Abstract
For the two dimensional Euler equations, a classical result by Yudovich states that solutions are unique in the class of bounded vorticity; it is a celebrated open problem whether this uniqueness result can be extended in other integrability spaces. We prove in this note that such uniqueness theorem fails in the class of vector fields u with uniformly bounded kinetic energy and vorticity in the Lorentz space \(L^{1, \infty }\).
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1 Introduction
Let us consider the 2-dimensional Euler equations
where \(u: [0,1] \times {{\mathbb {T}}}^2 \rightarrow {\mathbb {R}}^2\) is the velocity of a fluid and \(p: [0,1] \times {{\mathbb {T}}}^2 \rightarrow {\mathbb {R}}\) the pressure. This system can be equivalently rewritten as the two dimensional Euler system in vorticity formulation, which is a transport equation for the vorticity \(\omega = \textrm{curl} (u)\), i.e.
In the latter formulation, it is clear that \(L^p\) norms of the vorticity are formally conserved for any \(p \in [1,\infty ]\). For \(p>1\), this was used in [11] to prove the existence of distributional solutions starting from an initial datum with vorticity in \(L^p\). A similar existence result is much more involved for \(p=1\), and it was obtained by Delort [10] (see also [11, 12, 22]), improving the existence theory up to measure initial vorticities in \(H^{-1}\) (this latter condition guarantees finiteness of the energy) whose positive (or negative) part is absolutely continuous. As regards uniqueness, the classical result of Yudovich [15, 16] (see also the proof in [17]) states that, given an initial datum \(\omega _0 \in L^\infty \), there exists a unique bounded solution to (2) starting from \(\omega _0\). However, the classical problem raised by Yudovich about the sharpness of his result is still open. Let \(u_0\) be an initial datum in \(L^2\) with \({{\,\textrm{curl}\,}}u_0\) in some function space X. Is the solution of the Euler equations in vorticity formulation unique in the class \(L^\infty (X)\)?
The main result of this paper provides a negative answer when X is the Lorentz space \(L^{1,\infty }\).
Theorem 1.1
There exists a nontrivial solution \(u\in C^0([0,1]; L^2({{\mathbb {T}}}^2))\) to (1) satisfying
-
(i)
\(\omega = {{\,\textrm{curl}\,}}u\in C^0([0,1]; L^{1,\infty }({{\mathbb {T}}}^2))\);
-
(ii)
\(u(0,\cdot ) = 0\).
Moreover, \(u\in C^0([0,1]; W^{s,p}({{\mathbb {T}}}^2))\) for any \(s\in (0,1)\) and \(p\in (1, \frac{2}{1+s})\).
Remark 1.2
The conclusion (i) in Theorem 1.1 has to be interpreted as follows. There exist a function \({{\,\textrm{curl}\,}}u\in C^0([0,1]; L^{1,\infty }({{\mathbb {T}}}^2))\) and a sequence \(u_n\in C^\infty ([0,1]\times {{\mathbb {T}}}^2)\) solving the Euler equations with an error term \(R_n\) in the right hand side (see (3)) such that
as \(n \rightarrow 0\).
Recently, there have been formidable attempts to disprove this conjecture for \(X= L^p\), none of which has by now fully solved it. Vishik [23, 24], see also [1], proposed a complex line of approach to this problem, which however has the price of showing nonuniqueness only with an additional degree of freedom, namely a forcing term in the right-hand side of the equation (2) in the integrability space \(L^1(L^p)\). The nonuniqueness suggested by this work is of symmetry breaking type and, in contrast with the ideas of this paper, his nonuniqueness stems from the linear part of the equation, by carefully choosing an initial datum that sees the instability directions of a linearized operator.
A second attempt has been pursued by Bressan and Shen [2], based on numerical experiments which share the symmetry breaking type of nonuniqueness of Vishik. Their work is a first step in the direction of a computer assisted proof.
Our approach is instead of different nature and stems from the convex integration technique. The latter was introduced by De Lellis and Székelyhidi [9] in the context of nonlinear PDEs, inspired by the work of Nash on isometric embeddings [20], which found striking applications in recent years to different PDEs (see for instance [5,6,7, 14, 18, 19] and the references quoted therein). As such, our proof would probably be less constructive with respect to the strategies of [2, 23, 24], where an initial datum for which nonuniqueness is expected is described fairly explicitly as well as the mechanism for the creation of two different singularities. Conversely, the latter approaches see the drawbacks described above and are by no means “generic” in the initial data, whereas it is known (see for instance [8, 21]) that convex integration methods yield not only the lack of uniqueness/smoothness for certain specific initial data, but also that solutions are typical (in the Baire category sense).
