zbMATH — the first resource for mathematics

Well-posedness for the Navier-Stokes equations. (English) Zbl 0972.35084
The existence of a global solution to the Cauchy problem for the Navier-Stokes equations \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v-\Delta v+\nabla p=0,\qquad \text{div }v=0 \quad \text{in } \mathbb{R}^n\times \mathbb{R}^+\\ &v(x,0)=v_0(x), \qquad x\in \mathbb{R}^n \end{aligned} \tag{1} \] is discussed. It is proved that if \(\text{div }v=0\) and if the norm of \(v_0\) \[ \|v_0\|_1=\sup_{x,R}\Biggl[\text{mes}^{-1}(B(x,R))\int_{Q(x,R)} |w(y,t)|^2 dy dt\Biggr]^{\frac 12} \] is sufficiently small, then the problem (1) has a unique small global solution in the space \(X\) with a norm \[ \|v_0\|_X=\sup_{t>0}\sqrt{t}\|v(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)}+ \Biggl(\sup_{x,R} \text{mes}^{-1}(B(x,R))\int_{Q(x,R)} |u(y,t)|^2 dy dt\Biggr)^{\frac 12}. \] Here \(Q(x,R)=B(x,R)\times(0,R^2)\) and \(w(x,t)\) is the solution to the Cauchy problem to the heat equation \[ \frac{\partial w}{\partial t}-\Delta v,\quad w(x,0)=v_0. \] A similar result of existence of a local solution is obtained, too. These results are more general than earlier results.

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI
[1] Ben-Artzi, M., Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. rational mech. anal., 128, 329-358, (1994) · Zbl 0837.35110
[2] Brezis, H., Remarks on the preceding paper “global solutions of two-dimensional Navier-Stokes and Euler equations”, Arch. rational mech. anal., 128, 359-360, (1994) · Zbl 0837.35112
[3] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. pure appl. math., 35, 771-831, (1982) · Zbl 0509.35067
[4] Cannone, M., A generalization of a theorem by Kato on Navier-Stokes equations, Rev. mat. iberoam., 13, 515-541, (1997) · Zbl 0897.35061
[5] Giga, Y.; Miyakawa, T., Navier-Stokes flow in \(R\)^3 with measures as initial vorticity and Morrey spaces, Comm. partial differential equations, 14, 577-618, (1989) · Zbl 0681.35072
[6] Giga, Y.; Miyakawa, T.; Osada, H., Two-dimensional Navier-Stokes flow with measures as initial vorticity, Arch. rational mech. anal., 104, 223-250, (1988) · Zbl 0666.76052
[7] Iftimie, D., The resolution of the Navier-Stokes equations in anisotropic spaces, Rev. mat. iberoam., 15, 1-36, (1999) · Zbl 0923.35119
[8] Kato, T., Strong L^p-solutions of the Navier-Stokes equation in \(R\)m, with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073
[9] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. pure appl. math., 41, 891-907, (1988) · Zbl 0671.35066
[10] Lin, F., A new proof of the caffarelli-Kohn-Nirenberg theorem, Comm. pure appl. math., 51, 241-257, (1998) · Zbl 0958.35102
[11] Lions, P.-L.; Masmoudi, N., Unicité des solutions faibles de Navier-Stokes dans L^N(ω), C. R. acad. sci. Paris ser. I math., 327, 491-496, (1998) · Zbl 0990.35114
[12] Planchon, F., Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in R3, Ann. inst. Henri poincare, anal. non lineaire, 13, 319-336, (1996) · Zbl 0865.35101
[13] Stein, E.M., Harmonic analysis, Princeton mathematical series, 43, (1993), Princeton University Press Princeton
[14] Struwe, M., On partial regularity results for the Navier-Stokes equations, Comm. pure appl. math., 41, 437-458, (1988) · Zbl 0632.76034
[15] Taylor, M., Analysis on Morrey spaces and applications to Navier-Stokes equation, Comm. partial differential equations, 17, 1407-1456, (1992) · Zbl 0771.35047
[16] Wu, S., Analytic dependence of Riemann mappings for bounded domains and minimal surfaces, Comm. pure appl. math., 46, 1303-1326, (1993) · Zbl 0818.30005
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.