November  2019, 18(6): 2879-2903. doi: 10.3934/cpaa.2019129

Variable lorentz estimate for stationary stokes system with partially BMO coefficients

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

* Corresponding author

Received  December 2017 Revised  December 2018 Published  May 2019

Fund Project: This research was supported by NSFC grant 11371050

We prove a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solution pair (u,P ) to the Dirichlet problem of stationary Stokes system. It is mainly assumed that the leading coefficients are merely measurable in one spatial variable and have sufficiently small bounded mean oscillation (BMO) seminorm in the other variables, the boundary of underlying domain is Reifenberg flat, and the variable exponents p(x) satisfy the so-called log-Hölder continuity.

Citation: Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129
References:
[1]

G. AcostaR. G. Durán and M. A. Muschietti, Solutions of the divergence operator on John domains, Adv. Math., 206 (2006), 373-401. doi: 10.1016/j.aim.2005.09.004. Google Scholar

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. doi: 10.1515/crll.2005.2005.584.117. Google Scholar

[3]

E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320. doi: 10.1215/S0012-7094-07-13623-8. Google Scholar

[4]

K. Adimurthil and N. C. Phuc, Global Lorentz and Lorentz-Morrey estimates below the natural exponent for quasilinear equations, Calc. Var., 54 (2015), 3107-3139. doi: 10.1007/s00526-015-0895-1. Google Scholar

[5]

P. Baroni, Lorentz estimates for degenerate and singular evolutionary systems, J. Differential Equations, 255 (2013), 2927-2951. doi: 10.1016/j.jde.2013.07.024. Google Scholar

[6]

P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188. doi: 10.1016/j.na.2013.11.004. Google Scholar

[7]

D. Breit, Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials, Comment. Math. Univ. Carolin., 54 (2013), 493-508. Google Scholar

[8]

S. S. ByunJ. Ok and L. H. Wang, W1, p(x)-Regularity for elliptic equations with measurable coefficients in nonsmooth domains, Commun. Math. Phys., 329 (2014), 937-958. doi: 10.1007/s00220-014-1962-8. Google Scholar

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S. S. Byun and L. H. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310. doi: 10.1002/cpa.20037. Google Scholar

[10]

S. S. Byun and H. So, Weighted estimates for generalized steady Stokes systems in nonsmooth domains, J. Math. Phys., 58 (2017), 023101. doi: 10.1063/1.4976501. Google Scholar

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S. S. ByunY. Jang and H. So, Calderón-Zygmund estimate for homogenization of steady state Stokes systems in nonsmooth domains, J. Dyn. Diff. Equat., 30 (2018), 1945-1966. doi: 10.1007/s10884-017-9638-7. Google Scholar

[12]

L. A. Caffarelli and I. Peral, On W1,p estimates for elliptic equations in divergence form, Commun. Pure Appl. Math., 51 (1998), 1-21. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N. Google Scholar

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J. Choi and H. J. Dong, Gradient estimates for Stokes systems with Dini mean oscillation coefficients, J. Differential Equations, 266 (2019), 4451-4509. doi: 10.1016/j.jde.2018.10.001. Google Scholar

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[15]

J. Choi and K. Lee, The green function for the Stokes system with measurable coefficients, Commun. Pure Appl. Anal., 16 (2017), 1989-2022. doi: 10.3934/cpaa.2017098. Google Scholar

[16]

M. Costabel and M. Dauge, On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne, Arch. Ration. Mech. Anal., 217 (2015), 873-898. doi: 10.1007/s00205-015-0845-2. Google Scholar

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L. DieningD. Lengeler and M. Růžička, The stokes and poisson problem in variable exponent spaces, Complex Variables and Elliptic Equations, 56 (2011), 789-811. doi: 10.1080/17476933.2010.504843. Google Scholar

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H. J. Dong and D. Kim, Lq-estimates for stationary Stokes system with coefficients measurable in one direction, Bull. Math. Sci., 2018. doi: 10.1007/s13373-018-0120-6. Google Scholar

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H. J. Dong and D. Kim, Weighted Lq-estimates for stationary Stokes system with partially BMO coefficients, J. Differential Equations, 264 (2018), 4603-4649. doi: 10.1016/j.jde.2017.12.011. Google Scholar

[20]

G. P. GaldiC. G. Simader and H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl., 167 (1994), 147-163. doi: 10.1007/BF01760332. Google Scholar

[21]

S. Gu and Z. W. Shen, Homogenization of Stokes systems and uniform regularity estimates, SIAM J. Math. Anal., 47 (2015), 4025-4057. doi: 10.1137/151004033. Google Scholar

