Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics

We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the $N$-dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl.


Introduction
The evolution of the velocity u = u(t, x) and pressure P = P (t, x) fields of a perfect homogeneous incompressible fluid is governed by the following Euler equations: There is a huge literature concerning the well-posedness issue for Euler equations. Roughly, they may be solved locally in time in any reasonable Banach space embedded in the set C 0,1 of bounded Lipschitz functions (see e.g. [1,4,6,12,13,17,19,22]).
In the two-dimensional case, it is well known that Euler equations are globally well-posed for sufficiently smooth initial data. This noticeable fact relies on the conservation of the vorticity ω := ∂ 1 u 2 − ∂ 2 u 1 along the flow of the velocity field, and has been first proved rigorously in the pioneering works by W. Wolibner [20] and V. Yudovich [21].
This conservation property is no longer true, however, in more physically relevant contexts such as (1) the three-dimensional setting for (0.1), (2) nonhomogeneous incompressible perfect fluids, (3) inviscid fluids subjected to a buoyancy force which is advected by the velocity fluid (the so-called inviscid Boussinesq system below). As a consequence, the problem of global existence for general (even smooth or small) data is still open for the above three cases.
In a recent work [9], it has been shown that for slightly nonhomogeneous two-dimensional incompressible fluids, the lifespan tends to infinity when the nonhomogeneity tends to zero. The present paper is mainly dedicated to the study of the lifespan for the first and third item.
More precisely, in the first section of the paper, we shall consider the inviscid Boussinesq system: ∂ t u + u · ∇u + ∇P = θe N , div u = 0.
Here the relative temperature θ = θ(t, x) is a real valued function 1 and e N stands for the unit vertical vector.
2010 Mathematics Subject Classification. 35Q35,76B03. 1 It need not be nonnegative as it designates the discrepancy to some reference temperature.
As for the standard incompressible Euler equations, any functional space embedded in C 0,1 is a good candidate for the study of the well-posedness issue for (0.2). This stems from the fact that System (0.2) is a coupling between transport equations. Hence preserving the initial regularity requires the velocity field to be at least locally Lipschitz with respect to the space variable. By arguing as in [1], Chap. 7, one may show that, indeed, System (0.2) is locally well-posed in B s p,q whenever B s p,q is embedded in C 0,1 or, in other words, for any (s, p, q) ∈ R × [1, +∞] 2 satisfying As a by-product of estimates for transport equations, we shall get various continuation criteria which generalize those of [11] and of [16]. We shall finally establish lower bounds for the lifespan of the solutions to (0.2) which show that in the two-dimensional case and for small initial temperature, the solution tends to be global-in-time.
As pointed out in many works (see e.g. [11]), there is a formal similarity between the twodimensional Boussinesq system and general axisymmetric solutions to the three-dimensional Euler system -the so-called axisymmetric solutions with swirl. In the second part of this paper, we adapt the method of the first part so as to establish new lower bounds for the lifespan to those solutions in the case where the swirl is small. In particular, we find out that the solution tends to be global if the swirl goes to zero.
In the Appendix, we briefly recall the definition and a few basic properties of Besov space, and prove a commutator estimate.
Before going further into the description of our results, let us introduce a few notation.
• Throughout the paper, C stands for a harmless "constant" the meaning of which depends on the context. • The vorticity ω associated to a vector field u over R N is the matrix valued function with entries If N = 2 then the vorticity is identified with the scalar function ω := ∂ 1 u 2 − ∂ 2 u 1 and if N = 3, with the vector field ∇ × u. • For all Banach space X and interval I of R, we denote by C(I; X) the set of continuous functions on I with values in X. If X has predual X * then we denote by C w (I; X) the set of bounded measurable functions f : I → X such that for any φ ∈ X * , the function t → f (t), φ X×X * is continuous over I.

The inviscid Boussinesq system
This section is devoted to the well-posedness issue for the inviscid Boussinesq system (0.2). We first establish a local-in-time existence result and continuation criteria in the spirit of those for the incompressible Euler equation. Next, we provide a new lower bound for the lifespan. Roughly, we establish that if θ 0 is of order ε, then the lifespan is at least of order log | log ε|.
Before proving this result, a few comments are in order.
