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December  2015, 8(4): 765-775. doi: 10.3934/krm.2015.8.765

## Strong solutions to compressible barotropic viscoelastic flow with vacuum

 1 Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China 2 Department of Mathematics, Sichuan University, Chengdu, 610064, China

Received  November 2014 Revised  May 2015 Published  July 2015

We consider strong solutions to compressible barotropic viscoelastic flow in a domain $\Omega\subset\mathbb{R}^{3}$ and prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Inspired by the work of Kato and Lax, we use the contraction mapping principle to get the result.
Citation: Tong Tang, Yongfu Wang. Strong solutions to compressible barotropic viscoelastic flow with vacuum. Kinetic & Related Models, 2015, 8 (4) : 765-775. doi: 10.3934/krm.2015.8.765
##### References:
 [1] 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. doi: 10.1016/j.matpur.2003.11.004. [2] Y. M. Chu, X. G. Liu and X. Liu, Strong solutions to the compressible liquid crystal system,, Pacific J. Math., 257 (2012), 37. doi: 10.2140/pjm.2012.257.37. [3] M. Hieber, Y. Naito and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains,, J. Differential Equations, 252 (2012), 2617. doi: 10.1016/j.jde.2011.09.001. [4] X. P. Hu and D. H. Wang, Local strong solution to the compressible viscoelastic flow with large data,, J. Differential Equations, 249 (2010), 1179. doi: 10.1016/j.jde.2010.03.027. [5] X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows,, J. Differential Equations, 250 (2011), 1200. doi: 10.1016/j.jde.2010.10.017. [6] X. P. Hu and D. H. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids,, J. Differential Equations, 252 (2012), 4027. doi: 10.1016/j.jde.2011.11.021. [7] C. Guillopé and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law,, Nonlinear Anal., 15 (1990), 849. doi: 10.1016/0362-546X(90)90097-Z. [8] X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, Comm. Pure Appl. Math., 65 (2012), 549. doi: 10.1002/cpa.21382. [9] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181. doi: 10.1007/BF00280740. [10] R. Kupferman, C. Mangoubi and E. S. Titi, A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime,, Commun. Math. Sci., 6 (2008), 235. doi: 10.4310/CMS.2008.v6.n1.a12. [11] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, (1973). [12] Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Ration. Mech. Anal., 188 (2008), 371. doi: 10.1007/s00205-007-0089-x. [13] Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain,, Commun. Math. Sci., 5 (2007), 595. doi: 10.4310/CMS.2007.v5.n3.a5. [14] F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437. doi: 10.1002/cpa.20074. [15] F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system,, Comm. Pure Appl. Math., 61 (2008), 539. doi: 10.1002/cpa.20219. [16] P. L. Lions, Mathematical Topics in Fluid Mechanics,, vol. 2. Compressible Models, (1998). [17] A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, (1984). doi: 10.1007/978-1-4612-1116-7. [18] 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. [19] A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat- conductive fluids,, Comm. Math. Phys., 89 (1983), 445. doi: 10.1007/BF01214738. [20] J. G. Oldroyd, On the formation of rheological equations of state,, Proc. R. Soc. Lond. Ser. A, 200 (1950), 523. doi: 10.1098/rspa.1950.0035. [21] J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids,, Proc. R. Soc. Lond. Ser. A, 245 (1958), 278. doi: 10.1098/rspa.1958.0083. [22] J. Z. Qian and Z. F. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium,, Arch. Ration. Mech. Anal., 198 (2010), 835. doi: 10.1007/s00205-010-0351-5. [23] J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid,, J. Differential Equations, 250 (2011), 848. doi: 10.1016/j.jde.2010.07.026. [24] 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.

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##### References:
 [1] 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. doi: 10.1016/j.matpur.2003.11.004. [2] Y. M. Chu, X. G. Liu and X. Liu, Strong solutions to the compressible liquid crystal system,, Pacific J. Math., 257 (2012), 37. doi: 10.2140/pjm.2012.257.37. [3] M. Hieber, Y. Naito and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains,, J. Differential Equations, 252 (2012), 2617. doi: 10.1016/j.jde.2011.09.001. [4] X. P. Hu and D. H. Wang, Local strong solution to the compressible viscoelastic flow with large data,, J. Differential Equations, 249 (2010), 1179. doi: 10.1016/j.jde.2010.03.027. [5] X. P. Hu and D. H. Wang, Global existence for the multi-dimensional compressible viscoelastic flows,, J. Differential Equations, 250 (2011), 1200. doi: 10.1016/j.jde.2010.10.017. [6] X. P. Hu and D. H. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids,, J. Differential Equations, 252 (2012), 4027. doi: 10.1016/j.jde.2011.11.021. [7] C. Guillopé and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law,, Nonlinear Anal., 15 (1990), 849. doi: 10.1016/0362-546X(90)90097-Z. [8] X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,, Comm. Pure Appl. Math., 65 (2012), 549. doi: 10.1002/cpa.21382. [9] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181. doi: 10.1007/BF00280740. [10] R. Kupferman, C. Mangoubi and E. S. Titi, A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime,, Commun. Math. Sci., 6 (2008), 235. doi: 10.4310/CMS.2008.v6.n1.a12. [11] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, (1973). [12] Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Ration. Mech. Anal., 188 (2008), 371. doi: 10.1007/s00205-007-0089-x. [13] Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain,, Commun. Math. Sci., 5 (2007), 595. doi: 10.4310/CMS.2007.v5.n3.a5. [14] F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437. doi: 10.1002/cpa.20074. [15] F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system,, Comm. Pure Appl. Math., 61 (2008), 539. doi: 10.1002/cpa.20219. [16] P. L. Lions, Mathematical Topics in Fluid Mechanics,, vol. 2. Compressible Models, (1998). [17] A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,, Applied Mathematical Sciences, (1984). doi: 10.1007/978-1-4612-1116-7. [18] 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. [19] A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat- conductive fluids,, Comm. Math. Phys., 89 (1983), 445. doi: 10.1007/BF01214738. [20] J. G. Oldroyd, On the formation of rheological equations of state,, Proc. R. Soc. Lond. Ser. A, 200 (1950), 523. doi: 10.1098/rspa.1950.0035. [21] J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids,, Proc. R. Soc. Lond. Ser. A, 245 (1958), 278. doi: 10.1098/rspa.1958.0083. [22] J. Z. Qian and Z. F. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium,, Arch. Ration. Mech. Anal., 198 (2010), 835. doi: 10.1007/s00205-010-0351-5. [23] J. Z. Qian, Initial boundary value problems for the compressible viscoelastic fluid,, J. Differential Equations, 250 (2011), 848. doi: 10.1016/j.jde.2010.07.026. [24] 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.
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