zbMATH — the first resource for mathematics

Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. (English) Zbl 1104.35037
Summary: When estimating solutions of dissipative partial differential equations in \(L^{p}\)-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian \(- \Delta \), lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian \((-\Delta)^{\alpha}\), the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving \((- \Delta)^{\alpha}\) by combining pointwise inequalities for \((-\Delta)^{\alpha}\) with Bernstein’s inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing \(-\Delta\) in the Navier-Stokes equations by \((-\Delta)^{\alpha}\).

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
26A33 Fractional derivatives and integrals
76D05 Navier-Stokes equations for incompressible viscous fluids
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
Full Text: DOI
[1] Beirão da Veiga, H.: Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space. Indiana Univ. Math. J. 36, 149–166 (1987) · Zbl 0601.35093
[2] Beirão da Veiga, H., Secchi, P.: Lp-stability for the strong solutions of the Navier-Stokes equations in the whole space. Arch. Rat. Mech. Anal. 98, 65–69 (1987) · Zbl 0678.35076
[3] Bergh, J., Löfström, J.: Interpolation Spaces, An Introduction. Berlin-Heidelberg-New York: Springer-Verlag, 1976 · Zbl 0344.46071
[4] Chae, D., Lee, J.: On the global well-posedness and stability of the Navier-Stokes and the related equations. In: Contributions to current challenges in mathematical fluid mechanics, Adv. Math. Fluid Mech., Basel: Birkhäuser, 2004, pp. 31–51 · Zbl 1083.35086
[5] Chemin, J.-Y.: Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel. J. Anal. Math. 77, 27–50 (1999)
[6] Chemin, J.-Y., Lerner, N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differ. Eq. 121, 314–328 (1995) · Zbl 0878.35089
[7] Constantin, P.: Euler equations, Navier-Stokes equations and turbulence. In: M. Cannone, T. Miyakawa, eds., Mathematical foundation of turbulent viscous flows. CIME Summer School, Martina Francs, Italy 2003. Berlin-Heidelberg-New York: Springer, to appear
[8] Constantin, P., Córdoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) · Zbl 0989.86004
[9] Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999) · Zbl 0957.76093
[10] Córdoba, A., Córdoba, D.: A pointwise estimate for fractionary derivatives with applications to partial differential equations. Proc. Natl. Acad. Sci. USA 100, 15316–15317 (2003) · Zbl 1111.26010
[11] Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004) · Zbl 1309.76026
[12] Danchin, R.: Poches de tourbillon visqueuses. J. Math. Pures Appl. 76(9), 609–647 (1997) · Zbl 0903.76020
[13] Gallagher, I., Iftimie, D., Planchon, F.: Asymptotics and stability for global solutions to the Navier-Stokes equations. Ann. Inst. Fourier (Grenoble) 53, 1387–1424 (2003) · Zbl 1038.35054
[14] Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun. Math. Phys. 255, 161–181 (2005) · Zbl 1088.37049
[15] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. (French) Paris: Dunod / Gauthier-Villars, 1969
[16] Planchon, F.: Sur un inégalité de type Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 330, 21–23 (2000) · Zbl 0953.46020
[17] Wu, J.: Generalized MHD equations. J. Differ. Eq. 195, 284–312 (2003) · Zbl 1057.35040
[18] Wu, J.: The generalized incompressible Navier-Stokes equations in Besov spaces. Dyn. Partial Differ. Eq. 1, 381–400 (2004) · Zbl 1075.35043
[19] Wu, J.: Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36, 1014–1030 (2004/05)
[20] Yuan, J.-M., Wu, J.: The complex KdV equation with or without dissipation. Discrete Contin. Dyn. Syst. Ser. B 5, 489–512 (2005) · Zbl 1080.35129
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.