Well-posedness and ill-posedness of the stationary Navier–Stokes equations in toroidal Besov spaces

Abstract

We consider the stationary Navier-Stokes equations in the n-dimensional torus T n for n 3. We show the existence and uniqueness of solutions in homogeneous toroidal Besov spacesWe can show its well-posedness by a similar method to that of Kaneko-Kozono-Shimizu (Indiana Univ. Math. J.), which has investigated the same problem in homogeneous Besov spaces on R n . Our advantage is to prove the ill-posedness in the critical exponents like p = n, 2 < q ∞ and n < p ∞, 1 q ∞. Indeed in such cases of p and q, there exists a sequence of external forces which converges to zero in Ḃ−3+ n p p,q (T n ) and yields a sequence of solutions which does not converge to zero in Ḃ−1 ∞,∞ (T n ). We can show this ill-posedness by constructing the sequence of external forces, as similar to those of initial data proposed by Yoneda (2010 J. Funct. Anal. 258 3376-87) in the non-stationary problem.

Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces

“…Recently, Tsuruni gave a partial answer to the above problem. More precisely, Tsuruni [12] proved the ill-posedness of (SNS) in R d (see [13] for the Torus case T d ) except p = d and 1 ≤ q ≤ 2, namely, Theorem 1.2 (see [12])…”

The two‐dimensional stationary Navier–Stokes equations in toroidal Besov spaces

“…Then by (8), we obtain the estimate ( 12) from ( 15), (17), and (20). Now let us show the existence of 𝑟 0 , 𝑟 1 , 𝑟 2 , 𝑠 0 , 𝛼, and 𝜂 satisfying all of ( 13), ( 14), ( 16), (18), and (19). First of all, by (19) (the second sentence and the third one), an exponent 𝛼 is expressed by…”

“…Before stating our main results, we should define some function spaces on the torus. For definitions and properties of such spaces, this paper refers to the author's previous one [19].…”

“…where 𝐶 denotes a constant depending only on 𝑝, 𝑞, and 𝑛. (4), ( 5), and ( 6) yield the following propositions (for details, see [19]).…”

“…Following their idea, we define the homogeneous space Ḃ𝑠 𝑝,𝑞 (𝕋 𝑛 ) so that we may apply a similar argument of Kaneko-Kozono-Shimizu [11] to (SNS). Here, we notice that the author [19] investigated the solutions of (SNS) in Ḃ𝑠 𝑝,𝑞 (𝕋 𝑛 ) with 𝑛 ≥ 3.…”