1.1 Strategy of proof
The guiding thread of this construction is an iterative procedure, where one starts from a solution \((u_q, p_q, R_q)\) of the Euler equations with an error term in the right-hand side, namely
and iteratively corrects this error by adding a fastly oscillating perturbation to the approximate solution. The nonlinear interaction of this perturbation with itself generates a resonance which allows for the cancellation of the previous error; the other terms are mainly seen as new error terms, with smaller size with respect to the previous error. More precisely, we define the new solution \((u_{q+1}, p_{q+1}, R_{q+1})\) by setting
where \(\lambda _{q+1} \gg \lambda _q\) is a higher frequency with respect to the typical frequencies in \(u_q\), \(w_{q+1}\) is called building block of the construction and enjoys suitable integrability properties, \(a_q\) is a slowly varying coefficient.
The cancellation of error happens because the low frequency term in \(a^2_q w_{\lambda _{q+1}} \otimes w_{\lambda _{q+1}}\) satisfies
This forces us to require that
On the contrary, we wish to control the quantity \(\Vert D u_q\Vert _{L^{1, \infty }}\) and to this end we need
arbitrarily small. This will be achieved by designing a new family of intermittent building blocks with gradients small in Lorentz spaces, see Sect. 1.3 below.
1.2 Limitations of current convex integration schemes
We now justify why, with the current method, getting a vorticity in \(L^p\), \(p \ge 1\), cannot be expected. The new error \(R_{q+1}\), generated after correcting the old error \(R_q\) with the low frequency term in \(a^2_q w_{\lambda _{q+1}} \otimes w_{\lambda _{q+1}}\), contains the high frequencies of \(a^2_q w_{\lambda _{q+1}} \otimes w_{\lambda _{q+1}}\), hence its size is at least
Here we used (4), i.e. \(a^2\sim |R_q|\) and that the intermittent term \(w_{\lambda _{q+1}}\) has unitary norm in \(L^2\). In particular, we have that \(\Vert R_q \Vert _{L^1} \lambda _q\) is a nondecreasing sequence in q and hence
In order to control \(\Vert \nabla u_q\Vert _{L^1}\) we need \(\Vert \nabla (u_{q+1} - u_q) \Vert _{L^1}\) arbitrarily small, but the Sobolev inequality gives that
Hence,
where we used (5) in the last step.
In particular, with the current way to cancel the error in the iteration, we cannot expect \(\nabla u \in L^1\).
Note that the inequality (6), where we end up with a term of size \(\sim \sqrt{\lambda _{q+1}}\), is in agreement with a known limitation of the convex integration scheme, see [5, Section 2.4.1]. In fact, it is shown there that the quadratic error term allows to control at most half of the derivative of the solution u. The only way to overcome this is to exploit the intermittency, but in our context we cannot because of the Sobolev inequality.
1.3 Building blocks with Lorentz integrability
To push the convex integration scheme to its boundaries and obtain \(X=L^{1,\infty }\), we need to introduce a new family of building blocks \(\{W_i\}_{ i=1, \ldots , 4}\). The latter is the most important novelty of this paper, and its construction requires a new idea. In a nutshell, we design \(W_i\) so that its atomic decomposition, as a Lorentz function, is made up of “almost solutions” to the Euler equations. To this aim, we bundle together a family of intermittent jets [4] with different sizes and characteristic velocities. This structure allows sharpening the intermittency mechanism reaching the critical \(L^{1,\infty }\) integrability of \(\nabla W_i\).
To put forward this idea, there are several technical challenges to overcome, let us mention a few.
The high velocity of each jet, needed to force the bundle to almost solve the Euler equations, makes the term \(\partial _t W_i\) big. The latter should be treated as an error, hence we have to make its anti-divergence small. To do so, we exploit a special structure: we build the profiles of our jets in such a way that \(\partial _t W_i = \mathrm{{div}}\,(A_i)\) where \(A_i\) is a small symmetric potential.
The bundle structure and the 2-dimensional constraint make it very difficult to keep the supports of \(W_i\) disjoint in space-time. It requires a new, delicate, combinatorial argument.
We refer the reader to Sect. 4 for the precise construction and more explanations on our choice of building blocks.
Remark 1.3
The proof of Theorem 1.1 is flexible enough, due to the exponential convergence of the iterative sequence, to give \(\omega \in L^{1,q}\) for some \(q \gg 1\). A technical refinement of the current proof, based on Remark 4.4, would give \(q>4\).
