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September  2021, 41(9): 4421-4460. doi: 10.3934/dcds.2021042

Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L tensor coefficient under relaxed ellipticity condition

1. 

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogǎlniceanu Str., 400084 Cluj-Napoca, Romania

2. 

Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, United Kingdom

3. 

Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

*Corresponding author

Received  September 2020 Revised  January 2021 Published  September 2021 Early access  March 2021

The first aim of this paper is to show well-posedness of Dirichlet and transmission problems in bounded and exterior Lipschitz domains in $ {\mathbb R}^n $, $ n\geq 3 $, for the anisotropic Stokes system with $ L_{\infty } $ viscosity tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices with zero matrix trace, with data from standard and weighted Sobolev spaces. To this end we reduce the linear problems to equivalent mixed variational formulations and show that the variational problems are well-posed. Then we use the Leray-Schauder fixed point theorem and establish the existence of a weak solution for nonlinear Dirichlet and transmission problems for the anisotropic Navier-Stokes system in bounded Lipschitz domains in $ {\mathbb R}^3 $, with general (including large) data in Sobolev spaces. For exterior domains in $ {\mathbb R}^3 $, the analysis of the nonlinear Dirichlet and transmission problems in weighted Sobolev spaces relies on the existence result for the Dirichlet problem for the anisotropic Navier-Stokes system in a family of bounded Lipschitz domains. The obtained estimates for pressure in $ {\mathbb R}^3 $ look new also for the classical isotropic case.

Citation: Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L tensor coefficient under relaxed ellipticity condition. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4421-4460. doi: 10.3934/dcds.2021042
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C. AmroucheP. G. Ciarlet and C. Mardare, On a Lemma of Jacques-Louis Lions and its relation to other fundamental results, J. Math. Pures Appl., 104 (2015), 207-226.  doi: 10.1016/j.matpur.2014.11.007.  Google Scholar

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C. AmroucheV. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the $n$-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl., 76 (1997), 55-81.  doi: 10.1016/S0021-7824(97)89945-X.  Google Scholar

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I. Babuška, The finite element method with Lagrangian multipliers, Numer. Math., 20 (1973), 179-192.  doi: 10.1007/BF01436561.  Google Scholar

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M. E. Bogovskiǐ, Solution of some problems of vector analysis related with operators div and grad, in Teoriya Kubaturnyh Formul I Prilozheniya Funktsional'nogo Analiza k Zadacham Matematicheskoi Fiziki. Trudy Seminara S.L.Soboleva, No.1, Siberian brunch of the Academy of Sci. of USSR, Institute of Mathematics, Novosibirsk, (1980), 5–40 (in Russian).  Google Scholar

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K. BrewsterD. MitreaI. Mitrea and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally $(\epsilon, \delta)$-domains and applications to mixed boundary problems, J. Funct. Anal., 266 (2014), 4314-4421.  doi: 10.1016/j.jfa.2014.02.001.  Google Scholar

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O. ChkaduaS. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. Ⅱ. Solution regularity and asymptotics, J. Int. Equ. Appl., 22 (2010), 19-37.  doi: 10.1216/JIE-2010-22-1-19.  Google Scholar

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J. ChoiH. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.  Google Scholar

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J. ChoiH. Dong and D. Kim, Green functions of conormal derivative problems for stationary Stokes system, J. Math. Fluid Mech., 20 (2018), 1745-1769.  doi: 10.1007/s00021-018-0387-0.  Google Scholar

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J. Choi and M. Yang, Fundamental solutions for stationary Stokes systems with measurable coefficients, J. Diff. Equ., 263 (2017), 3854-3893.  doi: 10.1016/j.jde.2017.05.005.  Google Scholar

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M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

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M. Dindoš and M. Mitrea, The stationary Navier-Stokes system in nonsmooth manifolds: The Poisson problem in Lipschitz and $C^1$ domains, Arch. Rational Mech. Anal., 174 (2004), 1-47.  doi: 10.1007/s00205-004-0320-y.  Google Scholar

[20]

B. R. Duffy, Flow of a liquid with an anisotropic viscosity tensor, J. Nonnewton. Fluid Mech., 4 (1978), 177-193.  doi: 10.1016/0377-0257(78)80002-0.  Google Scholar

[21]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[22]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, Second Edition, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[24]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[25]

V. Girault and A. Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal., 114 (1991), 313-333.  doi: 10.1007/BF00376137.  Google Scholar

[26]

B. Hanouzet, Espaces de Sobolev avec poids – application au probléme de Dirichlet dans un demi-espace, Rend. Sere. Mat. Univ. Padova, 46 (1971), 227-272.   Google Scholar

