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Blow-up for a semi-linear advection-diffusion system with energy conservation

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Abstract

The authors study radial solutions to a model equation for the Navier-Stokes equations. It is shown that the model equation has self-similar singular solution if 5 ≤ n ≤ 9. It is also shown that the solution will blow up if the initial data is radial, large enough and n ≥ 5.

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References

  1. [1]

    Budd, C. J. and Qi, Y. W., The existence of bounded solutions of a semilinear elliptic equation, J. Diff. Eqs., 82(2), 1989, 207–218.

  2. [2]

    Escauriaza, L., Seregin, G. and Šverák, V., Backward uniqueness for the heat operator in half-space, Algebra i Analiz, 15(1), 2003, 201–214.

  3. [3]

    Giga, Y. and Kohn, R. V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38(3), 1985, 297–319.

  4. [4]

    Lepin, L. A., Self-similar solutions of a semilinear heat equation (in Russian), Mat. Modle., 2(3), 1990, 63–74.

  5. [5]

    Lepin, L. A., Spectra of eigenfunctions for a semilinear heat equation, Dokl. Akad. Nauk SSSR, 311(5), 1990, 1049–1051.

  6. [6]

    Plecháč, P. and Šverák, V., On self-similar singular solutions of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 54(10), 2001, 1215–1242.

  7. [7]

    Plecháč, P. and Šverák, V., Singular and regular solutions of a non-linear parabolic system, Nonlinearty, 16(6), 2003, 2083–2097.

  8. [8]

    Troy, W. C., The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18(2), 1987, 332–336.

  9. [9]

    Zhang, P., Remark on the regularities of Kato’s solutions to Navier-Stokes equations with initial data in L d(ℝd), Chin. Ann. Math., 29B(3), 2008, 265–272.

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Author information

Correspondence to Dapeng Du.

Additional information

Project supported by the China Postdoctoral Science Foundation (No. 20070410683).

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Du, D., Lü, J. Blow-up for a semi-linear advection-diffusion system with energy conservation. Chin. Ann. Math. Ser. B 30, 433–446 (2009). https://doi.org/10.1007/s11401-007-0474-7

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Keywords

  • Navier-Stokes equations
  • Self-similar singular solutions
  • Blow-up

2000 MR Subject Classification

  • 35K55
  • 35B05
  • 76A02