2 Iteration and Euler–Reynolds System
We consider the system of equations (3) in \([0,1] \times {{\mathbb {T}}}^2\), where R is a traceless symmetric tensor.
As already remarked, our solution to (1) is obtained by passing to the limit solutions of (3) with suitable constraints on u and R. The latter are built by means of an iterative procedure based on the following.
Proposition 2.1
There exists \(M>0\) such that the following holds. For any smooth solution \((u_0, p_0, R_0)\) of (3), there exists another smooth solution \((u_1,p_1,R_1)\) of (3) such that
-
(i)
\(\Vert R_1 \Vert _{L^\infty (L^1)} \le \frac{1}{3}\Vert R_0 \Vert _{L^\infty (L^1)}\);
-
(ii)
\(\Vert u_1 - u_0 \Vert _{C^0(L^2)} + \Vert {{\,\textrm{curl}\,}}(u_1 - u_0) \Vert _{C^0(L^{1,\infty })} \le M \Vert R_0 \Vert _{L^\infty (L^1)}\);
-
(iii)
for any \(s\in (0,1)\), \(p\in (1, \frac{2}{1+s})\) there exists \(c(p,s)>0\) s.t.
$$\begin{aligned} \Vert D^s(u_1 - u_0) \Vert _{C^0(L^p)} \le (M \Vert R_0 \Vert _{L^\infty (L^1)})^{c(p,s)}\,; \end{aligned}$$ -
(iv)
if \(R_0(\cdot ,t)=0\) in \([0,t_0]\), then \(R_1(\cdot ,t )=0\) and \(u_1(\cdot ,t) = u_0(\cdot ,t)\) in \([0,t_0/2]\).
Proof of Theorem 1.1, given Proposition 2.1
Fix \(\lambda >0\). We start the iteration scheme with
where \(\chi \in C_c^\infty ([0,1])\), \(\chi =0\) in [0, 1/2] and \(\chi = 1\) in [3/2, 1]. Notice that \(-\mathrm{{div}}\,R_0 = \chi '(t) \sin (x_2 \lambda ) e_1 + \nabla p\), hence we can choose a traceless symmetric tensor \(R_0\) such that \(\Vert R_0 \Vert _{L^1} \le C\lambda ^{-1}\).
Applying iteratively Proposition 2.1 with \(t_0 = 1/2\) we build a sequence \(\{(u_n, p_n, R_n)\,: \, n\in {\mathbb {N}}\}\) of smooth solutions to (1) such that, for any \(n\ge 0\), it holds
and \(u_n(\cdot , t) = 0\) for any \(t\in [0,2^{-n-1}]\). Moreover, for any \(s\in (0,1)\) and \(p\in \left( 1, \frac{2}{1+s}\right) \) it holds
It follows that \(R_n \rightarrow 0\) in \(L^\infty (L^1)\) and \(u_n \rightarrow u\) in \(C^0(L^2)\), where u satisfies the assumptions of Theorem 1.1. Moreover \(u\in C^0(W^{s,p})\) for \(s\in (0,1)\) and \(p\in (1, \frac{2}{1+s})\) as a consequence of (7).
We now prove that there exists \({{\,\textrm{curl}\,}}u\in C^0(L^{1,\infty })\) with the property that \(\Vert {{\,\textrm{curl}\,}}u_n - {{\,\textrm{curl}\,}}u\Vert _{C^0(L^{1,\infty })} \rightarrow 0\) as \(n\rightarrow \infty \). A bit of care is needed since only the weak triangle inequality \( \Vert f + g \Vert _{L^{1,\infty }} \le 2 \Vert f\Vert _{L^{1,\infty }} + 2 \Vert g \Vert _{L^{1,\infty }} \) holds true. However, the latter is enough for our purposes
hence setting \({{\,\textrm{curl}\,}}u:= {{\,\textrm{curl}\,}}u_0 + \sum _{n=0}^\infty ({{\,\textrm{curl}\,}}u_{n+1} - {{\,\textrm{curl}\,}}u_n)\) we get the sought conclusion. \(\square \)
The remaining part of this note is devoted to the proof of Proposition 2.1. In Sect. 4 we introduce the building blocks of our construction, in Sect. 5 we use them to define the perturbation \(u_1-u_0\), finally in Sect. 6, we introduce the new error term \(R_1\) and show that it can be made arbitrarily small.
