Advertisement

Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 591–636 | Cite as

Partial Regularity for Type Two Doubly Nonlinear Parabolic Systems

  • Ryan HyndEmail author
Article

Abstract

We consider weak solutions v : \({U \times (0, T ) \rightarrow \mathbb{R}^{m}}\) of the nonlinear parabolic system
$${D\psi({v}_{t} ) = {\rm div} DF({D}_{v}),}$$
where \({\psi}\) and F are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of F are Hölder continuous, we show that D2v and vt are locally Hölder continuous except for possibly on a lower dimensional subset of \({U \times (0, T )}\). Our approach relies on two integral identities, decay of the local space-time energy of solutions, and fractional time derivative estimates for D2v and vt.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn Lectures inMathematics ETH Zrich. Birkhuser Verlag, Basel (2008)zbMATHGoogle Scholar
  2. 2.
    Arai T.: On the existence of the solution for \(\partial \phi(u'(t))+\partial \psi(u(t))\ni f(t)\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26(1), 75–96 (1979)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blanchard D., Damlamian A., Ghidouche H.: A nonlinear system for phase change with dissipation. Differ. Integral Equ. 2, 344–362 (1989)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bonfanti G., Frémond M., Luterotti F.: Global solution to a nonlinear system for irreversible phase changes. Adv. Math. Sci. Appl. 10, 1–24 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Campanato S.: Proprietà di hölderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa (3) 17, 175–188 (1963)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Campanato S.: On the nonlinear parabolic systems in divergence form Hölder continuity and partial Hölder continuity of the solutions. . Ann. Mat. Pura Appl. 137(4), 83–122 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colli P.: On some doubly nonlinear evolution equations in Banach spaces. Jpn. J. Ind.Appl. Math. 9(2), 181–203 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colli P., Visintin A.: On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15, 737–756 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Colli P., Luterotti F., Schimperna G., Stefanelli U.: Global existence for a class of generalized systems for irreversible phase which contradicts changes. NoDEA Nonlinear Differ. Equ. Appl. 9, 255–276 (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Da Prato G.: Spazi \({\mathcal{L}^{p, \theta}(\Omega,\delta) }\) e loro proprietà. Ann.Mat. Pura Appl.(4) 69, 383–392 (1965)MathSciNetCrossRefGoogle Scholar
  11. 11.
    De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. Ital.(4) 1, 135–137 (1968)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duzaar F., Mingione G.: Second order parabolic systems, optimal regularity, and singular sets of solutions. Ann. I. H. Poincaré AN 22, 705–751 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Duzaar, F., Mingione, G., Steffen, K.: Parabolic systems with polynomial growth and regularity. Mem. Am. Math. Soc. (2011). http://doi.org/10.1090/S0065-9266-2011-00614-3
  15. 15.
    Evans, L.C.: Partial Differential Equations , 2nd edn.Graduate Studies in Mathematics , vol. 19. American Mathematical Society, Providence, 2010Google Scholar
  16. 16.
    Evans L. C., Gariepy R.: Measure Theory and Fine Properties of Functions Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  17. 17.
    Francfort G., Mielke A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595, 55–91 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Giaquinta M., Martinazzi L.: An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edn. Edizioni della Normale, Pisa (2012)CrossRefzbMATHGoogle Scholar
  19. 19.
    Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co.Inc, River Edge (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Giusti E., Miranda M.: Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari. Arch. Ration. Mech. Anal. 31, 173–184 (1968)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hynd, R.: Partial regularity for doubly nonlinear parabolic systems of the first type. Indiana Univ. Math. J. (to appear) Google Scholar
  22. 22.
    Hynd R.: Compactness methods for doubly nonlinear parabolic systems. Trans. Am. Math. Soc. 369(7), 5031–5068 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hynd, R., Lindgren, E.: Approximation of the least Rayleigh quotient for degree p homogeneous functionals. J. Funct. Anal. 272(12), 4873–4918Google Scholar
  24. 24.
    Larsen C., Ortiz M., Richardson C.: Fracture paths from front kinetics: relaxation and rate independence. Arch. Ration. Mech. Anal. 193(3), 539–583 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lawson H.B., Osserman R.: Non-existence, non-uniqueness and irregularity of so1269 lutions to the minimal surface system. Acta Math. 139, 1–17 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Manfredi J., Vespri V.: Large time behavior of solutions to a class of doubly nonlinear parabolic equations. Electron. J. Differ. Equ. 02, 1–17 (1994)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mielke A., Rossi R.: Existence and uniqueness results for a class of rate-independent hysteresis problems. Math. Models Methods Appl. Sci. 17(1), 81–123 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mielke A., Rossi R., Savarè G.: Nonsmooth analysis of doubly nonlinear evolution equations. Calc. Var. Partial Differ. Equ. 46(1-2), 253–310 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mielke A., Ortiz M.: A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM Control Optim. Calc. Var. 14(3), 494–516 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mingione G.: The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166(4), 287–301 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rogers C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970)zbMATHGoogle Scholar
  32. 32.
    Schimperna G., Segatti A., Stefanelli U.: Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete Contin. Dyn. Syst. 18(1), 15–38 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Simon J.: Compact sets in the space L p(0, T ; B). Ann.Mat. Pura Appl. (4) 146, 65–96 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Visintin A.: Models of Phase Transitions Progress in NonlinearDifferential Equations and their Applications, vol. 28. . Birkhuser Boston, Inc., Boston (1996)Google Scholar
  35. 35.
    Visintin A.: Differential Models of Hysteresis Applied Mathematical Sciences, vol. 111. Springer, Berlin (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

Personalised recommendations