# American Institute of Mathematical Sciences

May  2018, 38(5): 2349-2374. doi: 10.3934/dcds.2018097

## Conormal derivative problems for stationary Stokes system in Sobolev spaces

 1 Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea 2 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA

* Corresponding author: Doyoon Kim

Received  August 2017 Published  March 2018

Fund Project: J. Choi was supported by a Korea University Grant. H. Dong was partially supported by the NSF under agreement DMS-1600593. D. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03934369)

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

Citation: Jongkeun Choi, Hongjie Dong, Doyoon Kim. Conormal derivative problems for stationary Stokes system in Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2349-2374. doi: 10.3934/dcds.2018097
##### References:
 [1] G. Acosta, R. 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] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, volume 183 of Applied Mathematical Sciences, Springer, New York, 2013. Google Scholar [3] S. Byun and H. So, Weighted estimates for generalized steady Stokes systems in nonsmooth domains, J. Math. Phys., 58 (2017), 023101, 19 pp. Google Scholar [4] S. Byun and L. Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc.(3), 90 (2005), 245-272. doi: 10.1112/S0024611504014960. Google Scholar [5] S. Byun and L. Wang, $L^p$ estimates for parabolic equations in Reifenberg domains, J. Funct. Anal., 223 (2005), 44-85. doi: 10.1016/j.jfa.2004.10.014. Google Scholar [6] S. Byun and L. Wang, $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in Reifenberg domains, Discrete Contin. Dyn. Syst., 20 (2008), 617-637. Google Scholar [7] J. Choi and S. 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. Google Scholar [8] 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 [9] J. Choi and M. Yang, Fundamental solutions for stationary Stokes systems with measurable coefficients, J. Differential Equations, 263 (2017), 3854-3893. doi: 10.1016/j.jde.2017.05.005. Google Scholar [10] B. E. J. Dahlberg, C. E. Kenig and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J., 57 (1988), 795-818. Google Scholar [11] H. Dong and D. Kim, $L_q$ -estimates for stationary Stokes system with coefficients measurable in one direction, Bull. Math. Sci. in press, arXiv: 1604.02690.Google Scholar [12] H. Dong and D. Kim, Weighted $L_q$ -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 [13] H. Dong and D. Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal., 43 (2011), 1075-1098. doi: 10.1137/100794614. Google Scholar [14] L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. Google Scholar [15] E. B. Fabes, C. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793. Google Scholar [16] M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214. Google Scholar [17] M. 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. Google Scholar [18] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869. Google Scholar [19] D. Kim, Global regularity of solutions to quasilinear conormal derivative problem with controlled growth, J. Korean Math. Soc., 49 (2012), 1273-1299. doi: 10.4134/JKMS.2012.49.6.1273. Google Scholar [20] V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, volume 85 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2001. Google Scholar [21] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175,239. Google Scholar [22] A. Lemenant, E. Milakis and L. V. Spinolo, On the extension property of Reifenberg-flat domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 51-71. doi: 10.5186/aasfm.2014.3907. Google Scholar [23] V. Maz'ya and J. 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. Google Scholar [24] V. G. Maz'ya and J. Rossmann, Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains, Math. Methods Appl. Sci., 29 (2006), 965-1017. doi: 10.1002/mma.695. Google Scholar [25] M. Mitrea, S. Monniaux and M. Wright, The Stokes operator with Neumann boundary conditions in Lipschitz domains, J. Math. Sci. (N.Y.), 176 (2011), 409-457. doi: 10.1007/s10958-011-0400-0. Google Scholar [26] M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, 344 (2012), viii+241pp. Google Scholar [27] 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), 272-287,312. Google Scholar [28] Y. Shibata and S. Shimizu, On a resolvent estimate for the Stokes system with Neumann boundary condition, Differential Integral Equations, 16 (2003), 385-426. Google Scholar [29] Y. Shibata and S. Shimizu, On the Stokes equation with Neumann boundary condition, Regularity and other aspects of the Navier-Stokes equations, volume 70 of Banach Center Publ., 239–250, Polish Acad. Sci. Inst. Math., Warsaw, 2005. Google Scholar

show all references

##### References:
 [1] G. Acosta, R. 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] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, volume 183 of Applied Mathematical Sciences, Springer, New York, 2013. Google Scholar [3] S. Byun and H. So, Weighted estimates for generalized steady Stokes systems in nonsmooth domains, J. Math. Phys., 58 (2017), 023101, 19 pp. Google Scholar [4] S. Byun and L. Wang, The conormal derivative problem for elliptic equations with BMO coefficients on Reifenberg flat domains, Proc. London Math. Soc.(3), 90 (2005), 245-272. doi: 10.1112/S0024611504014960. Google Scholar [5] S. Byun and L. Wang, $L^p$ estimates for parabolic equations in Reifenberg domains, J. Funct. Anal., 223 (2005), 44-85. doi: 10.1016/j.jfa.2004.10.014. Google Scholar [6] S. Byun and L. Wang, $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in Reifenberg domains, Discrete Contin. Dyn. Syst., 20 (2008), 617-637. Google Scholar [7] J. Choi and S. 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. Google Scholar [8] 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 [9] J. Choi and M. Yang, Fundamental solutions for stationary Stokes systems with measurable coefficients, J. Differential Equations, 263 (2017), 3854-3893. doi: 10.1016/j.jde.2017.05.005. Google Scholar [10] B. E. J. Dahlberg, C. E. Kenig and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J., 57 (1988), 795-818. Google Scholar [11] H. Dong and D. Kim, $L_q$ -estimates for stationary Stokes system with coefficients measurable in one direction, Bull. Math. Sci. in press, arXiv: 1604.02690.Google Scholar [12] H. Dong and D. Kim, Weighted $L_q$ -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 [13] H. Dong and D. Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal., 43 (2011), 1075-1098. doi: 10.1137/100794614. Google Scholar [14] L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010. Google Scholar [15] E. B. Fabes, C. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793. Google Scholar [16] M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214. Google Scholar [17] M. 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. Google Scholar [18] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869. Google Scholar [19] D. Kim, Global regularity of solutions to quasilinear conormal derivative problem with controlled growth, J. Korean Math. Soc., 49 (2012), 1273-1299. doi: 10.4134/JKMS.2012.49.6.1273. Google Scholar [20] V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, volume 85 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2001. Google Scholar [21] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175,239. Google Scholar [22] A. Lemenant, E. Milakis and L. V. Spinolo, On the extension property of Reifenberg-flat domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 51-71. doi: 10.5186/aasfm.2014.3907. Google Scholar [23] V. Maz'ya and J. 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. Google Scholar [24] V. G. Maz'ya and J. Rossmann, Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains, Math. Methods Appl. Sci., 29 (2006), 965-1017. doi: 10.1002/mma.695. Google Scholar [25] M. Mitrea, S. Monniaux and M. Wright, The Stokes operator with Neumann boundary conditions in Lipschitz domains, J. Math. Sci. (N.Y.), 176 (2011), 409-457. doi: 10.1007/s10958-011-0400-0. Google Scholar [26] M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, 344 (2012), viii+241pp. Google Scholar [27] 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), 272-287,312. Google Scholar [28] Y. Shibata and S. Shimizu, On a resolvent estimate for the Stokes system with Neumann boundary condition, Differential Integral Equations, 16 (2003), 385-426. Google Scholar [29] Y. Shibata and S. Shimizu, On the Stokes equation with Neumann boundary condition, Regularity and other aspects of the Navier-Stokes equations, volume 70 of Banach Center Publ., 239–250, Polish Acad. Sci. Inst. Math., Warsaw, 2005. Google Scholar
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