Optimality of Serrin type extension criteria to the Navier-Stokes equations

Definition 2.1

Let s ∈ ℝ, 1 ≤ p, q, θ ≤ ∞ and let {ϕj}j=−∞∞ be the Littlewood-Paley decomposition. Then, V˙p,q,θs(Rn):={f∈Z′;∥f∥V˙p,q,θs<∞} is introduced by the norm

Definition 2.2

Let s, β ∈ ℝ, 1 ≤ p, σ ≤ ∞ and let {ϕj}j=−∞∞ be the Littlewood-Paley decomposition. Then, U˙p,β,σs(Rn):={f∈Z′;∥f∥U˙p,β,σs<∞} is equipped with the norm

Proposition 2.3

1. Let s ∈ ℝ, 1 ≤ p, q ≤ ∞ and 1 ≤ θ₁ ≤ θ₂ ≤ q < θ₃. Then, it holds that

   {0}=V˙p,q,θ3s⊂B˙p,qs=V˙p,q,qs⊂V˙p,q,θ2s⊂V˙p,q,θ1s.

2. Let s ∈ ℝ, 1 ≤ p, σ ≤ ∞ and β₁ < 0 ≤ β₂ ≤ β₃. Then, it holds that

   {0}=U˙p,β1,σs⊂B˙p,σs=U˙p,0,σs⊂U˙p,β2,σs⊂U˙p,β3,σs.

3. Let s, β ∈ ℝ, 1 ≤ p, q, θ ≤ ∞, β~=1θ−1q and 1 ≤ σ₁ ≤ σ₂ ≤ ∞. Then, it holds that

   V˙p,q,θs=U˙p,β~,θsandU˙p,β,σ1s⊂U˙p,β,σ2s.

Proof

We easily prove V˙p,q,θ2s⊂V˙p,q,θ1s in (i) by the standard and the reverse Hölder’s inequality. The others follow from the definitions of B˙p,qs , V˙p,q,θs and U˙p,β,σs . □

Example 2.4

1. The continuous embedding B˙∞,∞s⊂V˙∞,∞,θs is proper if s > –n and 1 ≤ θ < ∞. We now introduce a distribution f∈V˙∞,∞,θs∖B˙∞,∞s for s > –n and 1 ≤ θ < ∞. Let f ∈ 𝓩^′ defined as

   f^(ξ):=k2−(n+s)[kθ+1],2[kθ+1]−1≤|ξ|≤2[kθ+1]+1(k=1,2,⋯),0,otherwise.

   Indeed, since f̂ ∈ L^∞ holds, we obtain f ∈ 𝓩^′. We easily see that

   ∥ϕj∗f∥∞=∫Rnϕ^j(ξ)f^(ξ)dξ=∫2j−1≤|ξ|≤2j+1ϕ^j(ξ)f^(ξ)dξ=2−s[kθ+1]k∥ϕ^∥1for j=[kθ+1](k=1,2,⋯),≤2−s[kθ+1]2nk∥ϕ^∥1for j=[kθ+1]±1(k=1,2,⋯),=0for j∈Z∖⋃k=1,2,⋯{[kθ+1],[kθ+1]±1}.

   Hence, it holds that

   ∥f∥B˙∞,∞s≥supk=1,2,⋯2s[kθ+1]∥ϕ[kθ+1]∗f∥∞=supk=1,2,⋯k∥ϕ^∥1=∞. (2.2)

   On the other hand, for any N = 1, 2, ⋯, there exists k_N ∈ ℕ such that kNθ+1≤N<(kN+1)θ+1. Therefore, we obtain

   ∑|j|≤N2jsθ∥ϕj∗f∥∞θ≤∑k=1kN+1∑j=[kθ+1]−1[kθ+1]+12jsθ∥ϕj∗f∥∞θ≤∑k=1kN+1∑j=[kθ+1]−1[kθ+1]+12jsθ(2−s[kθ+1]2nk∥ϕ^∥1)θ=C∑k=1kN+1kθ≤C(kN+1)θ+1≤CkNθ+1≤CN,

   where C is dependent only on n, s and θ. Thus, it follows that

   ∥f∥V˙∞,∞,θs=supN=1,2,⋯∑|j|≤N2jsθ∥ϕj∗f∥∞θ1θN1θ≤supN=1,2,⋯C1θN1θN1θ<∞. (2.3)

   From (2.2) and (2.3), we get f∈V˙∞,∞,θs∖B˙∞,∞s.