[22] M. Giaquinta, Multiple Integral in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, 1983.
[23]

F. J. HuD. S. Li and L. H. Wang, A new proof of Lp estimates of Stokes equations, J. Math. Anal. Appl., 420 (2014), 1251-1264. doi: 10.1016/j.jmaa.2014.06.039. Google Scholar

[24]

C. E. Kenig and T. Toro, Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551. doi: 10.1215/S0012-7094-97-08717-2. Google Scholar

[25]

D. Kim and N. V. Krylov, Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others, SIAM J. Math. Anal., 32 (2007), 489-506. doi: 10.1137/050646913. Google Scholar

[26]

N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Commun. Partial Differential Equations, 32 (2007), 453-475. doi: 10.1080/03605300600781626. Google Scholar

[27]

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. akad. nauk Ssr Ser. mat, 26 (1980), 345-361. doi: 10.1007/s11118-007-9042-8. Google Scholar

[28]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Rational Mech. Anal., 203 (2012), 189-216. doi: 10.1007/s00205-011-0446-7. Google Scholar

[29]

G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 6 (2007), 195-261. Google Scholar

[30]

G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486. doi: 10.4171/JEMS/258. Google Scholar

[31]

M. V. Safonov, Harnack's inequality for elliptic equations and Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 96 (1980), 272-287. Google Scholar

[32]

H. Tian and S. Z. Zheng, Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients, Bound. Value Probl, 2017 (2017), 128. doi: 10.1186/s13661-017-0859-9. Google Scholar

[33]

H. Tian and S. Z. Zheng, Uniformly nondegenerate elliptic equations with partially BMO coefficients in nonsmooth domains, Nonlinear Anal., 156 (2017), 90-110. doi: 10.1016/j.na.2017.02.013. Google Scholar

[34]

C. Zhang and S. L. Zhou, Global weighted estimates for quasilinear elliptic equations with non-standard growth, J. Funct. Anal., 267 (2014), 605-642. doi: 10.1016/j.jfa.2014.03.022. Google Scholar

[35]

J. J. Zhang and S. Z. Zheng, Lorentz estimates for fully nonlinear parabolic and elliptic equations, Nonlinear Anal., 148 (2017), 106-125. doi: 10.1016/j.na.2016.09.012. Google Scholar

[36]

J. J. Zhang and S. Z. Zheng, Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients, Commun. Pure Appl. Anal., 16 (2017), 899-914. doi: 10.3934/cpaa.2017043. Google Scholar

show all references

References:
[1]

G. AcostaR. G. Durán and M. A. Muschietti, Solutions of the divergence operator on John domains, Adv. Math., 206 (2006), 373-401. doi: 10.1016/j.aim.2005.09.004. Google Scholar

[2]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. Reine Angew. Math., 584 (2005), 117-148. doi: 10.1515/crll.2005.2005.584.117. Google Scholar

[3]

E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320. doi: 10.1215/S0012-7094-07-13623-8. Google Scholar

[4]

K. Adimurthil and N. C. Phuc, Global Lorentz and Lorentz-Morrey estimates below the natural exponent for quasilinear equations, Calc. Var., 54 (2015), 3107-3139. doi: 10.1007/s00526-015-0895-1. Google Scholar

[5]

P. Baroni, Lorentz estimates for degenerate and singular evolutionary systems, J. Differential Equations, 255 (2013), 2927-2951. doi: 10.1016/j.jde.2013.07.024. Google Scholar

[6]

P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188. doi: 10.1016/j.na.2013.11.004. Google Scholar

[7]

D. Breit, Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials, Comment. Math. Univ. Carolin., 54 (2013), 493-508. Google Scholar

[8]

S. S. ByunJ. Ok and L. H. Wang, W1, p(x)-Regularity for elliptic equations with measurable coefficients in nonsmooth domains, Commun. Math. Phys., 329 (2014), 937-958. doi: 10.1007/s00220-014-1962-8. Google Scholar

[9]

S. S. Byun and L. H. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310. doi: 10.1002/cpa.20037. Google Scholar

[10]

S. S. Byun and H. So, Weighted estimates for generalized steady Stokes systems in nonsmooth domains, J. Math. Phys., 58 (2017), 023101. doi: 10.1063/1.4976501. Google Scholar

[11]

S. S. ByunY. Jang and H. So, Calderón-Zygmund estimate for homogenization of steady state Stokes systems in nonsmooth domains, J. Dyn. Diff. Equat., 30 (2018), 1945-1966. doi: 10.1007/s10884-017-9638-7. Google Scholar

[12]