[27]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer-Verlag, Heidelberg 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[28]

M. Kohr, M. Lanza de Cristoforis, S. E. Mikhailov and W. L. Wendland, Integral potential method for transmission problem with Lipschitz interface in ${\mathbb R}^3$ for the Stokes and Darcy-Forchheimer-Brinkman PDE systems,, Z. Angew. Math. Phys., 67 (2016), Art. 116, 30 pp. doi: 10.1007/s00033-016-0696-1.  Google Scholar

[29]

M. KohrM. Lanza de Cristoforis and W. L. Wendland, Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains, Potential Anal., 38 (2013), 1123-1171.  doi: 10.1007/s11118-012-9310-0.  Google Scholar

[30]

M. KohrM. Lanza de Cristoforis and W. L. Wendland, Poisson problems for semilinear Brinkman systems on Lipschitz domains in ${\mathbb R}^3$, Z. Angew. Math. Phys., 66 (2015), 833-864.  doi: 10.1007/s00033-014-0439-0.  Google Scholar

[31]

M. KohrS. E. Mikhailov and W. L. Wendland, Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with $L_\infty $ strongly elliptic coefficient tensor, Complex Var. Elliptic Equ., 65 (2020), 109-140.  doi: 10.1080/17476933.2019.1631293.  Google Scholar

[32]

M. Kohr, S. E. Mikhailov and W. L. Wendland, Layer potential theory for the anisotropic Stokes system with variable $L_\infty$ symmetrically elliptic tensor coefficient, Math. Meth. Appl. Sci., to appear. Google Scholar

[33]

M. Kohr and W. L. Wendland, Variational approach for the Stokes and Navier-Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds, Calc. Var. Partial Differ. Equ., 57 (2018), Paper No. 165, 41 pp. doi: 10.1007/s00526-018-1426-7.  Google Scholar

[34]

M. Kohr and W. L. Wendland, Layer potentials and Poisson problems for the nonsmooth coefficient Brinkman system in Sobolev and Besov spaces, J. Math. Fluid Mech., 20 (2018), 1921-1965.  doi: 10.1007/s00021-018-0394-1.  Google Scholar

[35]

O. A. Ladyzhenskaya and V. A. Solonnikov, Some problems of vector analysis and generalized formulations of boundary value problems for Navier-Stokes equations, Zap. Nauchn. Sem. LOMI. Leningrad. Otdel. Mat. Inst. Steklov, 59 (1976), 81-116.   Google Scholar

[36]

A. L. Mazzucato and V. Nistor, Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks, Arch. Rational Mech. Anal., 195 (2010), 25-73.  doi: 10.1007/s00205-008-0180-y.  Google Scholar

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, UK, 2000.   Google Scholar
[38]

S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342.  doi: 10.1016/j.jmaa.2010.12.027.  Google Scholar

[39]

S. E. Mikhailov, Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains, J. Math. Anal. Appl., 400 (2013), 48-67.  doi: 10.1016/j.jmaa.2012.10.045.  Google Scholar

[40]

S. E. Mikhailov and C. Fresneda-Portillo, Boundary-domain integral equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs, Comm. Pure and Applied Analysis, 18 (2019), 3059-3088.  doi: 10.3934/cpaa.2019137.  Google Scholar

[41]

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

[42]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.  Google Scholar

[43]

E. Otárola and A. J. Salgado, A weighted setting for the stationary Navier-Stokes equations under singular forcing, Appl. Math. Letters, 99 (2020), 105933, 7pp. doi: 10.1016/j.aml.2019.06.004.  Google Scholar

[44]

T. Runst and W. Sickel, Sobolev Spaces of Fractional order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter, Berlin, 1996. doi: 10.1515/9783110812411.  Google Scholar

[45]

F.-J. Sayas and V. Selgas, Variational views of Stokeslets and stresslets, SeMA, 63 (2014), 65-90.  doi: 10.1007/s40324-014-0013-x.  Google Scholar

[46]

G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations, World Scientific, London, 2015.  Google Scholar

[47]

L. Tartar, Topics in Nonlinear Analysis, Publications Mathéatiques d'Orsay, 1978.  Google Scholar

[48]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 2001. doi: 10.1090/chel/343.  Google Scholar

show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Academic Press, 2003.   Google Scholar
[2]

F. Alliot and C. Amrouche, The Stokes problem in ${\mathbb R}^n$: An approach in weighted Sobolev spaces, Math. Models Meth. Appl. Sci., 9 (1999), 723-754.  doi: 10.1142/S0218202599000361.  Google Scholar

[3]