2. The continuous embedding V˙∞,∞,θs=U˙∞,1/θ,θs⊂U˙∞,1/θ,∞s is also proper if s > –n and 1 ≤ θ < ∞. We now introduce a distribution g∈U˙∞,1/θ,∞s∖V˙∞,∞,θs for s > –n and 1 ≤ θ < ∞. Let g ∈ 𝓩^′ defined as

   g^(ξ):=kθ+1θ2−(n+s)[kθ+1],2[kθ+1]−1≤|ξ|≤2[kθ+1]+1(k=1,2,⋯),0,otherwise.

   Indeed, since ĝ ∈ L^∞ holds, we obtain g ∈ 𝓩^′. We easily see that

   ∥ϕj∗g∥∞=∫Rnϕ^j(ξ)g^(ξ)dξ=∫2j−1≤|ξ|≤2j+1ϕ^j(ξ)g^(ξ)dξ=2−s[kθ+1]kθ+1θ∥ϕ^∥1for j=[kθ+1](k=1,2,⋯),≤2−s[kθ+1]2nkθ+1θ∥ϕ^∥1for j=[kθ+1]±1(k=1,2,⋯),=0for j∈Z∖⋃k=1,2,⋯{[kθ+1],[kθ+1]±1}.

   For any N = 1, 2, ⋯, we take k_N ∈ ℕ such that kNθ+1≤N<(kN+1)θ+1. Then, it holds that

   ∑|j|≤N2jsθ∥ϕj∗g∥∞θ≥∑1≤k≤kN2s[kθ+1]θ∥ϕ[kθ+1]∗g∥∞θ=C1∑1≤k≤kNkθ+1≥C1kNθ+2≥C1(kN+1)θ+2≥C1Nθ+2θ+1,

   where C₁ is dependent only on n and θ. Hence, we have

   ∥f∥V˙∞,∞,θs=supN=1,2,⋯(1N∑|j|≤N2jsθ∥ϕj∗f∥∞θ)1θ≥supN=1,2,⋯C11θN1θ(θ+2θ+1−1)=∞. (2.4)

   On the other hand, it follows that

   max|j|≤N2js∥ϕj∗g∥∞≤max1≤k≤kN+1maxj=[kθ+1],[kθ+1]±12js∥ϕj∗g∥∞≤max1≤k≤kN+1maxj=[kθ+1],[kθ+1]±12js2−s[kθ+1]2nkθ+1θ∥ϕ^∥1≤C2max1≤k≤kN+1kθ+1θ=C2(kN+1)θ+1θ≤C2kNθ+1θ≤C2N1θ,

   where C₂ is dependent only on n and s. Thus, we obtain

   ∥g∥U˙∞,1/θ,∞s=supN=1,2,⋯max|j|≤N2js∥ϕj∗f∥∞N1θ≤supN=1,2,⋯C21θN1θN1θ<∞. (2.5)

   From (2.4) and (2.5), we get g∈U˙∞,1/θ,∞s∖V˙∞,∞,θs.

Theorem 2.5

1. Let s₀, s₁, s₂ ∈ ℝ satisfy s₁ < s₀ < s₂, let 0 ≤ β < ∞ and 1 ≤ p, σ ≤ ∞. Then there exists a positive constant C depending only on s₀, s₁, s₂, but not on p, β, σ such that

   ∥f∥B˙p,σs0≤C1+∥f∥U˙p,β,σs0logβ⁡(e+∥f∥B˙p,∞s1∩B˙p,∞s2) (2.6)

   for all f∈B˙p,∞s1∩B˙p,∞s2.

2. Let s₀ ∈ ℝ, 0 ≤ β < ∞ and 1 ≤ p, σ ≤ ∞, and let X be a normed space of distributions on 𝓩. Assume that X satisfies the following conditions:

   1. X ↪ 𝓩^′;

   2. there exists a constant K₁ > 0 such that

      ∥f(⋅−y)∥X≤K1∥f∥X for all f∈X and all y∈Rn;

      there exists a constant K₂ > 0 such that

      ∥ρ∗f∥X≤K2∥ρ∥1∥f∥X for all ρ∈Z  and all f∈X;

      there exist s₁, s₂ ∈ ℝ satisfy s₁ < s₀ < s₂ and K₃ > 0 such that

      ∥f∥B˙p,σs0≤K31+∥f∥Xlogβ⁡(e+∥f∥B˙p,∞s1∩B˙p,∞s2) for all f∈X∩Z.

   Then, X ↪ U˙p,β,σs0 holds.

Remark 2.6

1. In the first part of Theorem 2.5, the assumption s₁ < s₀ < s₂ is essential. If either of s₁ or s₂ tends to s₀, then the constant C appearing on the right hand side diverges to infinity.

