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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181.  doi: 10.1007/BF00280740.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics,, vol. 2. Compressible Models, (1998).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181.  doi: 10.1007/BF00280740.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. L. Lions, Mathematical Topics in Fluid Mechanics,, vol. 2. Compressible Models, (1998).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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