# 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  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 & Continuous Dynamical Systems - B, 2020, 25 (4) : 1517-1541. doi: 10.3934/dcdsb.2019238
##### References:
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Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.  Google Scholar [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.  Google Scholar [8] T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018), 99-128.  doi: 10.1007/s00245-016-9368-y.  Google Scholar [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.  Google Scholar [10] S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar [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.  Google Scholar [16] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089649.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] H. Komatsu, Fractional powers of operators, Pacific J. 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Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, 187. North-Holland Publishing Co., Amsterdam, 2001.  Google Scholar [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. Google Scholar [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.  Google Scholar [30] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. doi: 10.1097/00005768-199805001-01817.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [38] J. H. Wu, Dissipative quasi-geostrophic equations with Lp data, Electron. J. Differential Equations, (2001), 13 pp. doi: 10.1111/1468-0262.00185.  Google Scholar [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.  Google Scholar

show all references

##### References:
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Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.  Google Scholar [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.  Google Scholar [8] T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., 77 (2018), 99-128.  doi: 10.1007/s00245-016-9368-y.  Google Scholar [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.  Google Scholar [10] S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] L. Grafakos and S. Oh, The Kato-Ponce inequality, Comm. Partial Differential Equations, 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar [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.  Google Scholar [16] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089649.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.  Google Scholar [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.  Google Scholar [23] I. Lasiecka, Unified theory for abstract parabolic boundary problems-A semigroup approach, Appl. Math. Optim., 6 (1980), 287-333.  doi: 10.1007/BF01442900.  Google Scholar [24] J. Leray, Sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar [25] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext. Springer, New York, 2009.  Google Scholar [26] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [30] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. doi: 10.1097/00005768-199805001-01817.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [38] J. H. Wu, Dissipative quasi-geostrophic equations with Lp data, Electron. J. Differential Equations, (2001), 13 pp. doi: 10.1111/1468-0262.00185.  Google Scholar [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.  Google Scholar
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