1. If it is assumed that ω 0 ∈ L r instead of u 0 ∈ L r then the vorticity of the constructed solution is continuous in time with values in L r . 2. In the two-dimensional case and in the Hölder spaces framework, the above statement has been established in [7]. The critical Besov case (that is p = 1 + 2/p, p ∈]1, ∞[) has been investigated in [16]. 3. In [11], a continuation criterion involving the L ∞ norm of the vorticity only has been stated. However, as the first inequality below (7) therein fails if m ≥ 2, we do not know whether that criterion is correct. 4. The first item has been proved recently in [16] in the two dimensional case. 5. Let us finally mention that one may replace ω L ∞ with ω Ḃ0 ∞,∞ ∩L r in the second criterion.
Proof of Theorem 1.1. The proof of the local well-posedness in the Besov spaces framework is a straightforward adaptation to that of the corresponding result for the Euler system in B s p,q , and is thus omitted. The reader may refer to [1], Chap. 7 for more details.
So let us go for the proof of the continuation criteria. Let us first assume that 1 < p < ∞. In this case, the Marcinkiewicz theorem for Calderon-Zygmund operators ensures that Therefore, decomposing ω into low and high frequencies as follows 3 : and taking advantage of the remark that follows Proposition A.1 in the appendix, we gather that Hence applying ∆ j to the vorticity equation yields with ω j := ∆ j ω and θ j := ∆ j θ. Therefore, because div u = 0, Next, let us use (see the appendix) that whenever s > 0. 3 The notation ∆−1 is defined in the appendix If s > 1 + N/p then standard tame estimates (see e.g. [1], Chap. 2) imply that The last inequality remains true in the limit case s = 1 + N/p and q = 1, a consequence of the algebraic structure of A(∇u, ω) (see e.g. Inequality (52) in [9]).
Hence, multiplying (1.3) by 2 j(s−1) , taking the ℓ q norm with respect to j and taking advantage of (1.2) yields Given that, according to (a slight modification of) Lemma 2.100 of [1], we have Finally, from the equation for θ, we easily get where P stands for the Leray projector over divergence-free vector-fields. As it is continuous over L p (recall that 1 < p < ∞), we deduce that Now, the standard continuation criterion for hyperbolic PDEs ensures that the solution (θ, u) may be continued beyond T.
Let us now treat the case where s > 1 + N/p and We first bound ω and ∇θ in L ∞ ([0, T [; L p ) by taking advantage of (1.1) and of the vorticity and temperature equations. We get So Gronwall's lemma provides us with a bound for ω and ∇θ in L ∞ ([0, T [; L p ).
Next, we use the following classical logarithmic interpolation inequality (see e.g. [1]): Plugging this inequality in (1.5) and (1.8), and summing up, we get Let us finally assume that N = 2 and that T 0 ∇θ L ∞ dt < ∞.
Then Equation (1.16) gives Hence ω ∈ L ∞ ([0, T [×R 2 ) and the previous continuation criterion implies that the solution (θ, u) may be continued beyond T.
Let us end the proof with a few comments concerning the cases p = 1, ∞. If p = ∞ and the solution also satisfies (∇θ, ω) ∈ L ∞ ([0, T [; L r ) for some r ∈]1, ∞[, then arguing as for proving (1.2) yields ∇u B s−1 ∞,q ∩L r ≤ C ω B s−1 ∞,q ∩L r . From the vorticity and temperature equations, we get So one may conclude that (1.5) and (1.8) hold true if replacing the norm in B s−1 ∞,q by the norm in B s−1 ∞,q ∩ L r . In order to bound u B s ∞,q , one may write that (using Bernstein's inequality to get the second line), ∞,q . Now, according to (1.10), we have From this, it is easy to complete the proof.
Finally, if p = 1 then embedding ensures that ∇θ and u are in L ∞ ([0, T [; L r ) for some finite r, so that one may conclude as in the case p = ∞.
The above result is, obviously, independent of the dimension. At the same time, in the case θ 0 ≡ 0 (corresponding to the incompressible Euler equation) global existence holds true in dimension 2. In the case θ 0 ≡ 0, the question of global existence has remained unsolved, even in the two-dimensional case. We here want to study whether, nevertheless, dimension 2 is somehow "better". To answer this question, we shall take advantage of the fact that the vorticity equation in dimension 2 has no stretching term: it reduces to (1.16) ∂ t ω + u · ∇ω = ∂ 1 θ.
