GLOBAL SOLUTIONS TO THE 2D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SOME LARGE INITIAL DATA*

doi:10.1007/s10473-023-0315-0

Abstract

Abstract: We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in $\mathbb{R}^2$. By exploiting the global-in-time estimate to the two-dimensional (2D for short) classical incompressible Navier-Stokes equations and using techniques developed in (SIAM J Math Anal, 2020, 52(2): 1806-1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part $\mathbb{P} u_0$ and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.

Key words: compressible Navier-Stokes equations, global large solutions, Littlewood-Paley theory

Xiaoping Zhai, Xin Zhong. GLOBAL SOLUTIONS TO THE 2D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SOME LARGE INITIAL DATA^*[J].Acta mathematica scientia,Series B, 2023, 43(3): 1251-1274.

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References

[1] Bahouri H, Chemin J Y, Danchin R.Fourier Analysis and Nonlinear Partial Differential Equations. Hei- delberg: Springer, 2011
[2] Bresch D, Jabin P E.Global existence of weak solutions for compressible Navier-Stokes equations: ther- modynamically unstable pressure and anisotropic viscous stress tensor. Ann of Math, 2018, 188: 577-684
[3] Charve F, Danchin R.A global existence result for the compressible Navier-Stokes equations in the critical Lp framework. Arch Ration Mech Anal, 2010, 198: 233-271
[4] Chemin J Y, Gallagher I.Wellposedness and stability results for the Navier-Stokes equations in $\mathbb{R}$³. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26: 599-624
[5] Chemin J Y, Lerner N.Flot de champs de vecteurs no lipschitziens et équations de Navier-Stokes (French). J Differential Equations, 2010, 248: 2130-2170
[6] Chen Q, Miao C, Zhang Z.Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity. Comm Pure Appl Math, 2010, 63: 1173-1224
[7] Chen Q, Miao C, Zhang Z.On the ill-posedness of the compressible Navier-Stokes equations. Rev Mat Iberoam, 2015, 31: 1375-1402
[8] Chen Z M, Zhai X. Global large solutions and incompressible limit for the compressible Navier-Stokes equations. J Math Fluid Mech, 2019, 21: Art 26
[9] Danchin R.Global existence in critical spaces for compressible Navier-Stokes equations. Invent Math, 2000, 141: 579-614
[10] Danchin R.Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm Partial Differential Equations, 2001, 26: 1183-1233. Erratum: “Local theory in critical spaces for compressible viscous and heat-conductive gases”. Comm Partial Differential Equations, 2002, 27: 2531-2532
[11] Danchin R.Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Comm Partial Differential Equations, 2007, 32(9): 1373-1397
[12] Danchin R.Fourier analysis methods for the compressible Navier-Stokes equations. arXiv:1507.02637
[13] Danchin R, He L.The incompressible limit in L^p type critical spaces. Math Ann, 2016, 366(3/4): 1365-1402
[14] Danchin R, Mucha P.Compressible Navier-Stokes system: Large solutions and incompressible limit. Adv Math, 2017, 320: 904-925
[15] Danchin R, Xu J.Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical L^p Framework. Arch Ration Mech Anal, 2017, 224: 53-90
[16] Fang D, Zhang T, Zi R.Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data. SIAM J Math Anal, 2018, 50: 4983-5026
[17] Feireisl E.Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004
[18] Feireisl E, Gwiazda P, Świerczewska-Gwiazda A, Wiedemann E. Dissipative measure-valued solutions to the compressible Navier-Stokes system. Calc Var Partial Differential Equations, 2016, 55: Art 141
[19] Feireisl E, Novotný A, Petzeltová H.On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3: 358-392
[20] Feireisl E, Novotný A, Sun Y.Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ Math J, 2011, 60: 611-631
[21] Fujita H, Kato T.On the Navier-Stokes initial value problem. I. Arch Rational Mech Anal, 1964, 16: 269-315
[22] Haspot B.Existence of global strong solutions in critical spaces for barotropic viscous fluids. Arch Ration Mech Anal, 2011, 202: 427-460
[23] He L, Huang J, Wang C.Global stability of large solutions to the 3D compressible Navier-Stokes equations. Arch Ration Mech Anal, 2019, 234: 1167-1222
[24] Hoff D.Global solutions of the Navier-Stokes equations for multidimensional compressible flow with dis- continuous initial data. J Differential Equations, 1995, 120: 215-254
[25] Hoff D.Compressible flow in a half-space with Navier boundary conditions. J Math Fluid Mech, 2005, 7: 315-338
[26] Hoff D, Zumbrun K.Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ Math J, 1995, 44: 603-676
[27] Huang F, Wang T, Wang Y.Diffusive wave in the low Mach limit for compressible Navier-Stokes equations. Adv Math, 2017, 319: 348-395
[28] Huang J, Paicu M, Zhang P.Global solutions to 2-D inhomogeneous Navier-Stokes system with general velocity. J Math Pures Appl, 2013, 100: 806-831
[29] Huang X, Li J, Xin Z.Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Comm Pure Appl Math, 2012, 65: 549-585
[30] Jiang S, Zhang P.On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Comm Math Phys, 2001, 215: 559-581
[31] Jiang S, Zhang P.Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids. J Math Pure Appl, 2003, 82: 949-973
[32] Li H, Wang Y, Xin Z.Non-existence of classical solutions with finite energy to the Cauchy problem of the compressible Navier-Stokes equations. Arch Ration Mech Anal, 2019, 232: 557-590
[33] Li J, Xin Z. Global well-posedness and large time asymptotic behavior of classical solutions to the com- pressible Navier-Stokes equations with vacuum. Ann PDE, 2019, 5: Art 7
[34] Lions P L.Mathematical Topics in Fluid Mechanics: Vol 2: Compressible Models. Oxford: Oxford Univer- sity Press, 1998
[35] Matsumura A, Nishida T.The initial value problem for the equations of motion of viscous and heat- conductive gases. J Math Kyoto Univ, 1980, 89: 67-104
[36] Matsumura A, Nishida T.Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Comm Math Phys, 1983, 89: 445-464
[37] Sun W, Jiang S, Guo Z.Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids. J Differential Equations, 2006, 222: 263-296
[38] Villani C. Hypocoercivity. Mem Amer Math Soc, 2009, 202: Art 950
[39] Wang C, Wang W, Zhang Z.Global well-posedness of compressible Navier-Stokes equations for some classes of large initial data. Arch Ration Mech Anal, 2014, 213: 171-214
[40] Xin Z.Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm Pure Appl Math, 1998, 51: 229-240
[41] Xin Z, Yan W.On blowup of classical solutions to the compressible Navier-Stokes equations. Comm Math Phys, 2013, 321: 529-541
[42] Zhai X, Li Y, Zhou F.Global large solutions to the three dimensional compressible Navier-Stokes equations.

GLOBAL SOLUTIONS TO THE 2D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SOME LARGE INITIAL DATA^*

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