doi: 10.3934/dcdsb.2020348

Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions

Southwestern University of Finance and Economics, School of Economic Mathematics, Chengdu 611130, China

* Corresponding author: Zhilei Liang

Received  May 2020 Published  November 2020

We study the Cauchy problem for the equations describing a viscous compressible and heat-conductive fluid in two dimensions. By imposing a weight function to initial density to deal with Sobolev embedding in critical space, and constructing an ad-hoc truncation to control the quadratic nonlinearity appeared in energy equation, we establish the local in time existence of unique strong solution with large initial data. The vacuum state at infinity or the compactly supported density is permitted. Moreover, we provide a different approach and slightly improve the weighted $ L^{p} $ estimates in [19,Theorem B.1].

Citation: Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020348
References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[3]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Composition Math., 53 (1984), 259-275.   Google Scholar

[4]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[5]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[6]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[7]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differ. Eqs., 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[8] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.   Google Scholar
[9]

E. FeireislA. Novotny and H. Petzeltov$\acute{a}$, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[10]

E. Gagliardo, Ulteriori propriet$\grave{a}$ di alcune classi di funzioni in pi$\grave{u}$ variabili, Ricerche Mat., 8 (1959), 24-51.   Google Scholar

[11]

D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181.  doi: 10.2307/2000785.  Google Scholar

[12]

D. Hoff, Global existence of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Eqs., 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[13]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, , American Mathematical Society, Providence, RI, 1968.  Google Scholar

[15]

J. Li and Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.  Google Scholar

[16]

J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), Paper No. 7, 37 pp. http://arXiv.org/abs/1310.1673. doi: 10.1007/s40818-019-0064-5.  Google Scholar

[17]

Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differ. Eqs., 258 (2015), 2633-2654.  doi: 10.1016/j.jde.2014.12.015.  Google Scholar

[18]

Z. Liang and X. Shi, Classical solution to the Cauchy problem for the 2D viscous polytropic fluids with vacuum and zero heat-conduction, Comm. Math. Sci., 13 (2015), 327-345.   Google Scholar

[19]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1-2, Compressible Models, Oxford Science Publication, Oxford, 1996/1998.  Google Scholar

[20]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.   Google Scholar

[21]

C. Miao, Harmonic Analysis and its Application in Partical Differential Equations, (in chinese), Science Press, Beijing, 2004. Google Scholar

[22]

J. Nash, Le probl$\grave{e}$me de Cauchy pour les$\acute{e}$quations diff$\acute{e}$rentielles d'un fluide g$\acute{e}$n$\acute{e}$ral, Bull. Soc. Math. France., 90 (1962), 487-497.   Google Scholar

[23]

D. Serre, Solutions faibles globales des $\acute{e}$quations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris S'er. I Math., 303 (1986), 639-642.   Google Scholar

[24]

D. Serre, Sur l'$\acute{e}$quation monodimensionnelle d'un fluide visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris S'er. I Math., 303 (1986), 703-706.   Google Scholar

[25]

J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal., 3 (1959), 271-288.  doi: 10.1007/BF00284180.  Google Scholar

show all references

References:
[1]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pures Appl., 87 (2007), 57-90.  doi: 10.1016/j.matpur.2006.11.001.  Google Scholar

[3]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Composition Math., 53 (1984), 259-275.   Google Scholar

[4]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[5]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[6]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.  Google Scholar

[7]

Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum, J. Differ. Eqs., 228 (2006), 377-411.  doi: 10.1016/j.jde.2006.05.001.  Google Scholar

[8] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.   Google Scholar
[9]

E. FeireislA. Novotny and H. Petzeltov$\acute{a}$, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.  Google Scholar

[10]

E. Gagliardo, Ulteriori propriet$\grave{a}$ di alcune classi di funzioni in pi$\grave{u}$ variabili, Ricerche Mat., 8 (1959), 24-51.   Google Scholar

[11]

D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181.  doi: 10.2307/2000785.  Google Scholar

[12]

D. Hoff, Global existence of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Eqs., 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[13]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, , American Mathematical Society, Providence, RI, 1968.  Google Scholar

[15]

J. Li and Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.  Google Scholar

[16]

J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), Paper No. 7, 37 pp. http://arXiv.org/abs/1310.1673. doi: 10.1007/s40818-019-0064-5.  Google Scholar

[17]

Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differ. Eqs., 258 (2015), 2633-2654.  doi: 10.1016/j.jde.2014.12.015.  Google Scholar

[18]

Z. Liang and X. Shi, Classical solution to the Cauchy problem for the 2D viscous polytropic fluids with vacuum and zero heat-conduction, Comm. Math. Sci., 13 (2015), 327-345.   Google Scholar

[19]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1-2, Compressible Models, Oxford Science Publication, Oxford, 1996/1998.  Google Scholar

[20]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342.   Google Scholar

[21]

C. Miao, Harmonic Analysis and its Application in Partical Differential Equations, (in chinese), Science Press, Beijing, 2004. Google Scholar

[22]

J. Nash, Le probl$\grave{e}$me de Cauchy pour les$\acute{e}$quations diff$\acute{e}$rentielles d'un fluide g$\acute{e}$n$\acute{e}$ral, Bull. Soc. Math. France., 90 (1962), 487-497.   Google Scholar

[23]

D. Serre, Solutions faibles globales des $\acute{e}$quations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris S'er. I Math., 303 (1986), 639-642.   Google Scholar

[24]

D. Serre, Sur l'$\acute{e}$quation monodimensionnelle d'un fluide visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris S'er. I Math., 303 (1986), 703-706.   Google Scholar

[25]

J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal., 3 (1959), 271-288.  doi: 10.1007/BF00284180.  Google Scholar

[1]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[2]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[3]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[4]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[5]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408

[6]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[7]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[8]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128

[9]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[10]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[11]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

[12]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039

[13]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141

[14]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[15]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062

[16]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[17]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[18]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439

[19]

Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021002

[20]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (18)
  • HTML views (71)
  • Cited by (0)

Other articles
by authors

[Back to Top]