-
Previous Article
Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations
- DCDS-B Home
- This Issue
-
Next Article
An SICR rumor spreading model in heterogeneous networks
Critical and super-critical abstract parabolic equations
1. | Institute of Mathematics, University of Silesia, Katowice, Poland |
2. | School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China |
Our purpose is to formulate an abstract result, motivated by the recent paper [
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
![]() ![]() |
[2] |
H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995.
doi: 10.1007/978-3-0348-9221-6. |
[3] |
J. M. 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: 10.2307/118154. |
[4] |
J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404.![]() ![]() ![]() |
[5] |
J. W. Cholewa and T. Dlotko,
Fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst. Series B, 23 (2018), 2967-2988.
doi: 10.3934/dcdsb.2017149. |
[6] |
A. Córdoba and D. Córdoba,
A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528.
doi: 10.1007/s00220-004-1055-1. |
[7] |
A. Córdoba and D. Córdoba,
A pointwise estimate for fractionary derivatives with applications to partial differential equations, Proc. Natl. Acad. Sci. USA, 100 (2003), 15316-15317.
doi: 10.1073/pnas.2036515100. |
[8] |
T. Dlotko,
Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018), 99-128.
doi: 10.1007/s00245-016-9368-y. |
[9] |
T. Dlotko, M. B. Kania and C. Y. Sun,
Quasi-geostrophic equation in $ \mathbb{R}^2$, J. Differential Equations, 259 (2015), 531-561.
doi: 10.1016/j.jde.2015.02.022. |
[10] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003. |
[11] |
C. Foias, D. D. Holm and E. S. Titi,
The Navier-Stokes-alpha model of fluid turbulence, Physica D, 152/153 (2001), 505-519.
doi: 10.1016/S0167-2789(01)00191-9. |
[12] |
Y. Giga,
Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z., 178 (1981), 297-329.
doi: 10.1007/BF01214869. |
[13] |
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. |
[14] |
L. Grafakos and S. Oh,
The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.
doi: 10.1080/03605302.2013.822885. |
[15] |
B. L. Guo, D. W. Huang, Q. X. Li and C. Y. 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. |
[16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.
doi: 10.1007/BFb0089649. |
[17] |
D. B. Henry,
How to remember the Sobolev inequalities, Differential Equations, Lecture Notes in Math., Springer, Berlin-New York, 957 (1982), 97-109.
doi: 10.1007/BFb0066235. |
[18] |
N. Ju,
Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation, Math. Ann., 334 (2006), 627-642.
doi: 10.1007/s00208-005-0715-6. |
[19] |
T. Kato,
Strong Lp-solutions of the Navier-Stokes equation in $ \mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[20] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[21] |
H. Komatsu,
Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.
doi: 10.2140/pjm.1966.19.285. |
[22] |
S. G. Kre$\check{{\rm i}}$n, Linear Differential Equations in Banach Spaces, Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence, R.I., 1971. |
[23] |
I. Lasiecka,
Unified theory for abstract parabolic boundary problems-A semigroup approach, Appl. Math. Optim., 6 (1980), 287-333.
doi: 10.1007/BF01442900. |
[24] |
J. Leray,
Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[25] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext. Springer, New York, 2009. |
[26] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[27] |
C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, 187. North-Holland Publishing Co., Amsterdam, 2001. |
[28] |
A. Rodriguez-Bernal, Existence, Uniqueness and Regularity of Solutions of Nonlinear Evolution Equations in Extended Scales of Hilbert Spaces, CDSNS91-61 Report, Georgia Institute of Technology, Atlanta, 1991. |
[29] |
H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2001.
doi: 10.1007/978-3-0348-8255-2. |
[30] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. |
[31] |
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. |
[32] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
doi: 10.1115/1.3424338. |
[33] |
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. |
[34] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
doi: 10.1097/00005768-199805001-01817. |
[35] |
W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig/Wiesbaden, 1985.
doi: 10.1007/978-3-663-13911-9. |
[36] |
W. von Wahl,
Global solutions to evolution equations of parabolic type, Differential Equations in Banach Spaces, Lecture Notes in Math., Springer, Berlin, 1223 (1986), 254-266.
doi: 10.1007/BFb0099198. |
[37] |
Y. Wang and T. Liang,
Mild solutions to the time fractional Navier-Stokes delay differential inclusions, Discrete Contin. Dyn. Syst. Series B, 24 (2019), 3713-3740.
|
[38] |
J. H. Wu, Dissipative quasi-geostrophic equations with Lp data, Electron. J. Differential Equations, (2001), 13 pp.
doi: 10.1111/1468-0262.00185. |
[39] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
![]() ![]() |
[2] |
H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995.
