Well-Posedness and Ill-Posedness Problems of the Stationary Navier–Stokes Equations in Scaling Invariant Besov Spaces

Tsurumi, Hiroyuki

doi:10.1007/s00205-019-01404-6

Cited by 11 publications

(8 citation statements)

References 9 publications

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“…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])…”

Section: Known Well/ill-posedness Resultsmentioning

confidence: 99%

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

Li¹,

Yu²,

Zhu³

2022

Preprint

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This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in R d is well-posed from Ḃ−2 p,q to P Ḃ0 p,q with p = d and 1 ≤ q < 2. In this paper, we demonstrate that for the case 1 ≤ q < 2 the 4D stationary Navier-Stokes equations is ill-posed from Ḃ−2 4,q (R 4 ) to P Ḃ0 4,q (R 4 ) by showing that a sequence of external forces is constructed to show discontinuity of the solution map at zero. Indeed in such case of q, there exists a sequence of external forces which converges to zero in Ḃ−2 4,q and yields a sequence of solutions which does not converge to zero in Ḃ0 4,q .

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Section: Known Well/ill-posedness Resultsmentioning

confidence: 99%

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

Li¹,

Yu²,

Zhu³

2022

Preprint

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“…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).…”

Section: Proof Of Theorem 31mentioning

confidence: 92%

“…We should note here that Ḃ−3+ 𝑛 𝑝 𝑝,𝑞 (ℝ 𝑛 ) and Ḃ−1+ 𝑛 𝑝 𝑝,𝑞 (ℝ 𝑛 ) are scaling invariant for external forces and velocities in (SNS), respectively. Finally, in this direction, the author [20] showed the ill-posedness of (SNS) when (𝑝, 𝑞) ∈ ({𝑛} × (2, ∞]) ∪ ((𝑛, ∞] × [1, ∞]), in the sense that there exist solutions, which never continuously depend on given external forces.…”

Section: Introductionmentioning

confidence: 99%

“…We should note here that

\dot{B}^{-3+\frac{n}{p}}_{p,q}(\mathbb {R}^n)

and

\dot{B}^{-1+\frac{n}{p}}_{p,q}(\mathbb {R}^n)

are scaling invariant for external forces and velocities in (SNS), respectively. Finally, in this direction, the author [20] showed the ill‐posedness of (SNS) when

(p,q)\in (\lbrace n\rbrace \times (2,\infty ])\cup ((n,\infty ]\times [1,\infty ])

, in the sense that there exist solutions, which never continuously depend on given external forces.…”

Section: Introductionmentioning

confidence: 99%

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

Tsurumi

2023

Mathematische Nachrichten

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We consider the stationary Navier–Stokes equations in the two‐dimensional torus double-struckT2$\mathbb {T}^2$. For any ε>0$\varepsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces Ḃp+ε,q−1+2p(double-struckT2)$\dot{B}^{-1+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ for given small external forces in Ḃp+ε,q−3+2p(double-struckT2)$\dot{B}^{-3+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ when 1≤p<2$1\le p <2$. These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case false(p,qfalse)∈false(false{2false}×false(2,∞false]false)∪false(false(2,∞false]×false[1,∞false]false)$(p,q)\in (\lbrace 2\rbrace \times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from Ḃp,q−3+2p(double-struckT2)$\dot{B}^{-3+\frac{2}{p}}_{p ,q}(\mathbb {T}^2)$ to Ḃp,q−1+2p(double-struckT2)$\dot{B}^{-1+\frac{2}{p}}_{p, q}(\mathbb {T}^2)$.

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“…For more background of this model and related results, we refer to [12,13]. Many results with regard to the ill-posedness have been obtained for some important nonlinear PDEs including the incompressible Navier-Stokes equations [2,15,17], the stationary Navier-Stokes equations [11,14], the compressible Navier-Stokes equations [3,5] and so on.…”

Section: Introductionmentioning

confidence: 99%

Ill-posedness issue on a multidimensional chemotaxis equations in the critical Besov spaces

Li¹,

Yu²,

Zhu³

2022

Preprint

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Well-Posedness and Ill-Posedness Problems of the Stationary Navier–Stokes Equations in Scaling Invariant Besov Spaces

Cited by 11 publications

References 9 publications

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

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

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

Ill-posedness issue on a multidimensional chemotaxis equations in the critical Besov spaces

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