On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions

Joachim Naumann; Jörg Wolf; Michael Wolff

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 2, page 237-255
  • ISSN: 0010-2628

Abstract

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We prove the interior Hölder continuity of weak solutions to parabolic systems ∂ u j ∂ t - D α a j α ( x , t , u , ∇ u ) = 0 in Q ( j = 1 , ... , N ) ( Q = Ω × ( 0 , T ) , Ω ⊂ ℝ 2 ), where the coefficients a j α ( x , t , u , ξ ) are measurable in x , Hölder continuous in t and Lipschitz continuous in u and ξ .

How to cite

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Naumann, Joachim, Wolf, Jörg, and Wolff, Michael. "On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions." Commentationes Mathematicae Universitatis Carolinae 39.2 (1998): 237-255. <http://eudml.org/doc/248240>.

@article{Naumann1998,
abstract = {We prove the interior Hölder continuity of weak solutions to parabolic systems \[ \frac\{\partial u^j\}\{\partial t\}-D\_\alpha a\_j^\alpha (x,t,u,\nabla u)=0 \text\{ in \} Q \quad (j=1,\ldots ,N) \] ($Q=\Omega \times (0,T),\Omega \subset \mathbb \{R\}^2$), where the coefficients $a_j^\alpha (x,t,u,\xi )$ are measurable in $x$, Hölder continuous in $t$ and Lipschitz continuous in $u$ and $\xi $.},
author = {Naumann, Joachim, Wolf, Jörg, Wolff, Michael},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonlinear parabolic systems; Hölder continuity; Fourier transform; interior Hölder continuity; Fourier transform},
language = {eng},
number = {2},
pages = {237-255},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions},
url = {http://eudml.org/doc/248240},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Naumann, Joachim
AU - Wolf, Jörg
AU - Wolff, Michael
TI - On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 2
SP - 237
EP - 255
AB - We prove the interior Hölder continuity of weak solutions to parabolic systems \[ \frac{\partial u^j}{\partial t}-D_\alpha a_j^\alpha (x,t,u,\nabla u)=0 \text{ in } Q \quad (j=1,\ldots ,N) \] ($Q=\Omega \times (0,T),\Omega \subset \mathbb {R}^2$), where the coefficients $a_j^\alpha (x,t,u,\xi )$ are measurable in $x$, Hölder continuous in $t$ and Lipschitz continuous in $u$ and $\xi $.
LA - eng
KW - nonlinear parabolic systems; Hölder continuity; Fourier transform; interior Hölder continuity; Fourier transform
UR - http://eudml.org/doc/248240
ER -

References

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  4. Da Prato G., Spazi ℒ ( p , θ ) ( Ø m e g a , δ ) e loro proprietà, Ann. Mat. Pura Appl. (4) 69 (1965), 383-392. (1965) Zbl0145.16207MR0192330
  5. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, New Jersey, 1983. Zbl0516.49003MR0717034
  6. John O., Stará J., On the regularity of weak solutions to parabolic systems in two dimensions, to appear. 
  7. Kalita E., On the Hölder continuity of solutions of nonlinear parabolic systems, Comment. Math. Univ. Carolinae 35 (1994), 675-680. (1994) Zbl0814.35011MR1321237
  8. Koshelev A., Regularity of solutions for some quasilinear parabolic systems, Math. Nachr. 162 (1993), 59-88. (1993) Zbl0811.35064MR1239576
  9. Ladyzenskaja O.A., Solonnikov V.A., Ural'ceva N.N., Linear and quasilinear equations of parabolic type, Transl. Math. Monographs 28, Amer. Math. Soc., Providence, R.I., 1968. MR0241822
  10. Naumann J., Wolf J., Interior differentiability of weak solutions to parabolic systems with quadratic growth nonlinearities, to appear. Zbl0894.35050MR1066428
  11. Nečas J., Šverák V., On regularity of nonlinear parabolic systems, Ann. Scuola Norm. Sup. Pisa (4) 18 (1991), 1-11. (1991) MR1118218

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