Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^d$, $d=2,3$, with initial data $B_0\in H^s(\mathbb{R}^d)$ and $u_0\in H^{s-1+\varepsilon}(\mathbb{R}^d)$ for $s>d/2$ and any $0<\varepsilon<1$. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking $\varepsilon=0$ is explained by the failure of solutions of the heat equation with initial data $u_0\in H^{s-1}$ to satisfy $u\in L^1(0,T;H^{s+1})$; we provide an explicit example of this phenomenon.


Introduction
In this paper we consider the equations of MHD with zero magnetic resistivity, u t − ∆u + (u · ∇)u + ∇p = (B · ∇)B, ∇ · u = 0, (1.1a) B t + (u · ∇)B = (B · ∇)u, ∇ · B = 0, (1.1b) along with specified initial data u(0) = u 0 and B(0) = B 0 . Jiu & Niu (2006) established the local existence of solutions in 2D for initial data in H s for integer s ≥ 3, and in a previous paper (Fefferman et al., 2014) we proved a local existence result for these equations when u 0 , B 0 ∈ H s (R d ) with s > d/2 for d = 2, 3. However, given the presence of the diffusive term in the equation for u, it is natural to expect that such a local existence result should be possible with less regularity for u 0 than for B 0 .
To this end, it was shown in Chemin et al. (2016) that one can prove local existence when B 0 ∈ B d/2 2,1 (R d ) and u 0 ∈ B d/2−1 2,1 (R d ). The underlying observation that allowed for such a result is that when u 0 ∈ B d/2−1 2,1 (R d ) the solution u of the heat equation with initial data u 0 is an element of L 1 (0, T ; B d/2+1 2,1 (R d )): in these Besov spaces the solution regularises sufficiently that an additional two derivatives become integrable in time.
In Sobolev spaces this does not occur: in Lemma 2.1 we show that for for any 0 < q < 1, and this is in some sense the best possible. The failure of the estimate u ∈ L 1 (0, T ; H s+2 ) is 'well known', but it is not easy to find any explicit example in the literature, so we provide one here in Lemma 2.2. In this paper we therefore take B 0 ∈ H s (R d ), with s > d/2, and u 0 slightly more regular than H s−1 (R d ), namely u 0 ∈ H s−1+ε (R d ), with 0 < ε < 1. By making use of maximal regularity results for the heat equation (which we recall in the next section) we are then able to prove the local existence of a solution that remains bounded in these spaces (see Theorem 3.1 for a precise statement).
Throughout the paper we use the notation Λ s to denote the fractional derivative of order s, given in terms of the Fourier transform by We write u 2 H s = Λ s u 2 + u 2 , s > 0, which is equivalent to the standard H s norm when s is a positive integer.

Estimates for solutions of the heat equation in Sobolev spaces
2.1. Energy estimates. First we prove some standard estimates for solutions of the heat equation, including the L q (0, T ; H s+2 ) estimate for 0 < q < 1. We give the proofs, since we will need to keep careful track of the dependence of the estimates on T ; for simplicity we restrict to T ≤ 1, which will of course be sufficient for local existence arguments.
Lemma 2.1. If u 0 ∈ H s (R d ) and u denotes the solution of the heat equation since 0 < q < 1 ensures that q/(2 − q) < 1 and the first term is integrable. 2.
Lemma 2.2. There exists u 0 ∈ L 2 (R d ) such that the solution u of the heat equation with initial data u 0 is not an element of L 1 (0, T ; H 2 (R d )).
Proof. We let u 0 be the function with Fourier transform The solution u(t) of the heat equation with initial data u 0 has Fourier transform We therefore have In order to bound this from below we split the range of time integration, choosing j 0 such that j −2 0 ≤ T , and write

