# American Institute of Mathematical Sciences

November  2017, 16(6): 1989-2022. doi: 10.3934/cpaa.2017098

## The Green function for the Stokes system with measurable coefficients

 1 Department of Mathematics, Korea University, Seoul 02841, Republic of Korea 2 Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea 3 Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea

* Corresponding author

Received  September 2016 Revised  April 2017 Published  July 2017

Fund Project: J. Choi was supported by BK21 PLUS SNU Mathematical Sciences Division. Ki-Ahm Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2015R1A4A1041675).

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $Ω\subset \mathbb{R}^n$, $n≥ 3$. We construct the Green function in $Ω$ under the condition $(\bf{A1})$ that weak solutions of the system enjoy interior Hölder continuity. We also prove that $(\bf{A1})$ holds, for example, when the coefficients are $\mathrm{VMO}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(\bf{A2})$ that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori $L^q$-estimates for Stokes systems with $\mathrm{BMO}$ coefficients on a Reifenberg domain, we verify that $(\bf{A2})$ is satisfied when the coefficients are $\mathrm{VMO}$ and $Ω$ is a bounded $C^1$ domain.

Citation: Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
##### References:
 [1] Gabriel Acosta, Ricardo G. Durán and Marí a 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] Hiroaki Aikawa, Martin Boundary and Boundary {H}arnack Principle for Non-smooth Domains Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc. , Providence, RI, 2005. doi: 10.1090/trans2/215/03. [3] Sun-Sig Byun and Lihe Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037. [4] TongKeun Chang and Hi Jun Choe, Estimates of the Green's functions for the elasto-static equations and Stokes equations in a three dimensional Lipschitz domain, Potential Anal., 30 (2009), 85-99.  doi: 10.1007/s11118-008-9107-3. [5] Jongkeun Choi and Seick Kim, Neumann functions for second order elliptic systems with measurable coefficients, Trans. Amer. Math. Soc., 365 (2013), 6283-6307.  doi: 10.1090/S0002-9947-2013-05886-2. [6] Georg Dolzmann and Stefan. Müller, Estimates for Green's matrices of elliptic systems by $L^p$ theory, Manuscripta Math., 88 (1995), 261-273.  doi: 10.1007/BF02567822. [7] Hongjie Dong and Doyoon Kim, $L_q$-estimates for stationary Stokes system with coefficients measurable in one direction, arXiv: 1604.02690v2. [8] Hongjie Dong and Doyoon Kim, Higher order elliptic and parabolic systems with variably partially {BMO} coefficients in regular and irregular domains, J. Funct. Anal., 261 (2011), 3279-3327.  doi: 10.1016/j.jfa.2011.08.001. [9] Hongjie Dong and Doyoon Kim, The Conormal Derivative Problem for Higher Order Elliptic Systems with Irregular Coefficients volume 581 of Contemp. Math. Amer. Math. Soc. , Providence, RI, 2012. doi: 10.1090/conm/581/11534. [10] Martin Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients, Z. Anal. Anwendungen, 5 (1986), 507-531.  doi: 10.4171/ZAA/219. [11] Giovanni Paolo Galdi, Christian G. Simader and Hermann Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl., 167 (1994), 147-163.  doi: 10.1007/BF01760332. [12] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 of Annals of Mathematics Studies Princeton University Press, Princeton, NJ, 1983. [13] Mariano Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993. [14] Mariano Giaquinta and Giuseppe Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1992), 173-214. [15] Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342.  doi: 10.1007/BF01166225. [16] Steve Hofmann and Seick Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta math., 124 (2007), 139-172.  doi: 10.1007/s00229-007-0107-1. [17] Kyungkeun Kang, On regularity of stationary Stokes and Navier-Stokes equations near boundary, J. Math. Fluid Mech., 6 (2004), 78-101.  doi: 10.1007/s00021-003-0084-3. [18] Kyungkeun Kang and Seick Kim, Global pointwise estimates for Green's matrix of second order elliptic systems, J. Differential Equations, 249 (2010), 2643-2662.  doi: 10.1016/j.jde.2010.05.017. [19] Carlos E. Kenig and Tatiana Toro, Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551.  doi: 10.1215/S0012-7094-97-08717-2. [20] Nicolai V. Krylov and Mikhail V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175. [21] Walter Littman, Guido Stampacchia and Hans F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 43-77. [22] Vladimir Gilelevich Maz'ya and Jürgen Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr., 280 (2007), 751-793.  doi: 10.1002/mana.200610513. [23] Dorina Mitrea and Irina Mitrea, On the regularity of Green functions in Lipschitz domains, Comm. Partial Differential Equations, 36 (2011), 304-327.  doi: 10.1080/03605302.2010.489629. [24] Marius Mitrea and Matthew Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque 344 (2012), ⅷ+241. [25] Katharine A. Ott, Seick Kim and Russell Murray Brown, The Green function for the mixed problem for the linear Stokes system in domains in the plane, Math. Nachr., 288 (2015), 452-464.  doi: 10.1002/mana.201300281. [26] Mikhail 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.

