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. 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]

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.

[3]

S. Byun and H. So, Weighted estimates for generalized steady Stokes systems in nonsmooth domains, J. Math. Phys., 58 (2017), 023101, 19 pp.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

B. E. J. DahlbergC. E. Kenig and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J., 57 (1988), 795-818.

[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.

[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.

[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.

[14]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010.

[15]

E. B. FabesC. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793.

[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.

[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.

[18]

P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869.

[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.

[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.

[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.

[22]

A. LemenantE. 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.

[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.

[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.

[25]

M. MitreaS. 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.

[26]

M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, 344 (2012), viii+241pp.

[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.

[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.

[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.

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]

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.

[3]

S. Byun and H. So, Weighted estimates for generalized steady Stokes systems in nonsmooth domains, J. Math. Phys., 58 (2017), 023101, 19 pp.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

B. E. J. DahlbergC. E. Kenig and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J., 57 (1988), 795-818.

[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.

[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.

[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.

[14]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2010.

[15]

E. B. FabesC. E. Kenig and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793.

[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.

[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.

[18]

P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869.

[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.

[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.

[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.

[22]

A. LemenantE. 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.

[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.

[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.

[25]

M. MitreaS. 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.

[26]

M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, 344 (2012), viii+241pp.

[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.

[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.

[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.

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