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Positive solutions to the unstirred chemostat model with Crowley-Martin functional response
Fractional Navier-Stokes equations
Institute of Mathematics, University of Silesia in Katowice, 40-007 Katowice, 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.
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. Google Scholar |
[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 Google Scholar |
[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 Lr-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 Lr 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 Lr spaces, Arch. Rational Mech. Anal., 89 (1985), 251-265.
doi: 10.1007/BF00276874. |
[20] |
Y. Giga and T. Miyakawa,
Solutions in Lr 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. Google Scholar |
[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. Google Scholar |
[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 Google Scholar |
[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 Lr-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 Lr 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 Lr spaces, Arch. Rational Mech. Anal., 89 (1985), 251-265.
doi: 10.1007/BF00276874. |
[20] |
Y. Giga and T. Miyakawa,
Solutions in Lr 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. Google Scholar |
[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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