# American Institute of Mathematical Sciences

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

## On the Stokes approximation equations for two-dimensional compressible flows

 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.
Citation: Qing Yi. On the Stokes approximation equations for two-dimensional compressible flows. Kinetic and 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. [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. [3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. [4] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. [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. [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. [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. [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. [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. [10] D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216 (2001), 255-276. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. [28] A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Diff. Equations, 36 (2000), 701-716.

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##### 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. [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. [3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. [4] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. [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. [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. [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. [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. [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. [10] D. Hoff and J. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys., 216 (2001), 255-276. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. [28] A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Diff. Equations, 36 (2000), 701-716.
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