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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[30]

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

[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. 

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[30]

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

[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. 

[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.

[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.

[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.

[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.

[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.

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