3 Preliminary Lemmas
3.1 Lorentz spaces
For every measurable function \(f:{{\mathbb {T}}}^d \rightarrow {\mathbb {R}}\) we recall the definition
(see e.g. [13]) and we define the Lorentz space \(L^{r,q}\) with \(r\in [1,\infty )\), \(q\in [1,\infty ]\), as the space of those functions f such that \(\Vert f\Vert _{L^{r,q}}<\infty \). Note that, in spite of the notation, \(\Vert \cdot \Vert _{L^{r,q}}\) is in general not a norm but for \((r,q)\ne (1,\infty )\) the topological vector space \(L^{r,q}\) is locally convex and there exists a norm \(|||\cdot |||_{r,q}\) which is equivalent to \(\Vert \cdot \Vert _{L^{r,q}}\) in the sense that the inequality \(C^{-1} |||f|||_{r,q}\le \Vert f\Vert _{L^{r,q}} \le C |||f|||_{r,q}\) holds.
3.2 Improved Hölder inequality
We recall the following improved Hölder inequality, stated as in [18, Lemma 2.6] (see also [3, Lemma 3.7]). If \(\lambda \in {\mathbb {N}}\) and \(f,g:{{\mathbb {T}}}^2 \rightarrow {\mathbb {R}}\) are smooth functions, then we have
When \(\int _{{{\mathbb {T}}}^2} g = 0\), then
3.3 Anti-divergence operators
Let now us introduce the anti-divergence operator
Here \(\text {Sym}_2\) denotes the space of symmetric matrices in \(\mathbb {R}^2\). It is simple to check that \(\mathrm{{div}}\,(\mathcal {R}_0(v)) = v-\int _{{{\mathbb {T}}}^2} v\), and that \(D \mathcal {R}_0\) is a Calderon-Zygmund operator, in particular it holds
Notice that (10) and (11) imply
where \(v_\lambda (x):= v(\lambda x)\) for some \(\lambda \in {\mathbb {N}}\). The latter is immediate for \(p\in (1,\infty )\), since
in the case \(p=1\) and \(p=\infty \) we need to take advantage of the Sobolev embedding theorem:
The same argument applies to \(\mathcal {R}_0^*\), the adjoint of \(\mathcal {R}_0\), then case \(p=1\) follows by duality.
Lemma 3.1
Let \(\lambda \in {\mathbb {N}}\) and \(f \in C^\infty ( {{\mathbb {T}}}^2; {\mathbb {R}})\), \(v\in C^\infty ( {{\mathbb {T}}}^2; {\mathbb {R}}^2)\) with \(\int _{{{\mathbb {T}}}^2} v = 0\), and \(v_\lambda = v(\lambda x)\). If we set
then,
Proof
The verification of \(\mathrm{{div}}\,{{\mathcal {R}}}(f v_\lambda ) = f v_\lambda -\int _{{{\mathbb {T}}}^2} fv_\lambda \) is immediate. To prove (13) we use (12) and (9):
\(\square \)
Remark 3.2
The operator \(\mathcal {R}\) can be also defined on scalar functions \(f: {{\mathbb {T}}}^2 \rightarrow {\mathbb {R}}\), \(v: {{\mathbb {T}}}^2\rightarrow {\mathbb {R}}\) as
and arguing as in Lemma 3.1 we can easily show that \(\textrm{div} {{\mathcal {R}}}(f v_\lambda ) = f v_\lambda -\int _{{{\mathbb {T}}}^2} f v_\lambda \) and
Lemma 3.3
For any \(a\in C^\infty ({{\mathbb {T}}}^2)\) and \(A\in C^\infty ({{\mathbb {T}}}; {\mathbb {R}}^{2\times 2})\) with \(\int _{{{\mathbb {T}}}^2} A =0\), it holds
Proof
Set \(T(A):= \mathcal {R}(\nabla a \cdot \mathrm{{div}}\,A)\). By duality, it suffices to show that
where \(T^*\) and \(\mathcal {R}_0^*\) denote the adjoint of T and \(\mathcal {R}_0\), respectively. To this aim we employ the Sobolev embedding and the fact that \(D T^* \mathcal {R}_0^*(B)\) maps \(L^p\) into \(L^p\) for any \(p\in (1,\infty )\):
\(\square \)
4 Building Blocks
In this section we introduce the building blocks of our construction. They will be employed in Sect. 5 to define the principal term of \(u_1 - u_0\) in Proposition 2.1.