Hence, taking advantage of the special a priori estimates for the transport equation in Besov spaces with null regularity index (as discovered by M. Vishik in [19] and by T. Hmidi and S. Keraani in [13]), one may write This will be the key to our result below.
Theorem 1.2. Assume that N = 2. Let (θ 0 , u 0 ) be in B s p,q with (s, p, q) satisfying (0.3). If p ∈ {1, +∞}, suppose in addition that (∇θ 0 , ω 0 ) ∈ L r for some 1 < r < ∞. There exists a constant C depending only on r and such that (setting p = r if p ∈ (1, +∞)), the lifespan T * of (0.2) satisfies Proof. Let us first notice that, according to the continuation criteria derived in Theorem 1.1, it suffices to show that if the solution is defined on [0, T [×R n with T ≤ T * and T * as above, then ω and ∇θ are bounded in L ∞ (0, T ; B 0 ∞,1 ∩ L r ). Now, estimates for the transport equation in Besov spaces (see e.g. [1], Chap. 3) yield Of course, standard L r estimates for the transport equation imply that Let us finally notice that putting together embedding, Inequality (1.1) and the remark that follows Proposition A.1, we have ∞,1 ∩L r and taking advantage of (1.17), (1.18), (1.19), we conclude that Now, plugging the inequality for Θ(t) in the inequality for Ω(t), we get Let us assume for a while that (1.20) and Gronwall's lemma imply that Therefore, for Condition (1.21) to be satisfied, it suffices that Hence Inequality (1.23) is satisfied provided that exp 2(e X − 1) ≤ 1 + Y.
So we easily gather from a bootstrap argument that the lifespan T * satisfies which is exactly the desired inequality.
Remark 1.3. In the case where the solution is C 1,r for some r ∈ (0, 1) (an assumption which is not satisfied in the critical regularity case), one may first write estimates for ω L ∞ and ω C r , and next use the classical logarithmic inequality for bounding ∇u L ∞ in terms of ω L ∞ and ω C r . This does not improve the lower bound for the lifespan, though.

The axisymmetric incompressible Euler equations
We now consider the incompressible Euler equations (0.1). As recalled in the introduction, Euler equations are globally well-posed in dimension 2. In dimension d ≥ 3, the global wellposedness issue has remained unsolved unless some property of symmetry is satisfied : it is known that axisymmetric or helicoidal without swirl data generate global solutions (see e.g. [8] and the references therein for more details).
In the general case, an easy scaling argument similar to that of the Boussinesq system yields that for data of size ε, the lifespan is at least of order ε −1 .
Here we want to focus on the axisymmetric solutions to Euler equations with swirl, that is on solutions u to (0.1) such that, in cylindrical coordinates, (2.1) u(r, z) = u r (r, z)e r + u θ (r, z)e θ + u z (r, z)e z .
Recall that the corresponding vorticity reads ω(r, z) = ω r (r, z)e r + ω θ (r, z)e θ + ω z (r, z)e z with With this notation, axisymmetric solutions satisfy (see e.g. [4]) As pointed out in [11], there is a striking similarity between the two-dimensional Boussinesq system (0.2) satisfied by (θ, ω) in the previous section, and the equations satisfied by (u θ , ω θ ) here. Indeed, Therefore, up to the singular coefficient 1/r 4 , the functions Γ = Γ(r, z) and ζ = ζ(r, z) play the same role as the temperature and the vorticity, respectively, in the 2D Boussinesq system. Keeping in mind that data such that u θ 0 ≡ 0 generate global solutions, it is natural to study whether having r −1 ω θ 0 = O(1) and ru θ 0 = O(ε) gives rise to a family of solutions with lifespan going to infinity when ε goes to 0.
For technical reasons however, due to the singularity near the axis, we shall consider the axisymmetric Euler equations in a smooth bounded axisymmetric domain Ω of R 3 such that, for some given 0 < r 0 < R 0 , Proof. This statement has been essentially proved by A. Dutrifoy in [10] except in the critical case s = 1 + 3/p and r = 1. However, the critical case may be handled by the same method 4 as it relies on a priori estimates for transport equations which are also true in this case.
The last part of the statement is a classical consequence of the uniqueness and of the symmetry of the data u 0 .
One can now state the main result of this part.