F. Alliot and C. Amrouche, On the regularity and decay of the weak solutions to the steady-state Navier-Stokes equations in exterior domains, in Applied Nonlinear Analysis (eds. A. Sequeira, H. Beirao da Veiga and J. H. Videman), Kluwer Academic, Dordrecht, (1999), 1–18. doi: 10.1007/0-306-47096-9_1.  Google Scholar

[4]

F. Alliot and C. Amrouche, Weak solutions for the exterior Stokes problem in weighted Sobolev spaces, Math. Meth. Appl. Sci., 23 (2000), 575-600.  doi: 10.1002/(SICI)1099-1476(200004)23:6<575::AID-MMA128>3.0.CO;2-4.  Google Scholar

[5]

C. AmroucheP. G. Ciarlet and C. Mardare, On a Lemma of Jacques-Louis Lions and its relation to other fundamental results, J. Math. Pures Appl., 104 (2015), 207-226.  doi: 10.1016/j.matpur.2014.11.007.  Google Scholar

[6]

C. AmroucheV. Girault and J. Giroire, Dirichlet and Neumann exterior problems for the $n$-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl., 76 (1997), 55-81.  doi: 10.1016/S0021-7824(97)89945-X.  Google Scholar

[7]

I. Babuška, The finite element method with Lagrangian multipliers, Numer. Math., 20 (1973), 179-192.  doi: 10.1007/BF01436561.  Google Scholar

[8]

M. E. Bogovskiǐ, Solution of some problems of vector analysis related with operators div and grad, in Teoriya Kubaturnyh Formul I Prilozheniya Funktsional'nogo Analiza k Zadacham Matematicheskoi Fiziki. Trudy Seminara S.L.Soboleva, No.1, Siberian brunch of the Academy of Sci. of USSR, Institute of Mathematics, Novosibirsk, (1980), 5–40 (in Russian).  Google Scholar

[9]

W. Borchers and H. Sohr, The equations ${\rm rot}\, v = g$ and ${\rm div}\, u = f$ with zero boundary condition, Hokkaido Math. J., 19 (1990), 67-87.  doi: 10.14492/hokmj/1381517172.  Google Scholar

[10]

K. BrewsterD. MitreaI. Mitrea and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally $(\epsilon, \delta)$-domains and applications to mixed boundary problems, J. Funct. Anal., 266 (2014), 4314-4421.  doi: 10.1016/j.jfa.2014.02.001.  Google Scholar

[11]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers, R.A.I.R.O. Anal. Numer., 8 (1974), 129-151.   Google Scholar

[12]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Comput. Math., 15, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[13]

O. ChkaduaS. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I. Equivalence and invertibility, J. Int. Equ. Appl., 21 (2009), 499-542.  doi: 10.1216/JIE-2009-21-4-499.  Google Scholar

[14]

O. ChkaduaS. E. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. Ⅱ. Solution regularity and asymptotics, J. Int. Equ. Appl., 22 (2010), 19-37.  doi: 10.1216/JIE-2010-22-1-19.  Google Scholar

[15]

J. ChoiH. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.  Google Scholar

[16]

J. ChoiH. Dong and D. Kim, Green functions of conormal derivative problems for stationary Stokes system, J. Math. Fluid Mech., 20 (2018), 1745-1769.  doi: 10.1007/s00021-018-0387-0.  Google Scholar

[17]

J. Choi and M. Yang, Fundamental solutions for stationary Stokes systems with measurable coefficients, J. Diff. Equ., 263 (2017), 3854-3893.  doi: 10.1016/j.jde.2017.05.005.  Google Scholar

[18]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626.  doi: 10.1137/0519043.  Google Scholar

[19]

M. Dindoš and M. Mitrea, The stationary Navier-Stokes system in nonsmooth manifolds: The Poisson problem in Lipschitz and $C^1$ domains, Arch. Rational Mech. Anal., 174 (2004), 1-47.  doi: 10.1007/s00205-004-0320-y.  Google Scholar

[20]

B. R. Duffy, Flow of a liquid with an anisotropic viscosity tensor, J. Nonnewton. Fluid Mech., 4 (1978), 177-193.  doi: 10.1016/0377-0257(78)80002-0.  Google Scholar

[21]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[22]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, Second Edition, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[24]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[25]

V. Girault and A. Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal., 114 (1991), 313-333.  doi: 10.1007/BF00376137.  Google Scholar

[26]

B. Hanouzet, Espaces de Sobolev avec poids – application au probléme de Dirichlet dans un demi-espace, Rend. Sere. Mat. Univ. Padova, 46 (1971), 227-272.   Google Scholar

[27]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer-Verlag, Heidelberg 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[28]

