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Germany; Russia Krasnoyarsk; Potsdam
Section Mathematics
Title Inverse image of precompact sets and regular solutions to the Navier–Stokes equations
Author(-s) Shlapunov A.A.a, Tarkhanov N.N.b
Affiliations Siberian Federal Universitya, University of Potsdamb
Abstract We consider the initial value problem for the Navier–Stokes equations over ${\mathbb R}^3 \times [0,T]$ with time $T>0$ in the spatially periodic setting. We prove that it induces open injective mappings ${\mathcal A}_s\colon B^{s}_1 \to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner–Sobolev type parametrized with the smoothness index $s \in \mathbb N$. Finally, we prove that a map ${\mathcal A}_s$ is surjective if and only if the inverse image ${\mathcal A}_s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${\mathcal A}_s$ is bounded in the Bochner space $L^{\mathfrak s} ([0,T], L^{{\mathfrak r}} ({\mathbb T}^3))$ with the Ladyzhenskaya–Prodi–Serrin numbers ${\mathfrak s}$, ${\mathfrak r}$.
Keywords уравнения Навье–Стокса, регулярные решения
UDC 517
MSC 76N10, 35Q30, 76D05
DOI 10.35634/vm220208
Received 21 January 2022
Language English
Citation Shlapunov A.A., Tarkhanov N.N. Inverse image of precompact sets and regular solutions to the Navier–Stokes equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 278-297.
  1. Adams R.A. Sobolev spaces, Academic Press, 2003.
  2. Agranovich M.S. Elliptic operators on closed manifolds, Encyclopaedia of Mathematical Sciences, vol. 63, Berlin: Springer, 1994, pp. 1-130.
  3. Barker T., Seregin G., Šverák V. On stability of weak Navier-Stokes solutions with large $L$${3,\infty}$ initial data, Communications in Partial Differential Equations, 2018, vol. 43, issue 4, pp. 628-651.
  4. Barker T., Seregin G. A necessary condition of potential blowup for the Navier-Stokes system in half-space, Mathematische Annalen, 2017, vol. 369, issues 3-4, pp. 1327-1352.
  5. Choe H.J., Wolf J., Yang M. A new local regularity criterion for suitable weak solutions of the Navier-Stokes equations in terms of the velocity gradient, Mathematische Annalen, 2018, vol. 370, issues 1-2, pp. 629-647.
  6. da Veiga H.B. On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half-space case, Journal of Mathematical Analysis and Applications, 2017, vol. 453, issue 1, pp. 212-220.
  7. da Veiga H.B., Yang J. Navier-Stokes equations under slip boundary conditions: Lower bounds to the minimal amplitude of possible time-discontinuities of solutions with two components in $L$$\infty$ ($L$$3$ ), Science China Mathematics, 2021.
  8. Escauriaza L., Seregin G.A., Šverak V. $L_{3,\infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Russian Mathematical Surveys, 2003, vol. 58, no. 2, pp. 211-250.
  9. Farwig R., Giga Y., Hsu P.-Y. On the continuity of the solutions to the Navier-Stokes equations with initial data in critical Besov spaces, Annali di Matematica Pura ed Applicata, 2019, vol. 198, no. 5, pp. 1495-1511.
  10. Hamilton R.S. The inverse function theorem of Nash and Moser, Bulletin of the American Mathematical Society, 1982, vol. 7, no. 1, pp. 65-222.
  11. Hopf E. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Mathematische Nachrichten, 1950, vol. 4, issues 1-6, pp. 213-231.
  12. Jia H., Sverak V. Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?, Journal of Functional Analysis, 2015, vol. 268, issue 12, pp. 3734-3766.
  13. Jiu Q., Wang Y., Zhou D. On Wolf's regularity criterion of suitable weak solutions to the Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 2019, vol. 21, issue 2, article number: 22.
  14. Ladyzhenskaya O.A. Uniqueness and smoothness of generalized solutions of the Navier-Stokes equation, Sem. Math. V.A. Steklov Math. Inst. Leningrad, 1969, vol. 5, pp. 60-66.
  15. Ladyzhenskaya O.A. Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti (Mathematical problems of incompressible viscous fluid), Moscow: Nauka, 1970.
  16. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and quasilinear equations of parabolic type), Moscow: Nauka, 1967.
  17. Leray J. Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 1934, vol. 63, pp. 193-248.
  18. Lions J.-L. Équations différentielles opérationelles. Et problèmes aux limites, Berlin: Springer, 1961.
  19. Lions J.-L. Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris: Dunod, 1969.
  20. Mitrinović D.S., Pečarić J.E., Fink A.M. Inequalities involving functions and their integrals and derivatives, Dordrecht: Kluwer Academic Publishers, 1991.
  21. Nirenberg L. On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 3, 1959, vol. 13, no. 2, pp. 115-162.
  22. Polkovnikov A. An open mapping theorem for nonlinear operator equations associated with elliptic complexes, Applicable Analysis, 2021, 23 p.
  23. Prodi G. Un teorema di unicitá per le equazioni di Navier-Stokes, Annali di Matematica Pura ed Applicata, 1959, vol. 48, issue 1, pp. 173-182.
  24. Seregin G., Šverák V. On global weak solutions to the Cauchy problem for the Navier-Stokes equations with large $L_3$-initial data, Nonlinear Analysis: Theory, Methods and Applications, 2017, vol. 154, pp. 269-296.
  25. Serrin J. Mathematical principles of classical fluid mechanics, Fluid Dynamics. I, Berlin: Springer, 1959, pp. 125-263.
  26. Serrin J. On the interior regularity of weak solutions of the Navie-Stokes equations, Archive for Rational Mechanics and Analysis, 1962, vol. 9, issue 1, pp. 187-195.
  27. Shlapunov A.A., Tarkhanov N. An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over ${\mathbb R}$$n$ , Siberian Electronic Mathematical Reports, 2021, vol.18, issue 2, pp. 1433-1466.
  28. Smale S. An infinite dimensional version of Sard's theorem, American Journal of Mathematics, 1965, vol. 87, no. 4, pp. 861-866.
  29. Temam R. Navier-Stokes equations. Theory and numerical analysis, Amsterdam: North Holland, 1979.
  30. Temam R. Navier-Stokes equations and nonlinear functional analysis, Philadelphia: SIAM, 1995.
  31. Tao T. Finite time blow-up for an averaged three-dimensional Navier-Stokes equation, Journal of the American Mathematical Society, 2015, vol. 29, issue 3, pp. 601-674.
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