Fractional Navier-Stokes equations

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October 2018, 23(8): 2967-2988. doi: 10.3934/dcdsb.2017149

Fractional Navier-Stokes equations

Jan W. Cholewa ^, and Tomasz Dlotko

Institute of Mathematics, University of Silesia in Katowice, 40-007 Katowice, Poland

* Corresponding author:Jan W. Cholewa

Received December 2016 Revised February 2017 Published October 2018 Early access April 2017

Fund Project: J.C. partially supported by grant MTM2012-31298 from Ministerio de Economia y Competividad, Spain; T.D. partially supported by NCN grant DEC-2012/05/B/ST1/00546, Poland.

We consider fractional Navier-Stokes equations in a smooth bound-ed domain $Ω\subset\mathbb{R}^N$, $N≥2$. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solution. For the original Navier-Stokes problem we construct next global solution of the Leray-Hopf type satisfying also Duhamel's integral formula. Focusing finally on the 3-D model with zero external force we estimate a time after which the latter solution regularizes to strong solution. We also estimate a time such that if a local strong solution exists until that time, then it exists for ever.

Keywords: Navier-Stokes equations, abstract parabolic equations, initial-boundary value problems for pseudodifferential operators.

Mathematics Subject Classification: Primary: 35Q30, 35K90, 35S11.

Citation: Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

References:

[1]	H. Amann, Linear and Quasilinear Parabolic Problems, Volume Ⅰ, Abstract Linear Theory Birkhaüser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.
[2]	J. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310. doi: 10.1090/S0002-9947-99-02528-3.
[3]	J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Am. Math. Soc., 63 (1977), 370-373. doi: 10.2307/2041821.
[4]	H. Brezis, Analyse Fonctionelle. Théorie et Applications, Masson, Paris, 1983.
[5]	L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[6]	A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. doi: 10.1016/j.jmaa.2005.02.024.
[7]	J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.
[8]	J. W. Cholewa, C. Quesada and A. Rodriguez-Bernal, Nonlinear evolution equations in scales of Banach spaces and applications to PDEs, preprint.
[9]	T. Dlotko, Navier-Stokes equation and its fractional approximations Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9368-y.
[10]	T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in R², J. Differential Equations, 259 (2015), 531-561. doi: 10.1016/j.jde.2015.02.022.
[11]	C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation http://www.claymath.org/sites/default/files/navierstokes.pdf
[12]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.
[13]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[14]	H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.
[15]	D. Fujiwara and H. Morimoto, An L_r-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, 24 (1977), 685-700.
[16]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.
[17]	Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_r spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869.
[18]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[19]	Y. Giga, Domains of fractional powers of the Stokes operator in L_r spaces, Arch. Rational Mech. Anal., 89 (1985), 251-265. doi: 10.1007/BF00276874.
[20]	Y. Giga and T. Miyakawa, Solutions in L_r of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.
[21]	B. Guo, D. Huang, Q. Li and C. Sun, Dynamics for a generalized incompressible Navier-Stokes equations in $\mathbb{R}^2$, Adv. Nonlinear Stud., 16 (2016), 249-272. doi: 10.1515/ans-2015-5018.
[22]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
[23]	A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo, Sec. I, 24 (1977), 303-319.
[24]	T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Math. Univ. Padova, 32 (1962), 243-260.
[25]	H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.
[26]	O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Math. Sci., 115 (2003), 2789-2891.
[27]	O. A. Ladyzhenskaya, On the uniqueness and smoothness of generalized solutions of the Navier-Stokes equations, Zap. Nauchn. Semin. LOMI, (1967), 169-185.
[28]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
[29]	J. -L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris, 1969.
[30]	J. -L. Lions and E. Magenes, Probl}mes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris 1968.
[31]	G. Łukaszewicz and P. Kalita, Navier-Stokes Equations. An Introduction with Applications Advances in Mechanics and Mathematics 34, Springer International Publishing, 2016. doi: 10.1007/978-3-319-27760-8.
[32]	C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators Elsevier, Amsterdam, 2001.
[33]	K. Masuda, Weak solutions of Navier-Stokes equations, Tôhoku Math. Journ., 36 (1984), 623-646. doi: 10.2748/tmj/1178228767.
[34]	J. Mattingly and Y. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equation, Commun. Contemp. Math., 1 (1999), 497-516. doi: 10.1142/S0219199799000183.
[35]	T. Miyakawa, On the initial value problem for the Navier-Stokes equations in $L^p$ spaces, Hiroshima Math. J., 11 (1981), 9-20.
[36]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.
[37]	G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664.
[38]	F. Ribaud, A remark on the uniqueness problem for the weak solutions of Navier-Stokes equations, Ann. Fac. Sci. Toulouse Math., 11 (2002), 225-238. doi: 10.5802/afst.1024.
[39]	J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.
[40]	J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos. , Madison, Wis. , 1962), (1963), 69{98, Univ. of Wisconsin Press, Madison, Wisconsin.
[41]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[42]	P. E. Sobolevskii, On non-stationary equations of hydrodynamics for viscous fluid, Dokl. Akad. Nauk SSSR, 128 (1959), 45-48 (in Russian).
[43]	P. E. Sobolevskii, On equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obsc. , 10 (1961), 297-350 (in Russian); Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl. , 49 (1966), 1{62.
[44]	H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.
[45]	W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543.
[46]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.
[47]	R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43. doi: 10.1016/0022-1236(75)90052-X.
[48]	W. von Wahl, Equations of Navier-Stokes and Abstract Parabolic Equations Vieweg, Braunschweig/Wiesbaden, 1985. doi: 10.1007/978-3-663-13911-9.
[49]	W. von Wahl, Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985 (Eds. A. Favini, E. Obrecht), Springer-Verlag, Berlin, 1223 (1986), 254{266
[50]	F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.
[51]	J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
[52]	H. Wu and J. Fan, Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Math. Lett., 25 (2012), 423-428. doi: 10.1016/j.aml.2011.09.028.

