| Title:
|
Regularity and uniqueness in quasilinear parabolic systems (English) |
| Author:
|
Krejčí, Pavel |
| Author:
|
Panizzi, Lucia |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
341-370 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma. (English) |
| Keyword:
|
parabolic system |
| Keyword:
|
regularity |
| Keyword:
|
uniqueness |
| MSC:
|
35A02 |
| MSC:
|
35B65 |
| MSC:
|
35K51 |
| MSC:
|
35K59 |
| MSC:
|
35K60 |
| idZBL:
|
Zbl 1240.35234 |
| idMR:
|
MR2833166 |
| DOI:
|
10.1007/s10492-011-0020-5 |
| . |
| Date available:
|
2011-06-23T13:07:18Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141598 |
| . |
| Reference:
|
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics, Vol. 65.Academic Press New York-London (1975). MR 0450957 |
| Reference:
|
[2] Barrett, J. W., Deckelnick, K.: Existence, uniqueness and approximation of a doubly-degenerate nonlinear parabolic system modelling bacterial evolution.Math. Models Methods Appl. Sci. 17 (2007), 1095-1127. Zbl 1144.35026, MR 2337432, 10.1142/S0218202507002212 |
| Reference:
|
[3] Besov, O. V., Il'in, V. P., Nikol'skij, S. M.: Integral Representations of Functions and Imbedding Theorems. Scripta Series in Mathematics (Vol. I, Vol. II).V. H. Winston & Sons, John Wiley & Sons Washington/New York (1978), 1979; Russian version Nauka, Moscow, 1975. MR 0430771 |
| Reference:
|
[4] Fasano, A., Hömberg, D., Panizzi, L.: A mathematical model for case hardening of steel.Math. Models Methods Appl. Sci. 19 (2009), 2101-2126. Zbl 1180.35277, MR 2588960, 10.1142/S0218202509004054 |
| Reference:
|
[5] Griepentrog, J. A.: Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces.Adv. Differ. Equ. 12 (2007), 1031-1078. Zbl 1157.35023, MR 2351837 |
| Reference:
|
[6] Koshelev, A. I.: Regularity Problem for Quasilinear Elliptic and Parabolic Systems. Lecture Notes in Mathematics, 1614.Springer Berlin (1995). MR 1442954 |
| Reference:
|
[7] Ladyzhenskaya, O. A., Solonnikov, V. A., Ural'ceva, N. N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society Transl. 23.AMS Providence (1968). |
| Reference:
|
[8] Lions, J.-L.: Quelques méthodes de résolution des problemes aux limites non linéaires.Dunod/Gauthier-Villars Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[9] Lieberman, G. M.: Second Order Parabolic Differential Equations.World Scientific Publishing Co., Inc. Singapore (1996). Zbl 0884.35001, MR 1465184 |
| Reference:
|
[10] Nečas, J.: Les méthodes directes en théorie des équations elliptiques.Academia Prague (1967). MR 0227584 |
| Reference:
|
[11] Rodrigues, J. F.: A nonlinear parabolic system arising in thermomechanics and in thermomagnetism.Math. Models Methods Appl. Sci. 2 (1992), 271-281. Zbl 0763.35093, MR 1181337, 10.1142/S021820259200017X |
| Reference:
|
[12] Shilkin, T.: Classical solvability of the coupled system modelling a heat-convergent Poiseuille-type flow.J. Math. Fluid Mech. 7 (2005), 72-84. Zbl 1065.35135, MR 2127742, 10.1007/s00021-004-0112-z |
| . |
