×

A local and global well-posedness results for the general stress-assisted diffusion systems. (English) Zbl 1382.35296

Summary: We prove the local and global in time existence of the classical solutions to two general classes of the stress-assisted diffusion systems. Our results are applicable in the context of the non-Euclidean elasticity and liquid crystal elastomers.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35Q35 PDEs in connection with fluid mechanics
74B20 Nonlinear elasticity
76A15 Liquid crystals
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abels, H., Mora, M.G., Müller, S.: The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity. Calc. Var. Partial Differ. Equ. 41, 241-259 (2011) · Zbl 1346.74110 · doi:10.1007/s00526-010-0360-0
[2] Abels, H., Mora, M.G., Müller, S.: Large time existence for thin vibrating plates. Commun. Partial Differ. Equ. 36, 2062-2102 (2011) · Zbl 1247.74019 · doi:10.1080/03605302.2011.618209
[3] Andrews, G.: On the existence of solutions to the equation utt=uxxt+σ(ux)x \(u_{tt} = u_{xxt} + {\sigma}(u_x)_x\). J. Differ. Equ. 35, 200-231 (1980) · Zbl 0397.35011 · doi:10.1016/0022-0396(80)90040-6
[4] Antmann, S., Malek-Madani, R.: Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media. Q. Appl. Math. 46, 77-93 (1988) · Zbl 0677.73022
[5] Antman, S., Seidman, T.: Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity. J. Differ. Equ. 124, 132-184 (1996) · Zbl 0844.35021 · doi:10.1006/jdeq.1996.0005
[6] Barker, B., Lewicka, M., Zumbrun, K.: Existence and stability of viscoelastic shock profiles. Arch. Ration. Mech. Anal. 200(2), 491-532 (2011) · Zbl 1233.35023 · doi:10.1007/s00205-010-0363-1
[7] Barucq, H., Madaune-Tort, M., Saint-Macary, P.: Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models. Nonlinear Anal. 61(4), 591-612 (2005) · Zbl 1068.35003 · doi:10.1016/j.na.2004.10.023
[8] Bhattacharya, K., Lewicka, M., Schaffner, M.: Plates with incompatible prestrain. Arch. Ration. Mech. Anal. (2015, to appear) · Zbl 1382.74081
[9] Dafermos, C.: The mixed initial-boundary value problem for the equations of one-dimensional nonlinear viscoelasticity. J. Differ. Equ. 6, 71-86 (1969) · Zbl 0218.73054 · doi:10.1016/0022-0396(69)90118-1
[10] Demoulini, S.: Weak solutions for a class of nonlinear systems of viscoelasticity. Arch. Ration. Mech. Anal. 155(4), 299-334 (2000) · Zbl 0991.74021 · doi:10.1007/s002050000115
[11] Dervaux, J., Ciarletta, P., Ben Amar, M.: Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Foppl-von Karman limit. J. Mech. Phys. Solids 57(3), 458-471 (2009) · Zbl 1170.74354 · doi:10.1016/j.jmps.2008.11.011
[12] Efrati, E., Sharon, E., Kupferman, R.: Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57, 762-775 (2009) · Zbl 1306.74031 · doi:10.1016/j.jmps.2008.12.004
[13] Feireisl, E., Mucha, P.B., Novotny, A., Pokorny, M.: Time-periodic solutions to the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204(3), 745-786 (2012) · Zbl 1287.76187 · doi:10.1007/s00205-012-0492-9
[14] Friesecke, G., James, R., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal. 180(2), 183-236 (2006) · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7
[15] Friesecke, G., James, R., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris 336(8), 697-702 (2003) · Zbl 1140.74481 · doi:10.1016/S1631-073X(03)00028-1
[16] Goriely, A.; Amar, M. B. (ed.); Goriely, A. (ed.); Müller, M. M. (ed.); Cugliandolo, L. (ed.), New trends in the physics and mechanics of biological systems (2009)
[17] Govindjee, S., Simo, J.: Coupled stress-diffusion: Case II. J. Mech. Phys. Solids 41(5), 863-887 (1993) · Zbl 0783.73057 · doi:10.1016/0022-5096(93)90003-X
[18] Garikipati, K., Bassman, L., Deal, M.: A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids. J. Mech. Phys. Solids 49(6), 1209-1237 (2001) · Zbl 1015.74005 · doi:10.1016/S0022-5096(00)00081-8
[19] Hill, J.: Plane steady solutions for stress-assisted diffusion. Mech. Res. Commun. 6(3), 147-150 (1979) · Zbl 0416.73077 · doi:10.1016/0093-6413(79)90056-9
[20] Jiang, S.; Wang, Y. G., Global existence and exponential stability in nonlinear thermoelasticity, 1998-2006 (2014) · doi:10.1007/978-94-007-2739-7_249
[21] Jones, G., Chapman, S.: Modeling growth in biological materials. SIAM Rev. 54(1), 52-118 (2012) · Zbl 1247.74042 · doi:10.1137/080731785
[22] Klein, Y., Efrati, E., Sharon, E.: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116-1120 (2007) · Zbl 1226.74013 · doi:10.1126/science.1135994
[23] Korn, A.: 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., 705-724 (1909) · JFM 40.0884.02
[24] Kupferman, R., Shamai, Y.: Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space. Isr. J. Math. 190, 135-156 (2012) · Zbl 1260.53069 · doi:10.1007/s11856-011-0187-1
