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For the parabolic Lamé system on polygonal domains: the transport equation. (English) Zbl 1286.35130

Summary: We study the parabolic Lamé system with initial and boundary conditions on non-convex plane polygonal domains. We express the solution by the inverse of the sum of two operators taken from [G. da Prato and P. Grisvard, J. Math. Pures Appl. (9) 54, 305–387 (1975; Zbl 0315.47009)] and split the solution into a singular part and a regular part by applying to the inverse the corner singularity result of the Lamé system with parameter. We show that the remainder has the \(H^{2,q}\)-regularity and that the coefficients of the corner singularities, so-called the stress intensity factors, have the fractional order regularities on the time interval. Also we investigate the transport equation directed by the vector field having the corner singularity decomposition.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35B65 Smoothness and regularity of solutions to PDEs
74A10 Stress
74B05 Classical linear elasticity
74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics

Citations:

Zbl 0315.47009
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Full Text: DOI

References:

[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030
[2] Agranovitch, M. S.; Vishik, M. I., Elliptic problems with parameters and parabolic problems of general type, Russian Math. Surveys, 19, 53-157 (1964) · Zbl 0137.29602
[3] Bochniak, M.; Sändig, A.-M., Computation of generalized intensity factors for bonded elastic structures, M2AN, 33, 853-878 (1999) · Zbl 0961.74027
[4] Chorin, A. J.; Marsden, J. E., A Mathematical Introduction to Fluid Mechanics (1993), Springer-Verlag: Springer-Verlag New York, Berlin · Zbl 0774.76001
[5] Da Prato, G.; Grisvard, P., Sommes Dʼopérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., 54, 305-387 (1975) · Zbl 0315.47009
[6] Dauge, M., Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math., vol. 1341 (1988), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0668.35001
[7] Dauge, M., Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I. Linearized equations, SIAM J. Math. Anal., 20, 74-97 (1989) · Zbl 0681.35071
[8] Dautray, R.; Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Evolution Problems I (1985), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York
[9] Dore, G.; Venni, A., On the closedness of the sum of two closed operators, Math. Z., 196, 189-201 (1987) · Zbl 0615.47002
[10] Evans, L. C., Partial Differential Equations (1998), AMS
[11] Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations (1986), Springer-Verlag: Springer-Verlag Berlin, Heidelberg
[12] Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston, London, Melbourne · Zbl 0695.35060
[13] Grisvard, P., Singular behavior of elliptic problems in non Hilbertian Sobolev spaces, J. Math. Pures Appl., 74, 3-33 (1995) · Zbl 0854.35018
[14] Grisvard, P., Singularities in Boundary Value Problems, Recherches en Mathematiques Appliquees, vol. 22 (1992), Masson · Zbl 0766.35001
[15] Grisvard, P., Edge behavior of the solution of an elliptic problem, Math. Nachr., 132, 281-299 (1987) · Zbl 0639.35008
[16] Grisvard, P., Equations différentielles abstraites, Ann. Sci. École Norm. Sup. (4), 2, 311-395 (1969) · Zbl 0193.43502
[17] Kellogg, R. B., Corner singularities and singular perturbations, Ann. Univ. Ferrara Sez. VII Sci. Mat., XLVII, 177-206 (2001) · Zbl 1119.35318
[18] Kondratʼev, V. A., The smoothness of solutions of Dirichletʼs problem for second-order elliptic equation in a region with a piecewise-smooth boundary, Differ. Equ., 6, 1392-1401 (1976)
[19] Kondratʼev, V. A., Singularities of a solution of Dirichletʼs problem for a second-order equation in the neighborhood of an edge, Differ. Equ., 13, 1411-1415 (1977) · Zbl 0394.35027
[20] Kozlov, V. A.; Maźya, V. G.; Rossmann, J., Elliptic Boundary Value Problems in Domains with Point Singularities (1997), AMS · Zbl 0947.35004
[21] Kozlov, V. A.; Maźya, V. G.; Rossmann, J., Spectral Problems Associated Corner Singularities of Solutions to Elliptic Equations (2001), AMS · Zbl 0965.35003
[22] Kweon, J. R.; Kellogg, R. B., Regularity of solutions to the Navier-Stokes equations for compressible flows on a polygon, SIAM J. Math. Anal., 35, 1451-1485 (2004) · Zbl 1064.35137
[23] Kweon, J. R., An evolution compressible Stokes system in a polygon, J. Differential Equations, 199, 352-375 (2004) · Zbl 1080.35075
[24] Kweon, J. R., Regularity of solutions for the Navier-Stokes system of incompressible flows on a polygon, J. Differential Equations, 235, 166-198 (2007) · Zbl 1119.35055
[25] Kweon, J. R.; Song, M. S., A discontinuous solution for an evolution compressible Stokes system in a bounded domain, J. Differential Equations, 219, 202-220 (2005) · Zbl 1091.35060
[26] Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications I and II (1972), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0227.35001
[27] Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995), Birkhäuser Verlag · Zbl 0816.35001
[28] Mazzucato, A. L.; Nistor, V., Well-posedness and regularity for the elasticity equations with mixed boundary conditions on polyhedral domains and domains with cracks, Arch. Ration. Mech. Anal., 195, 25-73 (2010) · Zbl 1188.35189
[29] Nazarov, S. A.; Plamenevsky, B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries (1994), Walter de Gruyter: Walter de Gruyter Berlin, New York · Zbl 0806.35001
[30] Nicaise, S.; Sändig, A.-M., Dynamic crack propagation in a 2D elastic body: The out-of-plane case, J. Math. Anal. Appl., 329, 1-30 (2007) · Zbl 1342.74143
[31] Rössle, A., Corner singularities and regularity of weak solutions for the two-dimensional Lamé equations on domains with angular corners, J. Elasticity, 60, 57-75 (2000) · Zbl 0976.74019
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