Ill-posedness of the Navier-Stokes equations in a critical space in 3D. (English) Zbl 1161.35037
Summary: We prove that the Cauchy problem for the three-dimensional Navier-Stokes equations is ill-posed in \({\dot B}^{-1,\infty}_{\infty}\) in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class \(\mathcal S\) that are arbitrarily small in \({\dot B}^{-1,\infty}_{\infty}\) can produce solutions arbitrarily large in \({\dot B}^{-1,\infty}_{\infty}\) after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in \({\dot B}^{-1,\infty}_{\infty}\) at the origin.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
References:
| [1] | Bourgain, J., Periodic Korteweg-De Vries equations with measures as initial data, Selecta Math. (N.S.), 3, 115-159 (1993) · Zbl 0891.35138 |
| [2] | Cannone, M., A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13, 3, 515-541 (1997) · Zbl 0897.35061 |
| [3] | Cannone, M., Harmonic analysis tools for solving the incompressible Navier-Stokes equations, (Handbook of Mathematical Fluid Dynamics, vol. III (2004), North-Holland: North-Holland Amsterdam), 161-244 · Zbl 1222.35139 |
| [4] | J.-Y. Chemin, I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in \(\mathbb{R}^3\), Ann. Inst. H. Poincaré Anal. Non Linéaire, in press; J.-Y. Chemin, I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in \(\mathbb{R}^3\), Ann. Inst. H. Poincaré Anal. Non Linéaire, in press |
| [5] | M. Christ, J. Colliander, T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint, 2003; M. Christ, J. Colliander, T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint, 2003 |
| [6] | M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Ann. Henry Poincaré, in press; M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Ann. Henry Poincaré, in press · Zbl 1048.35101 |
| [7] | Germain, P., The second iterate for the Navier-Stokes equation (2008), preprint · Zbl 1173.35097 |
| [8] | Giga, Y.; Miyakawa, T., Navier-Stokes flow in \(R^3\) with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14, 5, 577-618 (1989) · Zbl 0681.35072 |
| [9] | Kato, T., Strong \(L^p\)-solutions of the Navier-Stokes equations in \(R^m\) with applications to weak solutions, Math. Z., 187, 471-480 (1984) · Zbl 0545.35073 |
| [10] | Kenig, C. E.; Ponce, G.; Vega, L., On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106, 617-633 (2001) · Zbl 1034.35145 |
| [11] | Koch, H.; Tataru, D., Well posedness for the Navier-Stokes equations, Adv. Math., 157, 22-35 (2001) · Zbl 0972.35084 |
| [12] | Lemarié-Rieusset, P. G., Une remarque sur l’analyticité des solutions milds des équations de Navier-Stokes dans \(R^3\), C. R. Acad. Sci. Paris Sér. I Math., 330, 183-186 (2000) · Zbl 0942.35131 |
| [13] | Meyer, Y., Wavelets, paraproducts and Navier-Stokes equations, (Current Developments in Mathematics, 1996 (1999), Internat. Press: Internat. Press Cambridge, MA), 105-212 · Zbl 0926.35115 |
| [14] | Montgomery-Smith, S., Finite time blow up for a Navier-Stokes like equation, Proc. Amer. Math. Soc., 129, 10, 3025-3029 (2001) · Zbl 0970.35100 |
| [15] | Planchon, F., Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in \(R^3\), Ann. Inst. H. Poincaré, Anal. Non Linéaire, 13, 3, 319-336 (1996) · Zbl 0865.35101 |
| [16] | Taylor, M., Analysis on Morrey spaces and applications to Navier-Stokes equation, Comm. Partial Differential Equations, 17, 1407-1456 (1992) · Zbl 0771.35047 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