It follows that
as since the sum is unbounded as N → ∞ it follows that u / ∈ L 1 (0, T ;Ḣ 2 ) as claimed.
2.3. Maximal regularity-type results. Usually, 'maximal regularity' results for the heat equation yield The results follows from maximal regularity when p = q, obtained as inequality (3.1) in Chapter IV, Section 3, pages 289-290 of Ladyzhenskaja et al. (1968) using the Mihlin Multiplier Theorem (Mihlin, 1957) and then an interpolation theorem due to Benedek et al. (1962); the procedure is clearly explained in Krylov (2001), for example. However, in this paper we will only require the following L 2 -based Sobolev-space result, for which the basic L 2 (0, T ; L 2 ) maximal regularity estimate can be obtained relatively easily.
The constant C r can be chosen uniformly for all 0 ≤ T ≤ 1.
Proof. First we treat the case s = 0, i.e. we bound u ∈ L r (0, T ; H 2 ) in terms of f ∈ L r (0, T ; L 2 ). The passage from estimates for the case r = 2 to the case 1 < r < ∞ is covered by Krylov (2001). We therefore only prove the estimates in the case r = 2.
The L 2 norm of u is bounded simply by taking the L 2 inner product with u, using the Cauchy-Schwarz inequality, and integrating in time, 1 2 i.e.
For u Ḣ2 we can argue directly from the Fourier transform of the solution u, since and we can use Young's inequality for convolutions to give since G(ξ, ·) L 1 = 1. Therefore Combined with (2.6) this yields u L 2 (0,T ;Ḣ 2 ) ≤ C 2 f L 2 (0,T ;L 2 ) , and hence (via the results of Benedek et al. (1962)) we obtain u L r (0,T ;Ḣ 2 ) ≤ C r f L r (0,T ;L 2 ) .
Letting C r be the constant for the choice T = 1, it can easily be seen that this constant is also valid for T ≤ 1 by extending f to be zero on the interval (T, 1].
We now apply this estimate to u = v and u = Λ s v: we obtain from which (2.5) follows.
We will apply this result in combination with the regularity results for the heat equation from Lemma 2.1 in the following form for solutions of the Stokes equation, allowing for non-zero initial data. Note that in order to obtain an L 1 -in-time estimate on u H s+1 we require L r integrability of f with r > 1, and the initial data to be in H s−1+ε . Considering the equations in Besov spaces as in Chemin et al. (2016) allows r = 1 and ε = 0 (and s = d/2 rather than s > d/2 in the final results).
Corollary 2.4. If f ∈ L r (0, T ; H s−1 ), 1 < r < ∞, s > 1, and where u 0 is divergence free, then for T ≤ 1 Proof. First we consider the solution v of Since u 0 is divergence free, if we apply the Leray projector P (orthogonal projection onto elements of L 2 whose weak divergence is zero) we obtain since P commutes with derivatives on the whole space. It follows that v is in fact the solution of the heat equation with initial data u 0 . We can therefore use Lemma 2.1 to ensure that v ∈ L ∞ (0, T ; H s−1+ε ) ∩ L 2 (0, T ; H s+ε ) ∩ L q (0, T ; H s+ε+1 ) for any q < 1, with all these norms depending only on the norm of the initial data in H s−1+ε . It follows by interpolation that v ∈ L 1 (0, T ; H s+1 ) with , using Hölder's inequality with exponents (2/(2 − ε), 2/ε). Noting that we can use Lemma 2.1 to obtain The difference w = u − v satisfies Again we can apply the Leray projector P, which is bounded from H s−1 into H s−1 , to obtain ∂ t w − ∆w = Pf, w(0) = 0.
By the maximal regularity results for the heat equation from Proposition 2.3, we know that for any r > 1 we have where (r, r ′ ) are conjugate. The inequality in (2.7) now follows by combining (2.8) and (2.9).

Proof of local existence
The main part of the proof consists of a priori estimates, which we prove formally. To make the proof rigorous requires some approximation procedure such as that employed in Fefferman et al. (2014), to which we refer for the details. Where the limiting process involved would turns equalities into inequalities, we write inequalities even in these formal estimates.
Note that the case ε = 1 was covered in a previous paper (Fefferman et al. 2014), and is specifically excluded here.
Proof. Throughout the proof various constants will depend on s, but we do not track this dependency.
We first obtain a basic energy estimate in L 2 . We take the L 2 inner product of the u equation with u and of the B equation with B to obtain 1 2 d dt u 2 + ∇u 2 = (B · ∇)B, u and 1 2 Since (B · ∇)u, B = − (B · ∇)u, B we can add the two equations to yield 1 2 d dt u 2 + B 2 + ∇u 2 ≤ 0 and so It is also helpful to have two other estimates for later use; observing that Substituting this into (3.10) ensures that B(t) H s < 2 B 0 H s for all t ∈ [0, T ], contradicting the maximality of T . It follows that T = T * and hence The result now follows from (3.15), (3.16), and (3.17).

Conclusion
In the scale of Sobolev spaces we suspect that the result that we have proved here is optimal. Bourgain & Li (2015) showed that the Euler equations on R d are ill posed in H 1+d/2 for n = 2, 3, and we have shown via an explicit example that for the heat equation we cannot gain the time integrability of two additional derivatives that is required in our local existence argument. It would be interesting to find a simpler model problem in which it is possible to demonstrate the failure of local existence for B 0 ∈ H s and u 0 ∈ H s−1 .