2. By Proposition 2.3 (ii), we observed that the following continuous embeddings hold for s₁ < s₀ < s₂ and β ≥ 0:

   B˙p,∞s1∩B˙p,∞s2⊂B˙p,σs0⊂U˙p,β,σs0.

   Thus, (2.6) may be regarded as an interpolation inequality.

3. From Theorem 2.5 (i), we see that U˙p,q,θs0 satisfies conditions (C1)-(C4). Hence, Theorem 2.5 (ii) implies that U˙p,q,θs0 is the weakest normed space that satisfies (C1)-(C4).

4. By Proposition 2.3 (iii), we see that Theorem 2.5 covers the result given by the author [12]. Indeed, by setting β=1θ−1q,σ=θ(1≤q≤∞,1≤θ≤q) in (2.6), it holds that

   ∥f∥B˙p,θs0≤C1+∥f∥V˙p,q,θs0log1θ−1q⁡(e+∥f∥B˙p,∞s1∩B˙p,∞s2)

   for all f∈B˙p,∞s1∩B˙p,∞s2.

Definition 2.7

Let s > n/2 – 1 and let u₀ ∈ Hσs . A measurable function u on ℝⁿ × (0, T) is called a strong solution to (N-S) in the class CL_s(0, T) if

1. u ∈ C([0, T); Hσs ) ∩ C¹((0, T); Hσs ) ∩ C((0, T); Hσs+2 );

2. u satisfies (N-S) with some distribution π such that ∇ π ∈ C((0, T); H^s).

Remark 2.8

For s > n/2 – 1, the existence of a strong solution to (N-S) in the class CL_s(0, T) has been proven in Fujita-Kato [9], Kato [14] and Giga [10].

Theorem 2.9

1. Let 0 < α < 1, s > n/2 – α and let u₀ ∈ Hσs . Assume that u is a strong solution to (N-S) in the class CL_s(0, T). If the solution u satisfies

   ∫0T∥u(t)∥U˙∞,1/θ,∞−αθdt<∞,2θ+α=1, (2.7)

   then u can be extended to a strong solution to (N-S) in the class CL_s(0, T^′) for some T^′ > T.

2. Let s > n/2 and let u₀ ∈ Hσs . Assume that u is a strong solution to (N-S) in the class CL_s(0, T). If the solution u satisfies

   ∫0T∥u(t)∥V˙∞,∞,202dt<∞, (2.8)

   then u can be extended to a strong solution to (N-S) in the class CL_s(0, T^′) for some T^′ > T.

Remark 2.10

1. Let 0 < α < 1. As is mentioned Example 2.4, we have proper embeddings B˙∞,∞−α ⊂ V˙∞,∞,θ−α ⊂ U˙∞,1/θ,∞−α and hence Thorem 2.9 (i) covers the extension criterion in B˙∞,∞−α given by Kozono-Shimada [17] for s > n/2 – α. Indeed, if the solution u satisfies either

   ∫0T∥u(τ)∥B˙∞,∞−αθdτ<∞,2θ+α=1,

   or

   ∫0T∥u(τ)∥V˙∞,∞,θ−αθdτ<∞,2θ+α=1,

   then the estimate (2.8) is easily obtained, so that the solution can be extended beyond t = T.

2. From Example 2.4, the proper embeddings B˙∞,∞0⊂V˙∞,∞,20⊂U˙∞,1/2,∞0 hold. Hence, Theorem 2.9 (ii) may be regarded as an extension of the B˙∞,∞0 -criterion given by Kozono-Ogawa-Taniuchi [16] for s > n/2. On the other hand, it seems to be difficult to obtain the same result as in Theorem 2.9 (ii) under the condition

   ∫0T∥u(τ)∥U˙∞,1/2,∞02dτ<∞.

   This stems from inapplicability of Lemma 4.1 with α = 0.

Corollary 2.11

1. Let 0 < α < 1, s > n/2 – α and let u₀ ∈ Hσs . Assume that u is a strong solution to (N-S) in the class CL_s(0, T). If T is maximal, i.e., u cannot be extended in the class CL_s(0, T^′) for any T^′ > T, then it holds that

   ∫0T∥u(t)∥U˙∞,1/θ,∞−αθdt=∞,2θ+α=1.

   In particular, we have lim supt→T∥u(t)∥U˙∞,1/θ,∞−α=∞.

2. Let s > n/2 and let u₀ ∈ Hσs . Assume that u is a strong solution to (N-S) in the class CL_s(0, T). If T is maximal, then it holds that

   ∫0T∥u(t)∥V˙∞,∞,202dt=∞.

   In particular, lim supt→T∥u(t)∥V˙∞,∞,20=∞.