show all references

References:

[1]	H. Amann, Linear and Quasilinear Parabolic Problems, Volume Ⅰ, Abstract Linear Theory Birkhaüser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.
[2]	J. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310. doi: 10.1090/S0002-9947-99-02528-3.
[3]	J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Am. Math. Soc., 63 (1977), 370-373. doi: 10.2307/2041821.
[4]	H. Brezis, Analyse Fonctionelle. Théorie et Applications, Masson, Paris, 1983.
[5]	L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[6]	A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. doi: 10.1016/j.jmaa.2005.02.024.
[7]	J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.
[8]	J. W. Cholewa, C. Quesada and A. Rodriguez-Bernal, Nonlinear evolution equations in scales of Banach spaces and applications to PDEs, preprint.
[9]	T. Dlotko, Navier-Stokes equation and its fractional approximations Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9368-y.
[10]	T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in R², J. Differential Equations, 259 (2015), 531-561. doi: 10.1016/j.jde.2015.02.022.
[11]	C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation http://www.claymath.org/sites/default/files/navierstokes.pdf
[12]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.
[13]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[14]	H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188.
[15]	D. Fujiwara and H. Morimoto, An L_r-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, 24 (1977), 685-700.
[16]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9.
[17]	Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_r spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869.
[18]	Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.
[19]	Y. Giga, Domains of fractional powers of the Stokes operator in L_r spaces, Arch. Rational Mech. Anal., 89 (1985), 251-265. doi: 10.1007/BF00276874.
[20]	Y. Giga and T. Miyakawa, Solutions in L_r of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875.
[21]	B. Guo, D. Huang, Q. Li and C. Sun, Dynamics for a generalized incompressible Navier-Stokes equations in $\mathbb{R}^2$, Adv. Nonlinear Stud., 16 (2016), 249-272. doi: 10.1515/ans-2015-5018.
[22]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
[23]	A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo, Sec. I, 24 (1977), 303-319.
[24]	T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Math. Univ. Padova, 32 (1962), 243-260.
[25]	H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285.
[26]	O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Math. Sci., 115 (2003), 2789-2891.
[27]	O. A. Ladyzhenskaya, On the uniqueness and smoothness of generalized solutions of the Navier-Stokes equations, Zap. Nauchn. Semin. LOMI, (1967), 169-185.
[28]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
[29]	J. -L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris, 1969.
[30]	J. -L. Lions and E. Magenes, Probl}mes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris 1968.
[31]	G. Łukaszewicz and P. Kalita, Navier-Stokes Equations. An Introduction with Applications Advances in Mechanics and Mathematics 34, Springer International Publishing, 2016. doi: 10.1007/978-3-319-27760-8.
[32]	C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators Elsevier, Amsterdam, 2001.
[33]	K. Masuda, Weak solutions of Navier-Stokes equations, Tôhoku Math. Journ., 36 (1984), 623-646. doi: 10.2748/tmj/1178228767.
[34]	J. Mattingly and Y. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equation, Commun. Contemp. Math., 1 (1999), 497-516. doi: 10.1142/S0219199799000183.
[35]	T. Miyakawa, On the initial value problem for the Navier-Stokes equations in $L^p$ spaces, Hiroshima Math. J., 11 (1981), 9-20.
[36]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.
[37]	G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664.
[38]	F. Ribaud, A remark on the uniqueness problem for the weak solutions of Navier-Stokes equations, Ann. Fac. Sci. Toulouse Math., 11 (2002), 225-238. doi: 10.5802/afst.1024.
[39]	J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.
[40]	J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos. , Madison, Wis. , 1962), (1963), 69{98, Univ. of Wisconsin Press, Madison, Wisconsin.
[41]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.
[42]	P. E. Sobolevskii, On non-stationary equations of hydrodynamics for viscous fluid, Dokl. Akad. Nauk SSSR, 128 (1959), 45-48 (in Russian).
[43]	P. E. Sobolevskii, On equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obsc. , 10 (1961), 297-350 (in Russian); Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl. , 49 (1966), 1{62.
[44]	H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2.
[45]	W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543.
[46]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.
[47]	R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43. doi: 10.1016/0022-1236(75)90052-X.
[48]	W. von Wahl, Equations of Navier-Stokes and Abstract Parabolic Equations Vieweg, Braunschweig/Wiesbaden, 1985. doi: 10.1007/978-3-663-13911-9.
[49]	W. von Wahl, Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985 (Eds. A. Favini, E. Obrecht), Springer-Verlag, Berlin, 1223 (1986), 254{266
[50]	F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539.
[51]	J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
[52]	H. Wu and J. Fan, Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Math. Lett., 25 (2012), 423-428. doi: 10.1016/j.aml.2011.09.028.

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