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A local existence result for a system of viscoelasticity with physical viscosity
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
P. Clement and S. Li, Abstract parabolic quasilinear equations and application to a ground-water flow problem, Adv. Math. Sci. Appl., 3 (1993/94), 17-32. |
[9] |
C. Dafermos, The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity, J. Diff. Eqs., 6 (1969), 71-86. |
[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), 1458-1480.
doi: 10.1002/cpa.21409. |
[12] |
S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity, Arch. Rat. Mech. Anal., 155 (2000), 299-334.
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, Springer-Verlag, New York, 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[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. |
[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. |
[18] |
O. Ladyzhenskaya, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translation of Mathematical Monographs, 23, AMS, 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), 402-426.
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, Optimisation and Calculus of Variations, 17 (2011), 1158-1173.
doi: 10.1051/cocv/2010039. |
[21] |
G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. |
[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. |
[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. |
[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. |
[25] |
P. B. Mucha, Limit of kinetic term for a Stefan problem, Dis. Cont. Dynamical Syst., 2007, Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, suppl., 741-750. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
P. Clement and S. Li, Abstract parabolic quasilinear equations and application to a ground-water flow problem, Adv. Math. Sci. Appl., 3 (1993/94), 17-32. |
[9] |
C. Dafermos, The mixed initial-boundary value problem for the equations of one- dimensional nonlinear viscoelasticity, J. Diff. Eqs., 6 (1969), 71-86. |
[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), 1458-1480.
doi: 10.1002/cpa.21409. |
[12] |
S. Demoulini, Weak solutions for a class of nonlinear systems of viscoelasticity, Arch. Rat. Mech. Anal., 155 (2000), 299-334.
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, Springer-Verlag, New York, 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[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. |
[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. |
[18] |
O. Ladyzhenskaya, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translation of Mathematical Monographs, 23, AMS, 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), 402-426.
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, Optimisation and Calculus of Variations, 17 (2011), 1158-1173.
doi: 10.1051/cocv/2010039. |
[21] |
G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. |
[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. |
[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. |
[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. |
[25] |
P. B. Mucha, Limit of kinetic term for a Stefan problem, Dis. Cont. Dynamical Syst., 2007, Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, suppl., 741-750. |
[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. |
[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. |
[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. |
[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. |
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