Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system. (English) Zbl 1236.35112
Summary: We consider the global well-posedness of the 3-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov spaces \(a_0 \in B^{3/q}_{q,1}(\mathbb R^3)\), \(u_0 = (u_0^h, u_0^3) \in B^{-1+3/p}_{p,1}(\mathbb R^3)\) for \(p, q\) satisfying \(1<q\leq p<6\) and \(\frac{1}{q}-\frac{1}{p}\leq 3\). More precisely, we prove that there exist two positive constants \(c_{0}, C_{0}\) such that if
\[
(\mu \| a_0 \|_{B^{3/q}_{q,1}} + \| u_0^h\|_{B^{-1+3/p}_{p,1}}) \exp (C_0 \| u_0^3\| ^2_{B^{-1+3/p}_{p,1}}/\mu^2) \leq c_0 \mu,
\]
then
\[
\begin{cases} \partial_t a + u \cdot\nabla a = 0, \qquad (t,x )\in \mathbb R^+ \times \mathbb R^3, \\ \partial_t u + u \cdot \nabla u + (1+a) (\nabla \Pi - \mu \Delta u) = 0, \\ \text{div} \, u = 0, \\ (a,u)|_{t=0} = (a_0, u_0) \end{cases}
\]
has a unique global solution \(a \in \tilde{L}^\infty (\mathbb R^+; B^{3/q}_{q,1}(\mathbb R^3))\), \(u\in \tilde{L}^\infty (\mathbb R^+; B^{-1+3/p}_{p,1}(\mathbb R^3))\, \cap \, \tilde{L}^1 (\mathbb R^+; B^{-1+3/p}_{p,1}(\mathbb R^3))\). In particular, this result implies the global well-posedness result in [H. Abidi and M. Paicu, Ann. Inst. Fourier 57, No. 3, 883–917 (2007; Zbl 1122.35091)] for the inhomogeneous Navier-Stokes system with small initial data.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 30H25 | Besov spaces and \(Q_p\)-spaces |
Citations:
Zbl 1122.35091References:
| [1] | Abidi, H., Équation de Navier-Stokes avec densité et viscosité variables dans lʼespace critique, Rev. Mat. Iberoam., 23, 2, 537-586 (2007) · Zbl 1175.35099 |
| [2] | Abidi, H.; Paicu, M., Existence globale pour un fluide inhomogéne, Ann. Inst. Fourier (Grenoble), 57, 883-917 (2007) · Zbl 1122.35091 |
| [3] | Abidi, H.; Gui, G.; Zhang, P., On the decay and stability to global solutions of the 3-D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64, 832-881 (2011) · Zbl 1222.35148 |
| [4] | Bahouri, H.; Chemin, J. Y.; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343 (2011), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1227.35004 |
| [5] | Bony, J. M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14, 209-246 (1981) · Zbl 0495.35024 |
| [6] | Chemin, J. Y., Théorèmes dʼunicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math., 77, 27-50 (1999) · Zbl 0938.35125 |
| [7] | Chemin, J. Y.; Gallagher, I., Wellposedness and stability results for the Navier-Stokes equations in \(R^3\), Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 599-624 (2009) · Zbl 1165.35038 |
| [8] | Chemin, J. Y.; Lerner, N., Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121, 314-328 (1995) · Zbl 0878.35089 |
| [9] | Danchin, R., Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differential Equations, 26, 1183-1233 (2001) · Zbl 1007.35071 |
| [10] | Danchin, R., Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133, 1311-1334 (2003) · Zbl 1050.76013 |
| [11] | Danchin, R., Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9, 353-386 (2004) · Zbl 1103.35085 |
| [12] | R. Danchin, P.B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math. (2012), in press.; R. Danchin, P.B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math. (2012), in press. · Zbl 1247.35088 |
| [13] | DiPerna, R. J.; Lions, P. L., Equations différentielles ordinaires et équations de transport avec des coefficients irréguliers, (Séminaire EDP 1988-1989 (1989), Ecole Polytechnique: Ecole Polytechnique Palaiseau) · Zbl 0708.34014 |
| [14] | Fujita, H.; Kato, T., On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301 |
| [15] | Gui, G.; Zhang, P., Stability to the global solutions of 3-D Navier-Stokes equations, Adv. Math., 225, 1248-1284 (2010) · Zbl 1202.35004 |
| [16] | Kazhikov, A. V., Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR, 216, 1008-1010 (1974), (in Russian) |
| [17] | Ladyženskaja, O. A.; Solonnikov, V. A., The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Boundary Value Problems of Mathematical Physics, and Related Questions of the Theory of Functions, 8. Boundary Value Problems of Mathematical Physics, and Related Questions of the Theory of Functions, 8, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 52, 52-109 (1975), 218-219 (in Russian) · Zbl 0376.76021 |
| [18] | Lions, P. L., Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3 (1996), Oxford Science Publications, The Clarendon Press, Oxford University Press: Oxford Science Publications, The Clarendon Press, Oxford University Press New York · Zbl 0866.76002 |
| [19] | Paicu, M.; Zhang, P., Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307, 713-759 (2011) · Zbl 1237.35129 |
| [20] | Peetre, J., New Thoughts on Besov Spaces, Duke Univers. Math. Ser., vol. 1 (1976), Duke University: Duke University Durham, NC · Zbl 0356.46038 |
| [21] | Planchon, F., Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris Sér. I Math., 330, 21-23 (2000) · Zbl 0953.46020 |
| [22] | Triebel, H., Theory of Function Spaces, Monogr. Math., vol. 78 (1983), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0546.46028 |
| [23] | Zhang, T., Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287, 211-224 (2009) · Zbl 1173.35633 |
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