Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions

Miao, Changxing; Zheng, Xiaoxin

doi:10.1090/tran/9148

Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions
HTML articles powered by AMS MathViewer

by Changxing Miao and Xiaoxin Zheng

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9148

Published electronically: April 11, 2024

HTML | PDF | Request permission

Abstract:

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, $L^{p}$-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures.

References

Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550, DOI 10.1007/978-3-642-16830-7
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 482275, DOI 10.1007/978-3-642-66451-9
Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246 (French). MR 631751, DOI 10.24033/asens.1404
Zachary Bradshaw and Patrick Phelps, Spatial decay of discretely self-similar solutions to the Navier-Stokes equations, Pure Appl. Anal. 5 (2023), no. 2, 377–407. MR 4605900, DOI 10.2140/paa.2023.5.377
Qionglei Chen, Changxing Miao, and Zhifei Zhang, A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys. 271 (2007), no. 3, 821–838. MR 2291797, DOI 10.1007/s00220-007-0193-7
E. B. Fabes, B. F. Jones, and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in $L^{p}$, Arch. Rational Mech. Anal. 45 (1972), 222–240. MR 316915, DOI 10.1007/BF00281533
Claus Gerhardt, Stationary solutions to the Navier-Stokes equations in dimension four, Math. Z. 165 (1979), no. 2, 193–197. MR 520820, DOI 10.1007/BF01182469
Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437, DOI 10.1007/978-0-387-09432-8
Hao Jia, Vladimir Šverák, and Tai-Peng Tsai, Self-similar solutions to the nonstationary Navier-Stokes equations, Handbook of mathematical analysis in mechanics of viscous fluids, Springer, Cham, 2018, pp. 461–507. MR 3916781, DOI 10.1007/978-3-319-13344-7_{9}
Mikhail Korobkov and Tai-Peng Tsai, Forward self-similar solutions of the Navier-Stokes equations in the half space, Anal. PDE 9 (2016), no. 8, 1811–1827. MR 3599519, DOI 10.2140/apde.2016.9.1811
Baishun Lai, Changxing Miao, and Xiaoxin Zheng, Forward self-similar solutions of the fractional Navier-Stokes equations, Adv. Math. 352 (2019), 981–1043. MR 3975031, DOI 10.1016/j.aim.2019.06.021
Baishun Lai, Changxing Miao, and Xiaoxin Zheng, Global regularity of weak solutions to the generalized Leray equations and its applications, Trans. Amer. Math. Soc. 374 (2021), no. 10, 7449–7497. MR 4315609, DOI 10.1090/tran/8455
Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
YanYan Li and Zhuolun Yang, Regular solutions of the stationary Navier-Stokes equations on high dimensional Euclidean space, Comm. Math. Phys. 394 (2022), no. 2, 711–734. MR 4469405, DOI 10.1007/s00220-022-04409-7
C. Miao, J. Wu, and Z. Zhang,Littlewood-Paley theory and applications to fluid dynamics equations, Monographs on Modern Pure Mathematics, No. 142, Science Press, Beijing, 2012.
J. Nečas, M. Růžička, and V. Šverák, On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math. 176 (1996), no. 2, 283–294. MR 1397564, DOI 10.1007/BF02551584
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Tai-Peng Tsai, Lectures on Navier-Stokes equations, Graduate Studies in Mathematics, vol. 192, American Mathematical Society, Providence, RI, 2018. MR 3822765, DOI 10.1090/gsm/192