence of Global Strong Solution for Korteweg System with Large Infinite Energy Initial Data. Journal of Mathematical Analysis and Applications, 438, 395-443. https://doi.org/10.1016/j.jmaa.2016.01.047
Danchin, R. and Desjardins, B. (2001) Existence of Solutions for Compressible Fluid Models of Korteweg Type. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 18, 97-133. https://doi.org/10.1016/S0294-1449(00)00056-1
Haspot, B. (2011) Existence of Global Weak Solution for Compressible Fluid Models of Korteweg Type. Journal of Mathematical Fluid Mechanics, 13, 223-249. https://doi.org/10.1007/s00021-009-0013-2
Bresch, D., Desjardins, B. and Lin, C.K. (2003) On Some Compressible Fluid Models: Korteweg, Lubrication and Shallow Water Systems. Communications in Partial Differential Equations, 28, 843-868. https://doi.org/10.1081/PDE-120020499
Germain, P. and LeFloch, P.G. (2016) Finite Energy Method for Compressible Fluids: The Navie-Stokes-Korteweg Model. Communications on Pure and Applied Mathematics, 69, 3-61. https://doi.org/10.1002/cpa.21622
Chen, Z., He, L. and Zhao, H. (2017) Global Smooth Solutions to the Nonisothermal Compressible Fluid Models of Korteweg Type with Large Initial Data. Zeitschrift für angewandte Mathematik und Physik, 68, Article No. 79. https://doi.org/10.1007/s00033-017-0822-8
Chen, Z., Chai, X.J., Dong, B.Q. and Zhao, H.J. (2015) Global Classical Solutions to the One-Dimensional Compressible Fluid Models of Korteweg Type with Large Initial Data. Journal of Differential Equations, 259, 4376-4411. https://doi.org/10.1016/j.jde.2015.05.023
Liu, Y., Wang, W. and Zheng, S.N. (2018) Strong Solutions to the Cauchy Problem of Two-Dimensional Incompressible Fluid Models of Korteweg Type. Journal of Mathematical Analysis and Applications, 465, 1075-1093. https://doi.org/10.1016/j.jmaa.2018.05.054
Lü, B.Q., Xu, Z.H. and Zhong, X. (2017) Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Promblem of 2D Density-Dependent Magnetohydrodynamic Equations with Vacuum. Journal de Mathématiques Pures et Appliquées, 108, 41-62. https://doi.org/10.1016/j.matpur.2016.10.009
Lü, B.Q., Shi, X.D. and Zhong, X. (2018) Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Promblem of 2D Density-Dependent Navier-Stokes Equations with Vacuum. Nonlinearity, 31, 2617-2632. https://doi.org/10.1088/1361-6544/aab31f
Li, J. and Xin, Z.P. (2019) Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum. Annals of PDE, 5, Article No. 37. https://doi.org/10.1007/s40818-019-0064-5
Huang, X.D. and Wang, Y. (2014) Global Strong Solution with Vacuum to the Two-Dimensional Density-Dependent Navier-Stokes System. SIAM Journal on Mathematical Analysis, 46, 1771-1788. https://doi.org/10.1137/120894865
Hoff, D. (1995) Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data. Journal of Differential Equations, 120, 215-254. https://doi.org/10.1006/jdeq.1995.1111
Desjardins, B. (1997) Regularity Results for Two-Dimensional Flows of Multiphase Viscous Fluids. Archive for Rational Mechanics and Analysis, 137, 135-158. https://doi.org/10.1007/s002050050025
Lü, B.Q. and Huang, B. (2015) On Strong Solutions to the Cauchy Problem of the Two-Dimensional Compressible Magnetohydrodynamic Equations with Vacuum. Nonlinearity, 28, 509-530. https://doi.org/10.1088/0951-7715/28/2/509
Beal, J.T., Kato, T. and Majda, A. (1984) Remarks on the Breakdown of Smooth Solutions for the 3D Euler Equations. Communications in Mathematical Physics, 94, 61-66. |https://doi.org/10.1007/BF01212349
Liu, S.Q. and Zhang, J.W. (2016) Global Well-Posedeness for the Two-Dimensional Equations of Nonhomogeneous Incompressible Liquid Crystal Flows with Nonnegative Density. Discrete & Continuous Dynamical Systems-B, 21, 2631-2648. https://doi.org/10.3934/dcdsb.2016065
Liu, Y. and Wang, W. (2018) Strong Solutions to the Cauchy Problem of Two-Dimensional Incompressible Fluid Models of Korteweg Type. Journal of Mathematical Analysis and Applications, 465, 1075-1093. https://doi.org/10.1016/j.jmaa.2018.05.054
Lions, P.L. (1996) Mathematical Topicas in Fluid Mechanics, Vol. I: Incompressible Models. Oxford University Press, Oxford.
Stein, E.M. (1993) Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton.
Coifman, R., Lions, P.L., Meyer, Y. and Semmes, S. (1993) Compensated Compactness and Hardy Spaces. Journal de Mathématiques Pures et Appliquées, 72, 247-286.
Temam, R. (2001) Navier-Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, New York. https://doi.org/10.1090/chel/343
Huang, X.D., Li, J. and Xin, Z.P. (2012) Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations. Communications on Pure and Applied Mathematics, 65, 549-585. https://doi.org/10.1002/cpa.21382