Blow-Up Criterion and Examples of Global Solutions of Forced Navier-Stokes Equations
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Blow-Up Criterion and Examples of Global Solutions of Forced Navier-Stokes Equations Di Wu1
Received: 15 September 2019 / Accepted: 16 March 2020 © Springer Nature B.V. 2020
Abstract In this paper we first show a blow-up criterion for solutions to the Navier-Stokes equations with a time-independent force by using the profile decomposition method. Based on the orthogonal properties related to the profiles, we give some examples of global solutions to the Navier-Stokes equations with a time-independent force, whose initial data are large. Keywords Navier-Stokes equation · Besov class · Long-time behavior · Regularity
1 Introduction We consider the incompressible Navier-Stokes equations with a time independent external force in R3 , ⎧ ⎨ ∂t uf − uf + uf · ∇uf = f − ∇p, ∇ · u = 0, (N Sf ) ⎩ uf |t=0 = u0 . Here uf is a three-component vector field uf = (u1,f , u2,f , u3,f ) representing the velocity of the fluid, p is a scalar denoting the pressure, and both are unknown functions of the space variable x ∈ R3 and of the time variable t > 0. Finally f = (f1 , f2 , f3 ) denotes a given timeindependent external force defined on R3 . We recall the Navier-Stokes scaling: ∀λ > 0, the vector field uf is a solution to (N Sf ) with initial data u0 if uλ,fλ is a solution to (N Sfλ ) with initial data u0,λ , where uλ,fλ (t, x) := λuf (λ2 t, λx), fλ (t, x) := λ3 f (λ2 t, λx), pλ (t, x) := λ2 p(λ2 t, λx) and u0,λ := λu0 (λx).
B D. Wu
[email protected]
1
Institute de Mathématiques de Jussieu-Paris Rive Gouche UMR CNRS 7586, Université Paris Diderot, 75205, Paris, France
D. Wu
Spaces which are invariant under the Navier-Stokes scaling are called critical spaces for the Navier-Stokes equation. Examples of critical spaces of initial data for the Navier-Stokes equation in 3D are: −1+ p3
L3 (R3 ) → B˙ p,q
−1 (R3 )(p < ∞, q ≤ ∞) → BMO−1 → B˙ ∞,∞ .
(See below for definitions).
1.1 Blow-Up Problem in Critical Spaces To put our results in perspective, we first recall the Navier-Stokes equations (without external force) blow-up problem in critical spaces. Consider the Navier-Stokes system: ⎧ ⎨ ∂t u − u + u · ∇u = −∇π, ∇ · u = 0, (N S) ⎩ uf |t=0 = u0 where u(t, ·) : R3 → R3 is the unknown velocity field. In the pioneering work [19], J. Leray introduced the concept of weak solutions to (N S) and proved global existence for datum u0 ∈ L2 . However, their uniqueness has remained an open problem. In 1984, T. Kato [15] initiated the study of (N S) with initial data belonging to the space L3 (R3 ) and obtained global existence in a subspace of C([0, ∞), L3 (R3 )) provided the norm u0 L3 (R3 ) is small enough. The existence result for initial data small in the −1+ p3
Besov space B˙ p,q
for p, ∈ [1, ∞) and q ∈ [1, ∞] can be found in [5, 11]. The function
−1+ 3 B˙ p,q p
spaces L (R ) and for (p, q) ∈ [1, ∞)2 both guarantee the existence of local-in-time solution for any initial data. In 2001, H. Koch and D. Tataru [18] showed that global wellposedness holds as well for small initial data in the space BMO−1 . This space consists o
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