On Blow-up of Regular Solutions to the Isentropic Euler and Euler-Boltzmann Equations with Vacuum

Abstract

In this paper, the authors study the Cauchy problem of n-dimensional isentropic Euler equations and Euler-Boltzmann equations with vacuum in the whole space. They show that if the initial velocity satisfies some condition on the integral J in the “isolated mass group” (see (1.13)), then there will be finite time blow-up of regular solutions to the Euler system with J ≤ 0 (n ≥ 1) and to the Euler-Boltzmann system with J < 0 (n ≥ 1) and J = 0 (n ≥ 2), no matter how small and smooth the initial data are. It is worth mentioning that these blow-up results imply the following: The radiation is not strong enough to prevent the formation of singularities caused by the appearance of vacuum, with the only possible exception in the case J = 0 and n = 1 since the radiation behaves differently on this occasion.

Keywords

Euler and Euler-Boltzmann equations / Finite time blow-up / Multidimensional / Regular solutions / Vacuum

References

[1] Blanc X, Ducomet B. Global weak solutions to 1D compressible Euler equations with radiation. Commun. Math. Sci., 2015, 13(7): 1905-1936

[2] Buet C, Després B. Asymptotic analysis of fluid models for the coupling of radiation hydrodynamics. J. Quant. Spectroscopy Rad. Trans., 2004, 85(3–4): 385-418

[3] Ducomet B, Feireisl E, Nečasová Š. On a model in radiation hydrodynamics. Ann. Inst. H. Poincaré. (c) Anal. Non Line., 2011, 28(6): 797-812

[4] Ducomet B, Nečasová Š. Global weak solutions to the 1D compressible Navier-Stokes equations with radiation. Commun. Math. Anal., 2010, 8(3): 23-65

[5] Ducomet B, Nečasová Š. Global existence of solutions for the one-dimensional motions of a compressible gas with radiation: An “infrarelativistic model”. Nonlinear Anal., 2010, 72(7–8): 3258-3274

[6] Ducomet B, Nečasová Š. Large time behavior of the motion of a viscous heat-conducting one-dimensional gas coupled to radiation. Ann. Mat. Pura Appl., 2012, 191(4): 219-260

[7] Ducomet B, Nečasová Š. Diffusion limits in a model of radiative flow. Ann. Univ. Ferrara, 2015, 61(1): 17-59

[8] Grassin M. Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J., 1998, 47(4): 1397-1432

[9] Grassin M, Serre D. Existence de solutions globales et reģuliéres aux équations d’Euler pour un gaz parfait isentropique. C. R. Acad. Sci. Paris Série. I Math., 1997, 325: 721-726

[10] Jiang P. Global weak solutions to the non-relativistic radiation hydrodynamical equations with isothermal fluids. J. Differential Equations, 2012, 253(7): 2191-2223

[11] Jiang P, Wang D. Formation of singularities of solutions to the three-dimensional Euler-Boltzmann equations in radiation hydrodynamics. Nonlinearity, 2010, 23(4): 809-821

[12] Jiang P, Wang D. Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics. Quart. Appl. Math., 2012, 70(1): 25-44

[13] Jiang P, Wang Y. Existence of solutions to an initial-boundary value problem of multidimensional radiation hydrodynamics. J. Differential Equations, 2011, 251(6): 1616-1636

[14] Jiang S, Zhong X. Local existence and finite-time blow-up in multidimensional radiation hydrodynamics. J. Math. Fluid Mech., 2007, 9(4): 543-564

[15] Kippenhahn R, Weigert A. Stellar Structure and Evolution, 1994, Berlin, Heideberg: Springer-Verlag

[16] Li Y, Pan R, Zhu S. On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum. Arch. Ration. Mech. Anal., 2019, 234(3): 1281-1334

[17] Li Y, Zhu S. Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum. J. Differential Equations, 2014, 256(12): 3943-3980

[18] Li Y, Zhu S. Existence results for the compressible radiation hydrodynamics equations with vacuum. Commun. Pure Appl. Anal., 2015, 14(3): 1023-1052

[19] Li Y, Zhu S. On regular solutions of the 3D compressible isentropic Euler-Boltzmann equations with vacuum. Discrete. Contin. Dyn. Syst., 2015, 35(7): 3059-3086

[20] Liu T, Yang T. Compressible Euler equations with vacuum. J. Differential Equations, 1997, 140(2): 223-237

[21] Lowrie R, Morel J, Hittinger J. The coupling of radiation and hydrodynamics. Astrophys. J., 1999, 521(1): 432-450

[22] Makino T, Ukai S, Kawashima S. Sur la solution à support compact de equations d’Euler compressible. Japan J. Appl. Math., 1986, 3(2): 249-257

[23] Milnor J. Analytic proofs of the “hairy ball theorem” and the Brouwer fixed point theorem. The American Mathematical Monthly, 1978, 85(7): 521-524

[24] Pomraning G. The Equations of Radiation Hydrodynamics, 1973, Oxford: Pergamon Press

[25] Serre D. Solutions classiques globales des equations d’Euler pour un fluide parfait compressible. Annales de l’Institut Fourier, 1997, 47(1): 139-153

[26] Serre D. Expansion of a compressible gas in vacuum. Bull. Inst. Math. Acad. Sin. (N.S.), 2015, 10(4): 695-716

[27] Sideris T. Formulation of singularities in three-dimensional compressible fluids. Commun. Math. Phys., 1985, 101(4): 475-485

[28] Wang X, Xiong A. Advanced Fluid Mechanics, 2004, Wuhan: Huazhong University of Science and Technology

[29] Xin Z. Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 1998, 51: 0229-0240

[30] Xin Z, Yan W. On blow-up of classical solutions to the compressible Navier-Stokes equations. Commun. Math. Phys., 2013, 321(2): 529-541