A conditional regularity result for p-harmonic flows
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- 作者：Krystian Kazaniecki ; Michał Łasica…
- 刊名：NoDEA : Nonlinear Differential Equations and Applications
- 出版年：2016
- 出版时间：April 2016
- 年：2016
- 卷：23
- 期：2
- 全文大小：560 KB
- 参考文献：1.Bögelein V., Duzaar F., Mingione G.: The regularity of general parabolic systems with degenerate diffusion. Mem. Am. Math. Soc. 221(1041), vi+143 (2013)MathSciNet MATH
2.Bögelein V., Duzaar F., Scheven C.: Global solutions to the heat flow for m-harmonic maps and regularity. Indiana Univ. Math. J. 61(6), 2157–2210 (2012)MathSciNet CrossRef MATH
3.Chang K.-C., Yue Ding W., Ye R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36(2), 507–515 1992 (1992)MathSciNet MATH
4.Chen Y.M., Struwe M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)MathSciNet CrossRef MATH
5.Coron J.-M., Ghidaglia J.-M.: Explosion en temps fini pour le flot des applications harmoniques. C. R. Acad. Sci. Paris Sér. I Math. 308(12), 339–344 (1989)MathSciNet MATH
6.DiBenedetto, E.: Degenerate Parabolic Equations. Universitext. Springer-Verlag, New York (1993)
7.DiBenedetto E., Friedman A.: Regularity of solutions of nonlinear degenerate parabolic systems. J. Reine Angew. Math. 349, 83–128 (1984)MathSciNet MATH
8.DiBenedetto E., Friedman A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357, 1–22 (1985)MathSciNet MATH
9.Duzaar F., Mingione G., Steffen K.: Parabolic systems with polynomial growth and regularity. Mem. Am. Math. Soc. 214(1005), x+118 (2011)MathSciNet MATH
10.Eells Jr. J., Sampson J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNet CrossRef MATH
11.Hardt R.M.: Singularities of harmonic maps. Bull. Am. Math. Soc. (N.S.) 34(1), 15–34 (1997)MathSciNet CrossRef MATH
12.Hélein F.: Harmonic Maps, Conservation Laws and Moving Frames, volume 150 of Cambridge Tracts in Mathematics, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRef
13.Hungerbühler N.: Global weak solutions of the p-harmonic flow into homogeneous spaces. Indiana Univ. Math. J. 45(1), 275–288 (1996)MathSciNet CrossRef MATH
14.Hungerbühler, N.: m-harmonic flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24(4), 593–631 (1997)
15.Karim C., Misawa M.: Hölder regularity for singular parabolic systems of p-Laplacian type. Adv. Differ. Equ. 20(7–8), 741–772 (2015)MathSciNet MATH
16.Kuusi T., Mingione G.: Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28(2), 535–576 (2012)MathSciNet MATH
17.Leone C., Misawa M., Verde A.: The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth. J. Differ. Equ. 256(8), 2807–2845 (2014)MathSciNet CrossRef MATH
18.Misawa, M.: Existence and regularity results for the gradient flow for p-harmonic maps. Electron. J. Differ. Equ. 36, 17(electronic) (1998)
19.Rivière T., Strzelecki P.: A sharp nonlinear Gagliardo–Nirenberg-type estimate and applications to the regularity of elliptic systems. Commun. Partial Differ. Equ. 30(4–6), 589–604 (2005)MathSciNet CrossRef MATH
20.Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(4), 558–581 (1985)MathSciNet CrossRef MATH
21.Struwe M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28(3), 485–502 (1988)MathSciNet MATH - 作者单位：Krystian Kazaniecki (1)
Michał Łasica (2)
Katarzyna Ewa Mazowiecka (1)
Paweł Strzelecki (1)
1. Institute of Mathematics, University of Warsaw, Banacha 2, 02–097, Warszawa, Poland
2. Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02–097, Warszawa, Poland - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis - 出版者：Birkh盲user Basel
- ISSN：1420-9004
文摘