Theorem 2.2. Let u 0 be an axisymmetric divergence-free vector-field in B s p,q (Ω) with (s, p, q) satisfying (0.3) and Ω a bounded domain satisfying (2.4). Suppose in addition that u 0 | ∂Ω is tangent to the boundary of Ω. Then the lifespan T * to the solution of (0.1) satisfies for some constant C depending only on Ω. 4 Proving a continuation criterion involving the vorticity was the main purpose of Dutrifoy's paper, and this requires that s > 1 + 3/p. This is probably the reason why the statement in the critical case is not given therein.
Proof. It suffices to bound the norm of u in B 1 ∞,1 as it controls high norms (see [10] and notice that B 1 ∞,1 embeds in C 0,1 ). Let u := u r e r +u z e z . Denote by ψ the solution given by Proposition A.4 to the elliptic equation Notice that div u = 0 = div (∇ ∧ ψ) and that ∇ ∧ u = ω θ e θ = ∇ ∧ (∇ ∧ ψ).
As, in addition, both u and ∇∧ ψ have null circulation on the components of ∂Ω (a consequence of the symmetry properties of those two functions and of the domain), they coincide. Hence, Proposition A.4 ensures that 1 . This inequality will enable us to adapt to the axisymmetric Euler equations the proof of lower bounds for the lifespan of solutions.
We proceed as follows. According to the work by A. Dutrifoy (see in particular Prop. 6 and Cor. 5 in [10]) for the transport equation in a smooth bounded domain, estimates in Besov spaces B s p,q (Ω) are the same as in the whole space case. From this, one may deduce by following the method of [13] that in the particular case s = 0, the estimates improve (as in (1.17)). So we get, bearing (2.3) in mind: General Dutrifoy's estimates for the transport equation also imply that Now, the important observation is that 1/r 4 is in C 0,1 (Ω) (for r ≥ r 0 in Ω). Hence Finally, according to (2.5) and classical embedding properties, we have As ω θ e θ = ζ re θ and, under our assumption on Ω, re θ is in C 0,1 , one may thus conclude that From this point, one may proceed exactly as for the Boussinesq system; we deduce the following lower bound for the lifespan of the solution: Of course, owing to the shape of Ω, up to an irrelevant constant, one may replace ζ 0 with ω θ 0 and ru θ 0 with u θ 0 , respectively. Remark 2.3. We believe Theorem 2.2 to be true in the case where Ω satisfying (2.4) is unbounded. However, we refrained from giving the statement as we did not find in the literature the counterpart of Theorem 2.1 and of Proposition A.4. Let us emphasize however that unbounded domains have been considered in [2] (Hölder spaces), and [14,15] (weighted Sobolev spaces). By following Dutrifoy's approach, we do not see any obstruction to get similar results in the Besov space framework. This is only a matter of having suitable extension operators available for the domain considered.
We also believe that Proposition A.4 may be extended to unbounded domains provided we prescribe some condition at infinity: the following inequality (2.6) ∇ u B 0 ∞,1 ∩L r ≤ C ω θ e θ B 0 ∞,1 ∩L r . for any r ∈]1, +∞[ seems to be reasonable. However, as proving such inequalities is not the point of this paper, we restricted ourselves to bounded domains.
Note that F(∆ j ω) is supported in an annulus of size 2 j . Hence Bernstein's Inequality ensures that R 2 so that we get if s > 0, R 2 j L p c j 2 −j(s−1) ω L ∞ u B s p,q . As for R 3 j , standard continuity results for the paraproduct operator (see e.g. [1], Chap. 2) imply that R 3 j L p c j 2 −j(s−1) ω L ∞ u B s p,q . For R 4 j , one may write that Hence, in view of Bernstein inequality, So we get R 4 j L p c j 2 −j(s−1) ω L ∞ u B s p,q . Next, standard continuity results for the remainder operator yield if s > 0, ∂ k R( u k , ω) B s−1 p,q ω L ∞ u B s p,q . Hence R 5 j L p c j 2 −j(s−1) ω L ∞ u B s p,q . Finally, let us notice that the operator ω → (Id − ∆ −1 )u satisfies the hypothesis of the last item of Proposition A.1 with m = −1, hence u B s p,q ω B s−1 p,q . So putting all the above inequalities together completes the proof of (1.4).