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July  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.  Google Scholar

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

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

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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.   Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.   Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

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