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##### References:
 [1] Gabriel Acosta, Ricardo G. Durán and Marí a 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] Hiroaki Aikawa, Martin Boundary and Boundary {H}arnack Principle for Non-smooth Domains Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc. , Providence, RI, 2005. doi: 10.1090/trans2/215/03. [3] Sun-Sig Byun and Lihe Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037. [4] TongKeun Chang and Hi Jun Choe, Estimates of the Green's functions for the elasto-static equations and Stokes equations in a three dimensional Lipschitz domain, Potential Anal., 30 (2009), 85-99.  doi: 10.1007/s11118-008-9107-3. [5] Jongkeun Choi and Seick Kim, Neumann functions for second order elliptic systems with measurable coefficients, Trans. Amer. Math. Soc., 365 (2013), 6283-6307.  doi: 10.1090/S0002-9947-2013-05886-2. [6] Georg Dolzmann and Stefan. Müller, Estimates for Green's matrices of elliptic systems by $L^p$ theory, Manuscripta Math., 88 (1995), 261-273.  doi: 10.1007/BF02567822. [7] Hongjie Dong and Doyoon Kim, $L_q$-estimates for stationary Stokes system with coefficients measurable in one direction, arXiv: 1604.02690v2. [8] Hongjie Dong and Doyoon Kim, Higher order elliptic and parabolic systems with variably partially {BMO} coefficients in regular and irregular domains, J. Funct. Anal., 261 (2011), 3279-3327.  doi: 10.1016/j.jfa.2011.08.001. [9] Hongjie Dong and Doyoon Kim, The Conormal Derivative Problem for Higher Order Elliptic Systems with Irregular Coefficients volume 581 of Contemp. Math. Amer. Math. Soc. , Providence, RI, 2012. doi: 10.1090/conm/581/11534. [10] Martin Fuchs, The Green matrix for strongly elliptic systems of second order with continuous coefficients, Z. Anal. Anwendungen, 5 (1986), 507-531.  doi: 10.4171/ZAA/219. [11] Giovanni Paolo Galdi, Christian G. Simader and Hermann Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl., 167 (1994), 147-163.  doi: 10.1007/BF01760332. [12] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, volume 105 of Annals of Mathematics Studies Princeton University Press, Princeton, NJ, 1983. [13] Mariano Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993. [14] Mariano Giaquinta and Giuseppe Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1992), 173-214. [15] Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342.  doi: 10.1007/BF01166225. [16] Steve Hofmann and Seick Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta math., 124 (2007), 139-172.  doi: 10.1007/s00229-007-0107-1. [17] Kyungkeun Kang, On regularity of stationary Stokes and Navier-Stokes equations near boundary, J. Math. Fluid Mech., 6 (2004), 78-101.  doi: 10.1007/s00021-003-0084-3. [18] Kyungkeun Kang and Seick Kim, Global pointwise estimates for Green's matrix of second order elliptic systems, J. Differential Equations, 249 (2010), 2643-2662.  doi: 10.1016/j.jde.2010.05.017. [19] Carlos E. Kenig and Tatiana Toro, Harmonic measure on locally flat domains, Duke Math. J., 87 (1997), 509-551.  doi: 10.1215/S0012-7094-97-08717-2. [20] Nicolai V. Krylov and Mikhail V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175. [21] Walter Littman, Guido Stampacchia and Hans F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 43-77. [22] Vladimir Gilelevich Maz'ya and Jürgen Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr., 280 (2007), 751-793.  doi: 10.1002/mana.200610513. [23] Dorina Mitrea and Irina Mitrea, On the regularity of Green functions in Lipschitz domains, Comm. Partial Differential Equations, 36 (2011), 304-327.  doi: 10.1080/03605302.2010.489629. [24] Marius Mitrea and Matthew Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque 344 (2012), ⅷ+241. [25] Katharine A. Ott, Seick Kim and Russell Murray Brown, The Green function for the mixed problem for the linear Stokes system in domains in the plane, Math. Nachr., 288 (2015), 452-464.  doi: 10.1002/mana.201300281. [26] Mikhail 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.
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