M. Kohr, M. Lanza de Cristoforis, S. E. Mikhailov and W. L. Wendland, Integral potential method for transmission problem with Lipschitz interface in ${\mathbb R}^3$ for the Stokes and Darcy-Forchheimer-Brinkman PDE systems,, Z. Angew. Math. Phys., 67 (2016), Art. 116, 30 pp. doi: 10.1007/s00033-016-0696-1.  Google Scholar

[29]

M. KohrM. Lanza de Cristoforis and W. L. Wendland, Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains, Potential Anal., 38 (2013), 1123-1171.  doi: 10.1007/s11118-012-9310-0.  Google Scholar

[30]

M. KohrM. Lanza de Cristoforis and W. L. Wendland, Poisson problems for semilinear Brinkman systems on Lipschitz domains in ${\mathbb R}^3$, Z. Angew. Math. Phys., 66 (2015), 833-864.  doi: 10.1007/s00033-014-0439-0.  Google Scholar

[31]

M. KohrS. E. Mikhailov and W. L. Wendland, Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with $L_\infty $ strongly elliptic coefficient tensor, Complex Var. Elliptic Equ., 65 (2020), 109-140.  doi: 10.1080/17476933.2019.1631293.  Google Scholar

[32]

M. Kohr, S. E. Mikhailov and W. L. Wendland, Layer potential theory for the anisotropic Stokes system with variable $L_\infty$ symmetrically elliptic tensor coefficient, Math. Meth. Appl. Sci., to appear. Google Scholar

[33]

M. Kohr and W. L. Wendland, Variational approach for the Stokes and Navier-Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds, Calc. Var. Partial Differ. Equ., 57 (2018), Paper No. 165, 41 pp. doi: 10.1007/s00526-018-1426-7.  Google Scholar

[34]

M. Kohr and W. L. Wendland, Layer potentials and Poisson problems for the nonsmooth coefficient Brinkman system in Sobolev and Besov spaces, J. Math. Fluid Mech., 20 (2018), 1921-1965.  doi: 10.1007/s00021-018-0394-1.  Google Scholar

[35]

O. A. Ladyzhenskaya and V. A. Solonnikov, Some problems of vector analysis and generalized formulations of boundary value problems for Navier-Stokes equations, Zap. Nauchn. Sem. LOMI. Leningrad. Otdel. Mat. Inst. Steklov, 59 (1976), 81-116.   Google Scholar

[36]

A. L. Mazzucato and V. Nistor, Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks, Arch. Rational Mech. Anal., 195 (2010), 25-73.  doi: 10.1007/s00205-008-0180-y.  Google Scholar

[37] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, UK, 2000.   Google Scholar
[38]

S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342.  doi: 10.1016/j.jmaa.2010.12.027.  Google Scholar

[39]

S. E. Mikhailov, Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains, J. Math. Anal. Appl., 400 (2013), 48-67.  doi: 10.1016/j.jmaa.2012.10.045.  Google Scholar

[40]

S. E. Mikhailov and C. Fresneda-Portillo, Boundary-domain integral equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs, Comm. Pure and Applied Analysis, 18 (2019), 3059-3088.  doi: 10.3934/cpaa.2019137.  Google Scholar

[41]

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

[42]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.  Google Scholar

[43]

E. Otárola and A. J. Salgado, A weighted setting for the stationary Navier-Stokes equations under singular forcing, Appl. Math. Letters, 99 (2020), 105933, 7pp. doi: 10.1016/j.aml.2019.06.004.  Google Scholar

[44]

T. Runst and W. Sickel, Sobolev Spaces of Fractional order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter, Berlin, 1996. doi: 10.1515/9783110812411.  Google Scholar

[45]

F.-J. Sayas and V. Selgas, Variational views of Stokeslets and stresslets, SeMA, 63 (2014), 65-90.  doi: 10.1007/s40324-014-0013-x.  Google Scholar

[46]

G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations, World Scientific, London, 2015.  Google Scholar

[47]

L. Tartar, Topics in Nonlinear Analysis, Publications Mathéatiques d'Orsay, 1978.  Google Scholar

[48]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 2001. doi: 10.1090/chel/343.  Google Scholar

Figure 1.  Bounded composite domain $ \Omega_{} = \overline{\Omega^0_+}\cup \Omega^0_- $
Figure 2.  Composite space $ \mathbb R^n = \overline{\Omega^0_+}\cup \Omega^0_- $
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P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691

[4]

Jongkeun Choi, Hongjie Dong, Doyoon Kim. Conormal derivative problems for stationary Stokes system in Sobolev spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2349-2374. doi: 10.3934/dcds.2018097

[5]

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