doi: 10.1007/978-3-0348-9221-6. |
[3] |
J. M. 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: 10.2307/118154. |
[4] |
J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278. Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404.![]() ![]() ![]() |
[5] |
J. W. Cholewa and T. Dlotko,
Fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst. Series B, 23 (2018), 2967-2988.
doi: 10.3934/dcdsb.2017149. |
[6] |
A. Córdoba and D. Córdoba,
A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528.
doi: 10.1007/s00220-004-1055-1. |
[7] |
A. Córdoba and D. Córdoba,
A pointwise estimate for fractionary derivatives with applications to partial differential equations, Proc. Natl. Acad. Sci. USA, 100 (2003), 15316-15317.
doi: 10.1073/pnas.2036515100. |
[8] |
T. Dlotko,
Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018), 99-128.
doi: 10.1007/s00245-016-9368-y. |
[9] |
T. Dlotko, M. B. Kania and C. Y. Sun,
Quasi-geostrophic equation in $ \mathbb{R}^2$, J. Differential Equations, 259 (2015), 531-561.
doi: 10.1016/j.jde.2015.02.022. |
[10] |
S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003. |
[11] |
C. Foias, D. D. Holm and E. S. Titi,
The Navier-Stokes-alpha model of fluid turbulence, Physica D, 152/153 (2001), 505-519.
doi: 10.1016/S0167-2789(01)00191-9. |
[12] |
Y. Giga,
Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z., 178 (1981), 297-329.
doi: 10.1007/BF01214869. |
[13] |
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. |
[14] |
L. Grafakos and S. Oh,
The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.
doi: 10.1080/03605302.2013.822885. |
[15] |
B. L. Guo, D. W. Huang, Q. X. Li and C. Y. 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. |
[16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.
doi: 10.1007/BFb0089649. |
[17] |
D. B. Henry,
How to remember the Sobolev inequalities, Differential Equations, Lecture Notes in Math., Springer, Berlin-New York, 957 (1982), 97-109.
doi: 10.1007/BFb0066235. |
[18] |
N. Ju,
Global solutions to the two dimensional quasi-geostrophic equation with critical or super-critical dissipation, Math. Ann., 334 (2006), 627-642.
doi: 10.1007/s00208-005-0715-6. |
[19] |
T. Kato,
Strong Lp-solutions of the Navier-Stokes equation in $ \mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[20] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[21] |
H. Komatsu,
Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.
doi: 10.2140/pjm.1966.19.285. |
[22] |
S. G. Kre$\check{{\rm i}}$n, Linear Differential Equations in Banach Spaces, Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence, R.I., 1971. |
[23] |
I. Lasiecka,
Unified theory for abstract parabolic boundary problems-A semigroup approach, Appl. Math. Optim., 6 (1980), 287-333.
doi: 10.1007/BF01442900. |
[24] |
J. Leray,
Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[25] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext. Springer, New York, 2009. |
[26] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[27] |
C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, 187. North-Holland Publishing Co., Amsterdam, 2001. |
[28] |
A. Rodriguez-Bernal, Existence, Uniqueness and Regularity of Solutions of Nonlinear Evolution Equations in Extended Scales of Hilbert Spaces, CDSNS91-61 Report, Georgia Institute of Technology, Atlanta, 1991. |
[29] |
H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2001.
doi: 10.1007/978-3-0348-8255-2. |
[30] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. |
[31] |
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. |
[32] |
R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
doi: 10.1115/1.3424338. |
[33] |
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. |
[34] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.
doi: 10.1097/00005768-199805001-01817. |
[35] |
W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig/Wiesbaden, 1985.
doi: 10.1007/978-3-663-13911-9. |
[36] |
W. von Wahl,
Global solutions to evolution equations of parabolic type, Differential Equations in Banach Spaces, Lecture Notes in Math., Springer, Berlin, 1223 (1986), 254-266.
doi: 10.1007/BFb0099198. |
[37] |
Y. Wang and T. Liang,
Mild solutions to the time fractional Navier-Stokes delay differential inclusions, Discrete Contin. Dyn. Syst. Series B, 24 (2019), 3713-3740.
|
[38] |
J. H. Wu, Dissipative quasi-geostrophic equations with Lp data, Electron. J. Differential Equations, (2001), 13 pp.
doi: 10.1111/1468-0262.00185. |
[39] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04631-5. |
[1] |
Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016 |
[2] |
Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 |
[3] |
Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152 |
[4] |
Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 |
[5] |
C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 |
[6] |
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 |
[7] |
Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3143-3168. doi: 10.3934/dcds.2022012 |
[8] |
I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 |
[9] |
Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67 |
[10] |
Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 |
[11] |
Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 |
[12] |
Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093 |
[13] |
T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171 |
[14] |
Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023 |
[15] |
May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179 |
[16] |
Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025 |
[17] |
Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336 |
[18] |
Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023 |
[19] |
Radjesvarane Alexandre, Jie Liao, Chunjin Lin. Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinetic and Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617 |
[20] |
Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]