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

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October 2021, 26(10): 5383-5405. doi: 10.3934/dcdsb.2020348

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

Zhilei Liang ^, and Jiangyu Shuai

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

^* Corresponding author: Zhilei Liang

Received May 2020 Published October 2021 Early access 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].

Keywords: Navier-Stokes equations, Cauchy problem, strong solution, existence.

Mathematics Subject Classification: 35A01, 35Q30, 76N05, 76N10.

Citation: Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348

References:

[1]	S. Agmon, A. 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.
[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.
[3]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Composition Math., 53 (1984), 259-275.
[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.
[5]	Y. Cho, H. 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.
[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.
[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.
[8]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.
[9]	E. Feireisl, A. 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.
[10]	E. Gagliardo, Ulteriori propriet$\grave{a}$ di alcune classi di funzioni in pi$\grave{u}$ variabili, Ricerche Mat., 8 (1959), 24-51.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1-2, Compressible Models, Oxford Science Publication, Oxford, 1996/1998.
[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.
[21]	C. Miao, Harmonic Analysis and its Application in Partical Differential Equations, (in chinese), Science Press, Beijing, 2004.
[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.
[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.
[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.
[25]	J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal., 3 (1959), 271-288. doi: 10.1007/BF00284180.

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References:

[1]	S. Agmon, A. 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.
[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.
[3]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Composition Math., 53 (1984), 259-275.
[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.
[5]	Y. Cho, H. 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.
[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.
[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.
[8]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.
[9]	E. Feireisl, A. 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.
[10]	E. Gagliardo, Ulteriori propriet$\grave{a}$ di alcune classi di funzioni in pi$\grave{u}$ variabili, Ricerche Mat., 8 (1959), 24-51.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1-2, Compressible Models, Oxford Science Publication, Oxford, 1996/1998.
[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.
[21]	C. Miao, Harmonic Analysis and its Application in Partical Differential Equations, (in chinese), Science Press, Beijing, 2004.
[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.
[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.
[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.
[25]	J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal., 3 (1959), 271-288. doi: 10.1007/BF00284180.

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