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June  2013, 2(2): 337-353. doi: 10.3934/eect.2013.2.337

A local existence result for a system of viscoelasticity with physical viscosity

Marta Lewicka 1, and  Piotr B. Mucha 2,

1. 

University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, United States
2. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02097 Warszawa, Poland

Received  September 2012 Revised  February 2013 Published  March 2013

We prove the local in time existence of regular solutions to the system of equations of isothermal viscoelasticity with clamped boundary conditions. We deal with a general form of viscous stress tensor $\mathcal{Z}(F,\dot F)$, assuming a Korn-type condition on its derivative $D_{\dot F}\mathcal{Z}(F, \dot F)$. This condition is compatible with the balance of angular momentum, frame invariance and the Claussius-Duhem inequality. We give examples of linear and nonlinear (in $\dot F$) tensors $\mathcal{Z}$ satisfying these required conditions.
Keywords: maximal regularity, well posedness., viscoelasticity, Local existence, viscosity.
Mathematics Subject Classification: Primary: 74B20; Secondary: 35Q7.
Citation: Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations & Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337
References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

G. Andrews, On the existence of solutions to the equation $u_{t t} = u_{x x t} + (u_x) _x$, J. Diff. Eqs., 35 (1980), 200-231. doi: 10.1016/0022-0396(80)90040-6.  Google Scholar

[3]

S. Antmann and R. Malek-Madani, Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media, Quart. Appl. Math., 46 (1988), 77-93.  Google Scholar

[4]

S. Antman and T. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity, J. Diff. Eqs., 124 (1996), 132-185. doi: 10.1006/jdeq.1996.0005.  Google Scholar

[5]

B. Barker, M. Lewicka and K. Zumbrun, Existence and stability of viscoelastic shock profiles, Arch. Rational Mech. Anal., 200 (2011), 491-532. doi: 10.1007/s00205-010-0363-1.  Google Scholar

[6]

O. Besov, V. Il'in and S. Nikol'skiĭ, "Integral Representations of Functions and Imbedding Theorems. Vol. I," Translated from the Russian, Scripta Series in Mathematics, Edited by Mitchell H. Taibleson, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978.  Google Scholar

[7]

R. Chill and S. Srivastava, $L^p$-maximal regularity for second order Cauchy problems, Math. Z., 251 (2005), 751-781. doi: 10.1007/s00209-005-0815-8.  Google Scholar

[8]

P. Clement and S. Li, Abstract parabolic quasilinear equations and application to a ground-water flow problem,, Adv. Math. Sci. Appl., 3 (): 17.   Google Scholar

[9]

C. Dafermos, The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity, J. Diff. Eqs., 6 (1969), 71-86.  Google Scholar

[10]

C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Springer-Verlag, 1999. Google Scholar

[11]

R. Danchin and P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480. doi: 10.1002/cpa.21409.  Google Scholar

[12]

S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity, Arch. Rat. Mech. Anal., 155 (2000), 299-334. doi: 10.1007/s002050000115.  Google Scholar

[13]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003).  Google Scholar

[14]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations," Springer Tracts in Natural Philosophy, 38, 39, Springer-Verlag, New York, 1994. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[15]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[16]

T. Hughes, T. Kato and J. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1977), 273-294.  Google Scholar

[17]

A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat., (1909), 705-724. Google Scholar

[18]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translation of Mathematical Monographs, 23, AMS, 1968. Google Scholar

[19]

M. Lewicka, L. Mahadevan and M. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains, Proceedings of the Royal Society A Math. Phys. Eng. Sci., 467 (2011), 402-426. doi: 10.1098/rspa.2010.0138.  Google Scholar

[20]

M. Lewicka and M. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 1158-1173. doi: 10.1051/cocv/2010039.  Google Scholar

[21]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[22]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[23]

M. G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Differential Equations, 252 (2012), 35-55. doi: 10.1016/j.jde.2011.09.009.  Google Scholar

[24]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.  Google Scholar

[25]

P. B. Mucha, Limit of kinetic term for a Stefan problem,, Dis. Cont. Dynamical Syst., 2007 (): 741.   Google Scholar

[26]

P. B. Mucha and W. Zajączkowski, On a $L^p$-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions, J. Differential Equations, 186 (2002), 377-393. doi: 10.1016/S0022-0396(02)00017-7.  Google Scholar

