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2016, 15(4): 1335-1350. doi: 10.3934/cpaa.2016.15.1335

Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids

1. 

School of Mathematical Science, Xiamen University, Xiamen 361005, Fujian

2. 

School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

Received  November 2015 Revised  January 2016 Published  April 2016

In this paper, we study the Dirichlet problem arising in the electrorheological fluids \begin{eqnarray} \begin{cases} -{\rm div}\ a(x,Du)=k(u^{\gamma-1}-u^{\beta-1}) & x\in \Omega, \\ u=0 & x\in \partial \Omega, \end{cases} \end{eqnarray} where $\Omega$ is a bounded domain in $R^n$ and ${\rm div}\ a(x,Du)$ is a $p(x)$-Laplace type operator with $1<\beta<\gamma<\inf_{x\in \Omega} p(x)$, $p(x)\in(1,2]$. By establish a reversed Hölder inequality, we show that for any suitable $\gamma,\beta$, the weak solution of previous equation has bounded $p(x)$ energy satisfies $|Du|^{p(x)}\in L_{\text{loc}}^{\delta}$ with some $\delta>1$.
Citation: Zhong Tan, Jianfeng Zhou. Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1335-1350. doi: 10.3934/cpaa.2016.15.1335
References:
[1]

S. Chen and Z. Tan, Optimal partial regularity of second order parabolic systems under controllable growth condition,, \emph{J. Funct. Anal.}, 66 (2014), 4908. doi: 10.1016/j.jfa.2014.02.022.

[2]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383. doi: 10.1137/050624522.

[3]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017). doi: 10.1007/978-3-642-18363-8.

[4]

D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent,, \emph{Stud. Math.}, 143 (2000), 267.

[5]

D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent II,, \emph{Math. Nachr.}, 246/247 (2002), 53. doi: 10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T.

[6]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton Univ. Press, (1983). doi: 86b:49003.

[7]

E. Giusti, Direct Methods in the Calculus of Variations,, World Scientific, (2003). doi: 10.1142/9789812795557.

[8]

H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz-Sobolev spaces $W^k_M(\Omega)$,, \emph{Comment. Math. Prace Mat.}, 21 (1980), 315.

[9]

O. Kovácik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 116 (1991), 592.

[10]

K. R. Rajagopal and M. Ruzicka, Mathematical modelling of electrorheological fluids,, \emph{Continuum. Mech. Thermdyn.}, 13 (2001), 59.

[11]

M. Růžička, A note on steady flow of fluids with shear dependent viscosity,, \emph{Nonlin. Anal. Theory, 30 (1997), 3029. doi: 10.1016/S0362-546X(97)00391-X.

[12]

M. Růžička, Electrorheological fluids: Modeling and Mathematical Theory,, Lecture Notes in Math., 1748 (2000). doi: 10.1007/BFb0104029.

[13]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions,, \emph{J. Differential Equations}, 90 (1991), 1. doi: 10.1016/0022-0396(91)90158-6.

[14]

M. Mihăliescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids,, \emph{Proc. R. Soc.}, 462 (2006), 2625. doi: 10.1098/rspa.2005.1633.

show all references

References:
[1]

S. Chen and Z. Tan, Optimal partial regularity of second order parabolic systems under controllable growth condition,, \emph{J. Funct. Anal.}, 66 (2014), 4908. doi: 10.1016/j.jfa.2014.02.022.

[2]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Appl. Math.}, 66 (2006), 1383. doi: 10.1137/050624522.

[3]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2017). doi: 10.1007/978-3-642-18363-8.

[4]

D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent,, \emph{Stud. Math.}, 143 (2000), 267.

[5]

D. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent II,, \emph{Math. Nachr.}, 246/247 (2002), 53. doi: 10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T.

[6]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton Univ. Press, (1983). doi: 86b:49003.

[7]

E. Giusti, Direct Methods in the Calculus of Variations,, World Scientific, (2003). doi: 10.1142/9789812795557.

[8]

H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz-Sobolev spaces $W^k_M(\Omega)$,, \emph{Comment. Math. Prace Mat.}, 21 (1980), 315.

[9]

O. Kovácik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$,, \emph{Czechoslovak Math. J.}, 116 (1991), 592.

[10]

K. R. Rajagopal and M. Ruzicka, Mathematical modelling of electrorheological fluids,, \emph{Continuum. Mech. Thermdyn.}, 13 (2001), 59.

[11]

M. Růžička, A note on steady flow of fluids with shear dependent viscosity,, \emph{Nonlin. Anal. Theory, 30 (1997), 3029. doi: 10.1016/S0362-546X(97)00391-X.

[12]

M. Růžička, Electrorheological fluids: Modeling and Mathematical Theory,, Lecture Notes in Math., 1748 (2000). doi: 10.1007/BFb0104029.

[13]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions,, \emph{J. Differential Equations}, 90 (1991), 1. doi: 10.1016/0022-0396(91)90158-6.

[14]

M. Mihăliescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids,, \emph{Proc. R. Soc.}, 462 (2006), 2625. doi: 10.1098/rspa.2005.1633.

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