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

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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

Abstract
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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 (1995). doi: 10.1007/978-3-0348-9221-6.
[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. doi: 10.1016/0022-0396(80)90040-6.
[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.
[4]	S. Antman and T. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity,, J. Diff. Eqs., 124 (1996), 132. doi: 10.1006/jdeq.1996.0005.
[5]	B. Barker, M. Lewicka and K. Zumbrun, Existence and stability of viscoelastic shock profiles,, Arch. Rational Mech. Anal., 200 (2011), 491. doi: 10.1007/s00205-010-0363-1.
[6]	O. Besov, V. Il'in and S. Nikol'skiĭ, "Integral Representations of Functions and Imbedding Theorems. Vol. I,", Translated from the Russian, (1978).
[7]	R. Chill and S. Srivastava, $L^p$-maximal regularity for second order Cauchy problems,, Math. Z., 251 (2005), 751. doi: 10.1007/s00209-005-0815-8.
[8]	P. Clement and S. Li, Abstract parabolic quasilinear equations and application to a ground-water flow problem,, Adv. Math. Sci. Appl., 3 (): 17.
[9]	C. Dafermos, The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity,, J. Diff. Eqs., 6 (1969), 71.
[10]	C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,", Springer-Verlag, (1999).
[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. doi: 10.1002/cpa.21409.
[12]	S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity,, Arch. Rat. Mech. Anal., 155 (2000), 299. doi: 10.1007/s002050000115.
[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).
[14]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,", Springer Tracts in Natural Philosophy, 38, 39 (1994). doi: 10.1007/978-0-387-09620-9.
[15]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).
[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.
[17]	A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen,, Bull. Int. Cracovie Akademie Umiejet, (1909), 705.
[18]	O. Ladyzhenskaya, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translation of Mathematical Monographs, 23 (1968).
[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. doi: 10.1098/rspa.2010.0138.
[20]	M. Lewicka and M. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics,, ESAIM: Control, 17 (2011), 1158. doi: 10.1051/cocv/2010039.
[21]	G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).
[22]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and their Applications, 16 (1995). doi: 10.1007/978-3-0348-9234-6.
[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. doi: 10.1016/j.jde.2011.09.009.
[24]	S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation,, Appl. Math. Optim., 64 (2011), 257. doi: 10.1007/s00245-011-9138-9.
[25]	P. B. Mucha, Limit of kinetic term for a Stefan problem,, Dis. Cont. Dynamical Syst., 2007 (): 741.
[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. doi: 10.1016/S0022-0396(02)00017-7.
[27]	R. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability,, Arch. Rational Mech. Anal., 97 (1987), 353. doi: 10.1007/BF00280411.
[28]	P. Rybka, Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions,, Proc. Roy. Soc. Edin. A, 121 (1992), 101. doi: 10.1017/S0308210500014177.
[29]	B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type,, Arch. Rat. Mech. Anal., 189 (2008), 237. doi: 10.1007/s00205-007-0109-x.

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References:

[1]	H. Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory,", Monographs in Mathematics, 89 (1995). doi: 10.1007/978-3-0348-9221-6.
[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. doi: 10.1016/0022-0396(80)90040-6.
[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.
[4]	S. Antman and T. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity,, J. Diff. Eqs., 124 (1996), 132. doi: 10.1006/jdeq.1996.0005.
[5]	B. Barker, M. Lewicka and K. Zumbrun, Existence and stability of viscoelastic shock profiles,, Arch. Rational Mech. Anal., 200 (2011), 491. doi: 10.1007/s00205-010-0363-1.
[6]	O. Besov, V. Il'in and S. Nikol'skiĭ, "Integral Representations of Functions and Imbedding Theorems. Vol. I,", Translated from the Russian, (1978).
[7]	R. Chill and S. Srivastava, $L^p$-maximal regularity for second order Cauchy problems,, Math. Z., 251 (2005), 751. doi: 10.1007/s00209-005-0815-8.
[8]	P. Clement and S. Li, Abstract parabolic quasilinear equations and application to a ground-water flow problem,, Adv. Math. Sci. Appl., 3 (): 17.
[9]	C. Dafermos, The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity,, J. Diff. Eqs., 6 (1969), 71.
[10]	C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,", Springer-Verlag, (1999).
[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. doi: 10.1002/cpa.21409.
[12]	S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity,, Arch. Rat. Mech. Anal., 155 (2000), 299. doi: 10.1007/s002050000115.
[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).
[14]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations,", Springer Tracts in Natural Philosophy, 38, 39 (1994). doi: 10.1007/978-0-387-09620-9.
[15]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).
[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.
[17]	A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen,, Bull. Int. Cracovie Akademie Umiejet, (1909), 705.
[18]	O. Ladyzhenskaya, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translation of Mathematical Monographs, 23 (1968).
[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. doi: 10.1098/rspa.2010.0138.
[20]	M. Lewicka and M. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics,, ESAIM: Control, 17 (2011), 1158. doi: 10.1051/cocv/2010039.
[21]	G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996).
[22]	A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and their Applications, 16 (1995). doi: 10.1007/978-3-0348-9234-6.
[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. doi: 10.1016/j.jde.2011.09.009.
[24]	S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation,, Appl. Math. Optim., 64 (2011), 257. doi: 10.1007/s00245-011-9138-9.
[25]	P. B. Mucha, Limit of kinetic term for a Stefan problem,, Dis. Cont. Dynamical Syst., 2007 (): 741.
[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. doi: 10.1016/S0022-0396(02)00017-7.
[27]	R. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability,, Arch. Rational Mech. Anal., 97 (1987), 353. doi: 10.1007/BF00280411.
[28]	P. Rybka, Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions,, Proc. Roy. Soc. Edin. A, 121 (1992), 101. doi: 10.1017/S0308210500014177.
[29]	B. Tvedt, Quasilinear equations for viscoelasticity of strain-rate type,, Arch. Rat. Mech. Anal., 189 (2008), 237. doi: 10.1007/s00205-007-0109-x.

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