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Finite Time Blowup for Parabolic Systems in Two Dimensions

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Abstract

We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.

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References

  1. Campanato S.: On the nonlinear parabolic systems in divergence form. Ann. Mat. Pura Appl. 137, 83–122 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. De Giorgi E.: Sulla differenziabilità e l’analicità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino cl. Sci. Fis. Fat. Nat. 3, 25–43 (1957)

    MATH  Google Scholar 

  3. De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. UMI 4, 135–137 (1968)

    MathSciNet  MATH  Google Scholar 

  4. Giusti E., Miranda M.: Un esempio di soluzione discontinua per un problem di minimo relativo ad un integrale regolare del calcolo delle variazioni. Boll. Un. Mat. Ital. 2, 1–8 (1968)

    MATH  Google Scholar 

  5. John O., Stará J.: On the regularity of weak solutions to parabolic systems in two spatial dimensions. Commun. PDE 23, 1159–1170 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Morrey C.B.: Multiple Integrals in the Calculus of Variations. Springer, Heidelberg (1966)

    MATH  Google Scholar 

  7. Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  8. Necas J., Šverák V.: On regularity of solutions of nonlinear parabolic systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 18, 1–11 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Stará J., John O.: Some (new) counterexamples of parabolic systems. Comment. Math. Univ. Carolin. 36, 503–510 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Stará J., John O., Malý J.: Counterexample to the regularity of weak solution of the quasilinear parabolic system. Comment. Math. Univ. Carolin. 27, 123–136 (1986)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, 8092, Zurich, Switzerland

    Connor Mooney

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  1. Connor Mooney
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Correspondence to Connor Mooney.

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Communicated by V. Šverák

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Mooney, C. Finite Time Blowup for Parabolic Systems in Two Dimensions. Arch Rational Mech Anal 223, 1039–1055 (2017). https://doi.org/10.1007/s00205-016-1052-5

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  • DOI: https://doi.org/10.1007/s00205-016-1052-5

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