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March  2013, 6(1): 205-218. doi: 10.3934/krm.2013.6.205

On the Stokes approximation equations for two-dimensional compressible flows

Qing Yi 1,

1. 

College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, China

Received  January 2012 Revised  September 2012 Published  December 2012

We deal with the unique global strong solution or classical solution to the Cauchy problem of the 2D Stokes approximation equations for the compressible flows with the density being some positive constant on the far field for arbitrarily large initial data, which may contain vacuum states. First, we prove that the density is bounded from above independently of time. Secondly, we show that if the initial density contains vacuum at least at one point, then the global strong (or classical) solution must blow up as time goes to infinity.
Keywords: blowup., Cauchy problem, uniform upper bound, Stokes approximation equations, vacuum.
Mathematics Subject Classification: Primary: 35Q30, 35Q20; Secondary: 76N10. .
Citation: Qing Yi. On the Stokes approximation equations for two-dimensional compressible flows. Kinetic & Related Models, 2013, 6 (1) : 205-218. doi: 10.3934/krm.2013.6.205
References:
[1]

F. J. Chatelon and P. Orenga, Some smoothness and uniqueness results for a shallow-water problem, Adv. Differential Equations, 3 (1998), 155-176.  Google Scholar

[2]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl. (9), 83 (2004), 243-275.  Google Scholar

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R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  Google Scholar

[4]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

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Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  Google Scholar

[6]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^ q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  Google Scholar

[7]

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

[8]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.  Google Scholar

[9]

D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow, SIAM J. Appl. Math., 51 (1991), 887-898.  Google Scholar

[10]

D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216 (2001), 255-276.  Google Scholar

[11]

F. Huang, J. Li and Z. Xin, Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data, J. Math. Pures Appl. (9), 86 (2006), 471-491.  Google Scholar

[12]

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.  Google Scholar

[13]

A. V. Kazhikhov and V. A. Weigant, Global solutions of equations of potential flows of a compressible viscous fluid for small Reynolds numbers, Differential Equations, 30 (1994), 935-947.  Google Scholar

[14]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.  Google Scholar

[15]

J. Li and Z. Xin, Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows, J. Differential Equations, 221 (2006), 275-308.  Google Scholar

[16]

P. L. Lions, Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1335-1340.  Google Scholar

[17]

P. L. Lions, Compacité des solutions des équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, Sér I Math., 317 (1993), 115–-120.  Google Scholar

[18]

P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. 2. Compressible models. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[19]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[20]

A. Matsumura and T. Nishida, The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  Google Scholar

[21]

L. Min, A. V. Kazhikhov and S. Ukai, Global solutions to the Cauchy problem of the Stokes approximation equations for two-dimensional compressible flows, Comm. Partial Differential Equations, 23 (1998), 985-1006.  Google Scholar

[22]

R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty.$, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 40 (1993), 17-51.  Google Scholar

[23]

D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 639-642.  Google Scholar

[24]

D. Serre, On the one-dimensional equation of a viscous, compressible, heat-conducting fluid, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 703-706.  Google Scholar

[25]

V. A. Solonnikov, On solvability of an initial boundary value problem for the equations of motion of viscous compressible fluid, Zap. Nauchn. Sem. LOMI, 56 (1976), 128-142.  Google Scholar

[26]

A. Valli and W. M. Zajaczkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296.  Google Scholar

[27]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  Google Scholar

[28]

A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Diff. Equations, 36 (2000), 701-716.  Google Scholar

show all references

References:
[1]

F. J. Chatelon and P. Orenga, Some smoothness and uniqueness results for a shallow-water problem, Adv. Differential Equations, 3 (1998), 155-176.  Google Scholar

[2]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl. (9), 83 (2004), 243-275.  Google Scholar

[3]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  Google Scholar

[4]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

[5]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  Google Scholar

[6]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^ q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  Google Scholar

[7]

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

[8]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.  Google Scholar

[9]

D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow, SIAM J. Appl. Math., 51 (1991), 887-898.  Google Scholar

[10]

D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216 (2001), 255-276.  Google Scholar

[11]

F. Huang, J. Li and Z. Xin, Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data, J. Math. Pures Appl. (9), 86 (2006), 471-491.  Google Scholar

[12]

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.  Google Scholar

[13]

A. V. Kazhikhov and V. A. Weigant, Global solutions of equations of potential flows of a compressible viscous fluid for small Reynolds numbers, Differential Equations, 30 (1994), 935-947.  Google Scholar

[14]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.  Google Scholar

[15]

J. Li and Z. Xin, Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows, J. Differential Equations, 221 (2006), 275-308.  Google Scholar

[16]

P. L. Lions, Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1335-1340.  Google Scholar

[17]

P. L. Lions, Compacité des solutions des équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, Sér I Math., 317 (1993), 115–-120.  Google Scholar

[18]

P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. 2. Compressible models. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[19]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[20]

A. Matsumura and T. Nishida, The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  Google Scholar

[21]

L. Min, A. V. Kazhikhov and S. Ukai, Global solutions to the Cauchy problem of the Stokes approximation equations for two-dimensional compressible flows, Comm. Partial Differential Equations, 23 (1998), 985-1006.  Google Scholar

[22]

R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty.$, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 40 (1993), 17-51.  Google Scholar

[23]

D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 639-642.  Google Scholar

[24]

D. Serre, On the one-dimensional equation of a viscous, compressible, heat-conducting fluid, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 703-706.  Google Scholar

[25]

V. A. Solonnikov, On solvability of an initial boundary value problem for the equations of motion of viscous compressible fluid, Zap. Nauchn. Sem. LOMI, 56 (1976), 128-142.  Google Scholar

[26]

A. Valli and W. M. Zajaczkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296.  Google Scholar

[27]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  Google Scholar

[28]

A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Diff. Equations, 36 (2000), 701-716.  Google Scholar

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