[27]

R. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal., 97 (1987), 353-394. doi: 10.1007/BF00280411.  Google Scholar

[28]

P. Rybka, Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions, Proc. Roy. Soc. Edin. A, 121 (1992), 101-138. doi: 10.1017/S0308210500014177.  Google Scholar

[29]

B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type, Arch. Rat. Mech. Anal., 189 (2008), 237-281. doi: 10.1007/s00205-007-0109-x.  Google Scholar

show all references

References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

G. Andrews, On the existence of solutions to the equation $u_{t t} = u_{x x t} + (u_x) _x$, J. Diff. Eqs., 35 (1980), 200-231. doi: 10.1016/0022-0396(80)90040-6.  Google Scholar

[3]

S. Antmann and R. Malek-Madani, Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media, Quart. Appl. Math., 46 (1988), 77-93.  Google Scholar

[4]

S. Antman and T. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity, J. Diff. Eqs., 124 (1996), 132-185. doi: 10.1006/jdeq.1996.0005.  Google Scholar

[5]

B. Barker, M. Lewicka and K. Zumbrun, Existence and stability of viscoelastic shock profiles, Arch. Rational Mech. Anal., 200 (2011), 491-532. doi: 10.1007/s00205-010-0363-1.  Google Scholar

[6]

O. Besov, V. Il'in and S. Nikol'skiĭ, "Integral Representations of Functions and Imbedding Theorems. Vol. I," Translated from the Russian, Scripta Series in Mathematics, Edited by Mitchell H. Taibleson, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978.  Google Scholar

[7]

R. Chill and S. Srivastava, $L^p$-maximal regularity for second order Cauchy problems, Math. Z., 251 (2005), 751-781. doi: 10.1007/s00209-005-0815-8.  Google Scholar

[8]

P. Clement and S. Li, Abstract parabolic quasilinear equations and application to a ground-water flow problem,, Adv. Math. Sci. Appl., 3 (): 17.   Google Scholar

[9]

C. Dafermos, The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity, J. Diff. Eqs., 6 (1969), 71-86.  Google Scholar

[10]

C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," Springer-Verlag, 1999. Google Scholar

[11]

R. Danchin and P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480. doi: 10.1002/cpa.21409.  Google Scholar

[12]

S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity, Arch. Rat. Mech. Anal., 155 (2000), 299-334. doi: 10.1007/s002050000115.  Google Scholar

[13]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003).  Google Scholar

[14]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations," Springer Tracts in Natural Philosophy, 38, 39, Springer-Verlag, New York, 1994. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[15]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[16]

T. Hughes, T. Kato and J. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1977), 273-294.  Google Scholar

[17]

A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat., (1909), 705-724. Google Scholar

[18]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translation of Mathematical Monographs, 23, AMS, 1968. Google Scholar

[19]

M. Lewicka, L. Mahadevan and M. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains, Proceedings of the Royal Society A Math. Phys. Eng. Sci., 467 (2011), 402-426. doi: 10.1098/rspa.2010.0138.  Google Scholar

[20]

M. Lewicka and M. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 1158-1173. doi: 10.1051/cocv/2010039.  Google Scholar

[21]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[22]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[23]

M. G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Differential Equations, 252 (2012), 35-55. doi: 10.1016/j.jde.2011.09.009.  Google Scholar

[24]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271. doi: 10.1007/s00245-011-9138-9.  Google Scholar

[25]

P. B. Mucha, Limit of kinetic term for a Stefan problem,, Dis. Cont. Dynamical Syst., 2007 (): 741.   Google Scholar

[26]

P. B. Mucha and W. Zajączkowski, On a $L^p$-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions, J. Differential Equations, 186 (2002), 377-393. doi: 10.1016/S0022-0396(02)00017-7.  Google Scholar

[27]

R. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal., 97 (1987), 353-394. doi: 10.1007/BF00280411.  Google Scholar

[28]

P. Rybka, Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions, Proc. Roy. Soc. Edin. A, 121 (1992), 101-138. doi: 10.1017/S0308210500014177.  Google Scholar

[29]

B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type, Arch. Rat. Mech. Anal., 189 (2008), 237-281. doi: 10.1007/s00205-007-0109-x.  Google Scholar

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