[25] Kupferman, R., Maor, C.: A Riemannian approach to the membrane limit in non-Euclidean elasticity. Commun. Contemp. Math. (2015, to appear) · Zbl 1407.74016
[26] Ladyzhenskaya, O., Solonnikov, V., Uralceva, N.: Linear and Quasilinear Eqs of Parabolic Type. Translation of Mathematical Monographs, vol. 23. AMS, Providence (1968)
[27] Larche, F., Cahn, J.: The interactions of composition and stress in crystalline solids. Acta Metall. 33, 331-357 (1985) · doi:10.1016/0001-6160(85)90077-X
[28] LeDret, H., Raoult, A.: The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 73, 549-578 (1995) · Zbl 0847.73025
[29] Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6, 59-84 (1996) · Zbl 0844.73045 · doi:10.1007/BF02433810
[30] Lewicka, M., Mahadevan, L., Pakzad, R.: The Foppl-von Karman equations for plates with incompatible strains. Proc. R. Soc. A 467, 402-426 (2011) · Zbl 1219.74027 · doi:10.1098/rspa.2010.0138
[31] Lewicka, M., Mahadevan, L., Pakzad, R.: Models for elastic shells with incompatible strains. Proc. R. Soc. A 470 (2014) · Zbl 1219.74027
[32] Lewicka, M., Mahadevan, L., Pakzad, R.: The Monge-Ampère constrained elastic theories of shallow shells. Ann. Inst. Henri Poincare (C) Non Linear Anal. (2015, to appear) · Zbl 1359.35085
[33] Lewicka, M., Mucha, P.B.: A local existence result for a system of viscoelasticity with physical viscosity. Evol. Equ. Control Theory 2(2), 337-353 (2013) · Zbl 1273.74024 · doi:10.3934/eect.2013.2.337
[34] Lewicka, M., Ochoa, P., Pakzad, R.: Variational models for prestrained plates with Monge-Ampere constraint. Differ. Integral Equ. 28(9-10), 861-898 (2015) · Zbl 1363.74063
[35] Lewicka, M., Pakzad, R.: Scaling laws for non-Euclidean plates and the \(W2,2W^{2,2}\) isometric immersions of Riemannian metrics. ESAIM Control Optim. Calc. Var. 17(4), 1158-1173 (2011) · Zbl 1300.74028 · doi:10.1051/cocv/2010039
[36] Lewicka, M., Pakzad, M.: The infinite hierarchy of elastic shell models; some recent results and a conjecture. Infinite Dimensional Dynamical Systems, Fields Inst. Commun. 64, 407-420 (2013) · Zbl 1263.74035 · doi:10.1007/978-1-4614-4523-4_16
[37] Lewicka, M., Raoult, A., Ricciotti, D.: Plates with incompatible prestrain of higher order (2015, to appear) · Zbl 1457.74121
[38] Liang, H., Mahadevan, L.: Growth, geometry and mechanics of the blooming lily. Proc. Natl. Acad. Sci. 108, 5516-5521 (2011) · doi:10.1073/pnas.1007808108
[39] Marder, M.: The shape of the edge of a leaf. Found. Phys. 33, 1743-1768 (2003) · doi:10.1023/A:1026229605010
[40] Modes, C.D., Bhattacharya, K., Warner, M.: Disclination-mediated thermo-optical response in nematic glass sheets. Phys. Rev. E 81 (2010) · Zbl 1346.74110
[41] Modes, C.D., Bhattacharya, K., Warner, M.: Gaussian curvature from flat elastica sheets. Proc. R. Soc. A 467, 1121-1140 (2011) · Zbl 1219.74026 · doi:10.1098/rspa.2010.0352
[42] Mora, M.G., Scardia, L.: Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density. J. Differ. Equ. 252, 35-55 (2012) · Zbl 1291.74128 · doi:10.1016/j.jde.2011.09.009
[43] Mucha, P.B., Pokorny, M., Zatorska, E.: Chemically reacting mixtures in terms of degenerated parabolic setting. J. Math. Phys. 54(7), 071501 (2013), 17 pp. · Zbl 1302.76207 · doi:10.1063/1.4811564
[44] Pawlow, I., Zajaczkowski, W.M.: Unique global solvability in two-dimensional non-linear thermoelasticity. Math. Methods Appl. Sci. 28(5), 551-592 (2005) · Zbl 1069.74013 · doi:10.1002/mma.582
[45] Pego, R.: Phase transitions in one-dimensional nonlinear viscoelasticity. Arch. Ration. Mech. Anal. 97, 353-394 (1987) · Zbl 0648.73017 · doi:10.1007/BF00280411
[46] Rodriguez, E. K.; Hoger, A.; McCulloch, A., No article title, J. Biomech., 27, 455 (1994) · doi:10.1016/0021-9290(94)90021-3
[47] Racke, R., Shibata, Y.: Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. Arch. Ration. Mech. Anal. 116(1), 1-34 (1991) · Zbl 0756.73012 · doi:10.1007/BF00375601
[48] Vandiver, R., Goriely, A.: Morpho-elastodynamics: the long-time dynamics of elastic growth. J. Biol. Dyn. 3(2-3), 180-195 (2009) · Zbl 1342.92035 · doi:10.1080/17513750802304885
[49] Weitsman, Y.: Stress assisted diffusion in elastic and viscoelastic materials. J. Mech. Phys. Solids 35(1), 73-94 (1987) · Zbl 0598.73117 · doi:10.1016/0022-5096(87)90029-9
[50] Wu, C.H.: The role of Eshelby stress in composition-generated and stress-assisted diffusion. J. Mech. Phys. Solids 49(8), 1771-1794 (2001) · Zbl 1024.74022 · doi:10.1016/S0022-5096(01)00011-4
[51] Zvyagin, V.G., Orlov, V.P.: Existence and uniqueness results for a coupled problem in continuum thermomechanics. Vestn.: Fiz. Mat. 2, 120-141 (2014) · Zbl 1352.74012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.