# American Institute of Mathematical Sciences

April  2020, 25(4): 1517-1541. doi: 10.3934/dcdsb.2019238

## 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

Received  December 2018 Published  April 2020 Early access  November 2019

Fund Project: This work was supported by NSF of China (Grants No. 41875084, 11571153), the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-ot03 and lzujbky-2018-it58.

Our purpose is to formulate an abstract result, motivated by the recent paper [8], allowing to treat the solutions of critical and super-critical equations as limits of solutions to their regularizations. In both cases we are improving the viscosity, making it stronger, solving the obtained regularizations with the use of Dan Henry's technique, then passing to the limit in the improved viscosity term to get a solution of the limit problem. While in case of the critical problems we will just consider a 'bit higher' fractional power of the viscosity term, for super-critical problems we need to use a version of the 'vanishing viscosity technique' that comes back to the considerations of E. Hopf, O.A. Oleinik, P.D. Lax and J.-L. Lions from 1950th. In both cases, the key to that method are the uniform with respect to the parameter estimates of the approximating solutions. The abstract result is illustrated with the Navier-Stokes equation in space dimensions 2 to 4, and with the 2-D quasi-geostrophic equation. Various technical estimates related to that problems and their fractional generalizations are also presented in the paper.

Citation: Tomasz Dlotko, Tongtong Liang, Yejuan Wang. Critical and super-critical abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1517-1541. doi: 10.3934/dcdsb.2019238
##### 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

2021 Impact Factor: 1.497