Well-posedness of the Navier-Stokes equations

Hideo Kozono Mathematical Institute, Tohoku University, Sendai, 980-8578 JAPAN

Abstract.

We investigate blow-up phenomena of strong solutions to the Navier-Stokes equations:

(N-S)  \{ \begin{matrix}\frac{∂u}{∂t} − Δu + u · ∇u + ∇p = 0, div u = 0 in x ∈ R^{n}, t > 0, \\ u |_{t=0} = a. \end{matrix}

Let a ∈ H_{σ}^{s} for s > n/2 − 1. For the strong solution of (N-S), we introduce the class CL_{s}(0, T) defined by

u ∈ C([0, T); H_{σ}^{s}) ∩ C^{1}((0, T); H_{σ}^{s}) ∩ C((0, T); H_{σ}^{s+2} ).

Theorem 1 Let s > n/2 − 1 and let a ∈ H_{σ}^{s}. Suppose that u is a strong solution of (N-S) in the class CL_{s}(0, T). If

∫_{ε_{0}}^{T} ∥u(t)∥_{BMO}^{2}dt < ∞ for some 0 < ε_{0} < T , (0.1)

then u can be continued to the strong solution in the class CL_{s}(0, T′) for some T′ > T.

Corollary 1 Let u be a strong solution of (N-S) in the class CL_{s}(0, T) for s > n/2 − 1. Suppose that T is maximal, i.e., u cannot be continued in the class CL_{s}(0, T′) for any T′ > T. Then

∫_{ε}^{T} ∥u(t)∥_{BMO}^{2}dt = ∞ for all 0 < ε < T. (0.2)

In particular, we have

lim sup_{t↑T} ∥u(t)∥_{BMO} = ∞. (0.3)

We next consider a criterion on uniqueness and regularity of weak solutions to (N-S). By the weak solution we mean u in L^{∞}(0, T; L_{σ}^{2}) ∩ L^{2}(0, T; H_{σ}^{1}) which satisfies (N-S) in the sense of distributions on R^{n} × (0, T).

Theorem 2 (1) (uniqueness) Let a ∈ L_{σ}^{2} and let u, v be two weak solutions of (N-S) on (0, T). Suppose that u ∈ L^{2}(0, T; BMO) (0.4) and that v satisfies the energy inequality

∥v(t)∥_{2}^{2} + 2 ∫_{0}^{t} ∥∇v∥_{2}^{2}dτ ≤ ∥a∥_{2}^{2}, 0 < t < T. (0.5)

Then we have u ≡ v on [0, T].

(2) (regularity) Let a ∈ L_{σ}^{2} and let u be a weak solution with the additional property (0.4). Then for every 0 < ε < T, u is actually a strong solution of (N-S) in the class CL_{s}(ε, T) for s > n/2 − 1.

Remark. 1. Theorem 2 may be regarded as an extension of Serrin’s criterion [4], [5] on uniqueness and regularity of weak solutions u in the class

u ∈ L^{κ}(0, T; L^{r}) for 2/κ + n/r = 1 with n < r ≤ ∞ . (0.6)

Our class (0.4) is larger than the marginal case L^{2}(0, T; L^{∞}) in (0.6). Moreover, by virtue of the estimate ∥u∥_{BMO} ≤ C∥∇u∥_{M^{n}} of John-Nirenberg, we see that the weak solution u with ∇u ∈ L^{2}(0, T; M^{n}) becomes regular, where M^{n} denotes the Morrey space which is larger than L^{n}. See Beirão da Veiga [2].

We shall next investigate continuation of the strong solution in terms of the vorticity ω = rot u ≡ (∂_{j}u^{k}−∂_{k}u^{j})_{1≤j,k≤n} and the deformation tensor Def u ≡ (∂_{j}u^{k}+∂_{k}u^{j})_{1≤j,k≤n}.

Theorem 3 Let s > n/2 − 1. Suppose that u is a strong solution of (N-S) in the class CL_{s}(0, T). If either

∫_{ε_{0}}^{T} ∥ω(t)∥_{BMO}dt < ∞ (0.7)

or ∫_{ε_{0}}^{T} ∥Def u(t)∥_{BMO}dt < ∞ (0.8)

holds for some 0 < ε_{0} < T, then u can be continued to the strong solution in the class CL_{s}(0, T′) for some T′ > T.

Corollary 2 Suppose that u is a strong solution of (N-S) in the class CL_{s}(0, T) for s > n/2 − 1. Assume that T is maximal in the same sense as in Corollary 1. Then both

∫_{ε}^{T} ∥ω(t)∥_{BMO}dt = ∞ (0.9)

and ∫_{ε}^{T} ∥Def u(t)∥_{BMO}dt = ∞ (0.10)

hold for all 0 < ε < T. In particular, we have

lim sup_{t↑T} ∥ω(t)∥_{BMO} = ∞ (0.11)

and (0.12) lim sup_{t↑T} ∥Def u(t)∥_{BMO} = ∞.

Remark. 1. Beale-Kato-Majda [1] and Ponce [3] considered the Euler equations in R^{3} and the same type of continuation principle as in Theorem 2 under the stronger assumption

that ∫_{0}^{T} ∥ω(t)∥_{∞}dt < ∞ and ∫_{0}^{T} ∥Def u(t)∥_{∞}dt < ∞, respectively.

Finally we are concerned with the regularity criterion on weak solutions by mean of rot u and Def u.

Theorem 4 Let a ∈ L_{σ}^{2}. Suppose that u is a weak solution of (N-S) on (0, T). If either

ω ∈ L^{1}(0, T; BMO) (0.13)

or Def u ∈ L^{1}(0, T; BMO) (0.14)

holds, then for every 0 < ε < T, u is actually a strong solution of (N-S) in the class CL_{s}(ε, T) for s > n/2 − 1.

Remark. 1. Beirão da Veiga [2] proved the regularity criterion in the class ∇u ∈ L^{κ}(0, T; L^{r}) for 2/κ + n/r = 2 with 1 < κ < ∞, n/2 < r < ∞. Theorem 4 covers the borderline case κ = 1 and r = ∞.

References

[1] Beale, J.T., Kato, T., Majda, A., Remarks on the breakdown of smooth solutions for the 3 − D Euler equations. Commu. Math. Phys. 94, 61–66 (1984).

[2] Beirão da Veiga, H., A new regularity class for the Navier-Stokes equations in R^{n}. Chin. Ann. of Math. 16 B, 407–412 (1995).

[3] Ponce, G., Remarks on a paper by J. T. Beale, T. Kato and A. Majda. Commun. Math. Phys. 98, 349–353 (1985).

[4] Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal. 9,187–195 (1962).

[5] Serrin, J., The initial value problem for the Navier-Stokes equations. Nonlinear Problems, R. E. Langer ed., Madison: University of Wisconsin Press, 69-98 (1963).