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Ill-posedness of the 3D-Navier-Stokes equations near $BMO^{-1}$

  • 35-Xx Partial Differential Equations
  • Mathematics


Ill-posedness of the 3D-Navier-Stokes equations near BMO−1 Tsuyoshi Yoneda University of Minnesota (IMA) August 23, 2010 We consider the nonstationary incompressible Navier-Stokes equations in R3: (0.1)  ∂u ∂t −∆u+ (u · ∇)u+∇p = 0, div u = 0 in x ∈ R3, t ∈ (0, T ), u|t=0 = u0, where u = u(t) = (u1(x, t), u2(x, t), u3(x, t)) and p = p(t) = p(x, t) denote the velocity vector field and the pressure of fluid at the point (x, t) ∈ R3 × (0, T ), respectively, while u0 = (u 1 0(x), u 2 0(x), u 3 0(x)) is a given initial velocity vector field. We are concerned with the ill-posedness of the Cauchy problem for (0.1). More pre- cisely for a given function space X = X(R3) we say that the Cauchy problem is well-posed in X if there exists a space Y ⊂ C([0, T ), X) such that for all u0 ∈ X there exists a unique solution u ∈ Y for (0.1) and the flow map u0 → u = Φ(u0) is continuous from X to C([0, T ), X). Also we say that the Cauchy problem is ill-posed in X if it is not. The classical results on the existence theorem of the mild solution were shown by Kato [6] and Giga-Miyakawa [3]. Making use of the iteration procedure, they constructed a global solution in the class C([0,∞);Ln(Rn))∩C((0,∞);Lp(Rn)) for n < p ≤ ∞, when an initial data u0 is small enough in L n(Rn). To construct a solution in more general classes of ini- tial data is very important problem. Giga-Miyakawa [4], Kato [7] and Taylor [12] proved the well-posedness in certain Morrey spaces. Cannone [2] and Kozono-Yamazaki [9] in- vestigated this problem in Besov spaces. In particular, Koch and Tataru [8] obtained the global solvability for (0.1), when the initial data u0 is small enough in BMO −1. BMO−1 includes above function spaces and it has been considered as the largest space of initial data (see Lemarie´-Rieusset [10]). On the other hand, Montgomery-Smith [11] introduce an equation similar to Navier-Stokes equation and proved ill-posedness in the Besov space B−1∞,∞, which is larger than BMO −1. In 2008, Bo

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