Quasi-linear parabolic systems in divergence form with weak monotonicity

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Quasi-linear parabolic systems in divergence form with weak monotonicity

Norbert Hungerbühler

Source: Duke Math. J. Volume 107, Number 3 (2001), 497-520.

Abstract

We consider the initial and boundary value problem for the quasi-linear parabolic system

$\begin{alignat*}{2} \frac{\partial u}{\partial t}-\di \sigma\big(x,t,u(x,t),Du(x,t)\big) &=f &\text{on }&\Omega\times(0,T),\\ u(x,t)&=0&\text{on }&\partial\Omega\times(0,T),\\ u(x,0)&=u_0(x)\quad&\text{on }&\Omega \end{alignat*}$

for a function u : Ω×[0,T)→ℝ^m with T>0. Here, f∈L^p′(0,T;W^p′(Ω;ℝ^m)) for some p∈(2n/2n+2),∞, and u₀∈L²(Ω, ℝ^m). We prove existence of a weak solution under classical regularity, growth, and coercivity conditions for σ but with only very mild monotonicity assumptions.

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1091737021
Mathematical Reviews number (MathSciNet): MR1828299
Digital Object Identifier: doi:10.1215/S0012-7094-01-10733-3
Zentralblatt MATH identifier: 1012.35037

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References

J. M. Ball, ``A version of the fundamental theorem for Young measures'' in PDEs and Continuum Models of Phase Transitions (Nice, 1988), Lecture Notes in Phys. 344, Springer, Berlin, 1989, 207--215.

Mathematical Reviews (MathSciNet): MR1036070

Digital Object Identifier: doi:10.1007/BFb0024945

Zentralblatt MATH: 0991.49500

L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), 581--597.

Mathematical Reviews (MathSciNet): MR1183665

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, Notas Mat. 50, American Elsevier, New York, 1973.

Mathematical Reviews (MathSciNet): MR348562

H. Brézis and F. E. Browder, Strongly nonlinear parabolic initial-boundary value problems, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 38--40.

Mathematical Reviews (MathSciNet): MR516141

Digital Object Identifier: doi:10.1073/pnas.76.1.38

JSTOR: links.jstor.org

Zentralblatt MATH: 0397.35030

F. E. Browder, ``Existence theorems for nonlinear partial differential equations'' in Global Analysis (Berkeley, Calif., 1968), Proc. Sympos. Pure Math. 16, Amer. Math. Soc., Providence, 1970, 1--60.

Mathematical Reviews (MathSciNet): MR269962

G. Dolzmann, N. Hungerbühler, and S. Müller, Non-linear elliptic systems with measure-valued right hand side, Math. Z. 226 (1997), 545--574.

Mathematical Reviews (MathSciNet): MR1484710

Digital Object Identifier: doi:10.1007/PL00004354

N. Dunford and J. T. Schwartz, Linear Operators, Parts I--III, Wiley Classics Lib., Wiley, New York, 1988.

Mathematical Reviews (MathSciNet): MR1009162

L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Stud. Adv. Math., CRC Press, Boca Raton, Fla., 1992.

Mathematical Reviews (MathSciNet): MR1158660

I. Fonseca, S. Müller, and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29 (1998), 736--756., http://epubs.siam.org/sam-bin/dbq/toclist/SIMA

Mathematical Reviews (MathSciNet): MR1617712

Digital Object Identifier: doi:10.1137/S0036141096306534

Zentralblatt MATH: 0920.49009

J. Frehse, private communication, 1999.

N. Hungerbühler, A refinement of Ball's theorem on Young measures, New York J. Math. 3 (1997), 48--53., http://nyjm.albany.edu:8000/nyjm.html

Mathematical Reviews (MathSciNet): MR1443136

--. --. --. --., Quasilinear elliptic systems in divergence form with weak monotonicity, New York J. Math. 5 (1999), 83--90., http://nyjm.albany.edu:8000/nyjm.html

Mathematical Reviews (MathSciNet): MR1701827

E. Kamke, Das Lebesgue-Stieltjes-Integral, Teubner, Leipzig, 1960.

Mathematical Reviews (MathSciNet): MR125193

D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal. 4 (1994), 59--90.

Mathematical Reviews (MathSciNet): MR1274138

J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions, Math. Ann. 313 (1999), 653--710.

Mathematical Reviews (MathSciNet): MR1686943

Digital Object Identifier: doi:10.1007/s002080050277

Zentralblatt MATH: 0924.49012

R. Landes, On weak solutions of quasilinear parabolic equations, Nonlinear Anal. 9 (1985), 887--904.

Mathematical Reviews (MathSciNet): MR804558

--. --. --. --., A note on strongly nonlinear parabolic equations of higher order, Differential Integral Equations 3 (1990), 851--862.

Mathematical Reviews (MathSciNet): MR1059334

R. Landes and V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Ark. Mat. 25 (1987), 29--40.

Mathematical Reviews (MathSciNet): MR918378

--. --. --. --., On parabolic initial-boundary value problems with critical growth for the gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 135--158.

Mathematical Reviews (MathSciNet): MR1267364

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969.

Mathematical Reviews (MathSciNet): MR259693

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341--346.

Mathematical Reviews (MathSciNet): MR169064

Digital Object Identifier: doi:10.1215/S0012-7094-62-02933-2

Project Euclid: euclid.dmj/1077470254

Zentralblatt MATH: 0111.31202

J. Simon, Compact sets in the space $L^ p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65--96.

Mathematical Reviews (MathSciNet): MR916688

M. I. Višik, Solubility of boundary-value problems for quasi-linear parabolic equations of higher orders (in Russian), Mat. Sb. (N.S.) 59 (101) (1962) suppl., 289--325.

Mathematical Reviews (MathSciNet): MR157124

K. Yosida, Functional Analysis, 6th ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1980.

Mathematical Reviews (MathSciNet): MR617913

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer, New York, 1990.

Mathematical Reviews (MathSciNet): MR1033498

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