L. A. Caffarelli and I. Peral, On W1,p estimates for elliptic equations in divergence form, Commun. Pure Appl. Math., 51 (1998), 1-21. doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N. Google Scholar

[13]

J. Choi and H. J. Dong, Gradient estimates for Stokes systems with Dini mean oscillation coefficients, J. Differential Equations, 266 (2019), 4451-4509. doi: 10.1016/j.jde.2018.10.001. Google Scholar

[14]

J. ChoiH. J. Dong and D. Kim, Conormal derivative problem for the stationary Stokes system in Sobolev spaces, Discrete and Continuous Dynamical Systems-A, 38 (2018), 2349-2374. doi: 10.3934/dcds.2018097. Google Scholar

[15]

J. Choi and K. Lee, The green function for the Stokes system with measurable coefficients, Commun. Pure Appl. Anal., 16 (2017), 1989-2022. doi: 10.3934/cpaa.2017098. Google Scholar

[16]

M. Costabel and M. Dauge, On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne, Arch. Ration. Mech. Anal., 217 (2015), 873-898. doi: 10.1007/s00205-015-0845-2. Google Scholar

[17]

L. DieningD. Lengeler and M. Růžička, The stokes and poisson problem in variable exponent spaces, Complex Variables and Elliptic Equations, 56 (2011), 789-811. doi: 10.1080/17476933.2010.504843. Google Scholar

[18]

H. J. Dong and D. Kim, Lq-estimates for stationary Stokes system with coefficients measurable in one direction, Bull. Math. Sci., 2018. doi: 10.1007/s13373-018-0120-6. Google Scholar

[19]

H. J. Dong and D. Kim, Weighted Lq-estimates for stationary Stokes system with partially BMO coefficients, J. Differential Equations, 264 (2018), 4603-4649. doi: 10.1016/j.jde.2017.12.011. Google Scholar

[20]

G. P. GaldiC. G. Simader and H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl., 167 (1994), 147-163. doi: 10.1007/BF01760332. Google Scholar

[21]

S. Gu and Z. W. Shen, Homogenization of Stokes systems and uniform regularity estimates, SIAM J. Math. Anal., 47 (2015), 4025-4057. doi: 10.1137/151004033. Google Scholar

[22] M. Giaquinta, Multiple Integral in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, 1983.
[23]

F. J. HuD. S. Li and L. H. Wang, A new proof of Lp estimates of Stokes equations, J. Math. Anal. Appl., 420 (2014), 1251-1264. doi: 10.1016/j.jmaa.2014.06.039. Google Scholar

[24]

C. E. Kenig and T. Toro, Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551. doi: 10.1215/S0012-7094-97-08717-2. Google Scholar

[25]

D. Kim and N. V. Krylov, Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others, SIAM J. Math. Anal., 32 (2007), 489-506. doi: 10.1137/050646913. Google Scholar

[26]

N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Commun. Partial Differential Equations, 32 (2007), 453-475. doi: 10.1080/03605300600781626. Google Scholar

[27]

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. akad. nauk Ssr Ser. mat, 26 (1980), 345-361. doi: 10.1007/s11118-007-9042-8. Google Scholar

[28]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Rational Mech. Anal., 203 (2012), 189-216. doi: 10.1007/s00205-011-0446-7. Google Scholar

[29]

G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 6 (2007), 195-261. Google Scholar

[30]

G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459-486. doi: 10.4171/JEMS/258. Google Scholar

[31]

M. V. Safonov, Harnack's inequality for elliptic equations and Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 96 (1980), 272-287. Google Scholar

[32]

H. Tian and S. Z. Zheng, Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients, Bound. Value Probl, 2017 (2017), 128. doi: 10.1186/s13661-017-0859-9. Google Scholar

[33]

H. Tian and S. Z. Zheng, Uniformly nondegenerate elliptic equations with partially BMO coefficients in nonsmooth domains, Nonlinear Anal., 156 (2017), 90-110. doi: 10.1016/j.na.2017.02.013. Google Scholar

[34]

C. Zhang and S. L. Zhou, Global weighted estimates for quasilinear elliptic equations with non-standard growth, J. Funct. Anal., 267 (2014), 605-642. doi: 10.1016/j.jfa.2014.03.022. Google Scholar

[35]

J. J. Zhang and S. Z. Zheng, Lorentz estimates for fully nonlinear parabolic and elliptic equations, Nonlinear Anal., 148 (2017), 106-125. doi: 10.1016/j.na.2016.09.012. Google Scholar

[36]

J. J. Zhang and S. Z. Zheng, Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients, Commun. Pure Appl. Anal., 16 (2017), 899-914. doi: 10.3934/cpaa.2017043. Google Scholar

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