We prove an \({\varepsilon}\)-regularity result for a wide class of parabolic systems $$u_t-\rm{div}\big(|\nabla u|^{p-2}\nabla u) = B(\cdot, u, \nabla u)$$with the right hand side B growing critically, like \({|\nabla u|^p}\). It is assumed a priori that the solution \({u(t,\cdot)}\) is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo–Nirenberg inequality which has been used earlier in the elliptic context by T. Rivière and the last named author. Mathematics Subject Classification Primary: 35K65 35K92 Secondary: 35K55 53C44 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (21) References1.Bögelein V., Duzaar F., Mingione G.: The regularity of general parabolic systems with degenerate diffusion. Mem. Am. Math. Soc. 221(1041), vi+143 (2013)MathSciNetMATH2.Bögelein V., Duzaar F., Scheven C.: Global solutions to the heat flow for m-harmonic maps and regularity. Indiana Univ. Math. J. 61(6), 2157–2210 (2012)MathSciNetCrossRefMATH3.Chang K.-C., Yue Ding W., Ye R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36(2), 507–515 1992 (1992)MathSciNetMATH4.Chen Y.M., Struwe M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989)MathSciNetCrossRefMATH5.Coron J.-M., Ghidaglia J.-M.: Explosion en temps fini pour le flot des applications harmoniques. C. R. Acad. Sci. Paris Sér. I Math. 308(12), 339–344 (1989)MathSciNetMATH6.DiBenedetto, E.: Degenerate Parabolic Equations. Universitext. Springer-Verlag, New York (1993)7.DiBenedetto E., Friedman A.: Regularity of solutions of nonlinear degenerate parabolic systems. J. Reine Angew. Math. 349, 83–128 (1984)MathSciNetMATH8.DiBenedetto E., Friedman A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357, 1–22 (1985)MathSciNetMATH9.Duzaar F., Mingione G., Steffen K.: Parabolic systems with polynomial growth and regularity. Mem. Am. Math. Soc. 214(1005), x+118 (2011)MathSciNetMATH10.Eells Jr. J., Sampson J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefMATH11.Hardt R.M.: Singularities of harmonic maps. Bull. Am. Math. Soc. (N.S.) 34(1), 15–34 (1997)MathSciNetCrossRefMATH12.Hélein F.: Harmonic Maps, Conservation Laws and Moving Frames, volume 150 of Cambridge Tracts in Mathematics, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRef13.Hungerbühler N.: Global weak solutions of the p-harmonic flow into homogeneous spaces. Indiana Univ. Math. J. 45(1), 275–288 (1996)MathSciNetCrossRefMATH14.Hungerbühler, N.: m-harmonic flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24(4), 593–631 (1997)15.Karim C., Misawa M.: Hölder regularity for singular parabolic systems of p-Laplacian type. Adv. Differ. Equ. 20(7–8), 741–772 (2015)MathSciNetMATH16.Kuusi T., Mingione G.: Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28(2), 535–576 (2012)MathSciNetMATH17.Leone C., Misawa M., Verde A.: The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth. J. Differ. Equ. 256(8), 2807–2845 (2014)MathSciNetCrossRefMATH18.Misawa, M.: Existence and regularity results for the gradient flow for p-harmonic maps. Electron. J. Differ. Equ. 36, 17(electronic) (1998)19.Rivière T., Strzelecki P.: A sharp nonlinear Gagliardo–Nirenberg-type estimate and applications to the regularity of elliptic systems. Commun. Partial Differ. Equ. 30(4–6), 589–604 (2005)MathSciNetCrossRefMATH20.Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(4), 558–581 (1985)MathSciNetCrossRefMATH21.Struwe M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28(3), 485–502 (1988)MathSciNetMATH About this Article Title A conditional regularity result for p-harmonic flows Journal Nonlinear Differential Equations and Applications NoDEA 23:9 Online DateMarch 2016 DOI 10.1007/s00030-016-0369-y Print ISSN 1021-9722 Online ISSN 1420-9004 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Analysis Keywords Primary: 35K65 35K92 Secondary: 35K55 53C44 Authors Krystian Kazaniecki (1) Michał Łasica (2) Katarzyna Ewa Mazowiecka (1) Paweł Strzelecki (1) Author Affiliations 1. Institute of Mathematics, University of Warsaw, Banacha 2, 02–097, Warszawa, Poland 2. Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02–097, Warszawa, Poland Continue reading... To view the rest of this content please follow the download PDF link above.