Proposition 4.1
(Building blocks) Set \(\xi _1:= e_1\), \(\xi _2:= e_2\), \(\xi _3:= e_1 + e_2\) and \(\xi _4: = e_1 - e_2\). Then, for any \({\varepsilon }>0\) there exist \(W^p_{i}, W^c_{i}, Q_i \in C^\infty ((-1,1)\times {{\mathbb {T}}}^2; {\mathbb {R}}^2)\), \(A_i \in C^\infty ((-1,1)\times {{\mathbb {T}}}^2; \textrm{Sym}_2)\) for \(i=1,\ldots , 4\), such that
-
(i)
\(\mathrm{{div}}\,(W_i^p + W_i^c) =0\), \(\partial _t Q_i = \mathrm{{div}}\,(W_i^p\otimes W_i^p)\), and \(\partial _t (W^p_i + W^c_i) = \mathrm{{div}}\,(A_i)\);
-
(ii)
\(\int _{{{\mathbb {T}}}^2} A_i =0\), \(\int _{{{\mathbb {T}}}^2} W_i^p = \int _{{{\mathbb {T}}}^2} W^c_i = 0\), and \(W_i^p\), \(W_i^c\), \(A_i\) are \(\lambda ^{-1}\)-periodic functions for some \(\lambda \in \mathbb {Z}\) with \(\lambda \ge {\varepsilon }^{-1}\);
-
(iii)
\(\int _{{{\mathbb {T}}}^2} W_i^p \otimes W_i^p = \frac{\xi _i}{|\xi _i|} \otimes \frac{\xi _i}{|\xi _i|}\);
-
(iv)
the following estimates hold
$$\begin{aligned}{} & {} {\varepsilon }\Vert W_i^p \Vert _{L^2} + \Vert W_i^p \Vert _{L^1} + \Vert W_i^c \Vert _{L^2} \le {\varepsilon }\,, \nonumber \\{} & {} \quad \Vert D (W_i^p + W_i^c) \Vert _{L^{1,\infty }} + \Vert Q_i \Vert _{L^2} + \Vert D Q_i \Vert _{L^{1,\infty }} + \Vert A_i \Vert _{L^1} \le {\varepsilon }\,, \nonumber \\{} & {} \quad \Vert D^s (W_i^p + W_i^c) \Vert _{L^p} + \Vert D^s Q_i \Vert _{L^p} \le {\varepsilon }^{c(p,s)} \quad \text {for any}\, s\in (0,1)\,\text { and}\, p\in \left( 1,\frac{2}{1+s}\right) \,; \nonumber \\ \end{aligned}$$(15) -
(v)
for \(i\ne i'\) the union of the supports of \(W^p_i\), \(W_i^c\), \(Q_i\), is disjoint in space-time from the union of the supports of \(W^p_{i'}\), \(W_{i'}^{c}\), \(Q_{i'}\).
The velocity field \(W^p_i\) is the principal term, it has zero mean, high frequency \(\lambda \ge {\varepsilon }^{-1}\), is controlled in the relevant norms (cf. (iv)), and satisfies the fundamental property (iii): the quadratic interaction \(W_i^p\otimes W_i^p\) produces the lower order term \(\frac{\xi _i}{|\xi _i|} \otimes \frac{\xi _i}{|\xi _i|}\). The latter, combined with slow coefficients \(a_i\in C^\infty ({{\mathbb {T}}}^2)\), is used to cancel the error \(R_0\) out. To achieve the crucial bound \(\Vert D W_i^p \Vert _{L^{1,\infty }}\) we design the principal term as
where \(K, n_0 \gg 1\) are big parameters and \(\xi _i \) is one of the four directions appearing in the statement of Proposition 4.1. In a first stage, we build \(W_i^p(x,t)\) for a fixed parameter i, ignoring the issue that, for different parameters, such functions will not have disjoint support; only in Sect. 4.6 we make sure to suitably time-translate them, making substantial use of their special structure, to guarantee that Proposition 4.1 (v) holds. The vector fields \(W_{(\xi _i)}^k(x,t)\), \(k=n_0+1, \ldots , n_0 + K\), are the 2-dimensional counterpart of the intermittent jets introduced in [4]. They have \(L^2\) norm equal to 1, and are supported on disjoint balls of radius \(2^{-k} r\), for some \(r\ll 1\), which move in direction \(\xi _i\) with speed \(\mu 2^k\), where \(\mu \gg 1\). The fast time translation is used to make \(W_{(\xi )}^k\) “almost divergence free” and “almost solutions to the Euler equations”. In more rigorous terms, it means that there exist vector fields \((W_{(\xi )}^{k})^c\), \((Q_{(\xi )}^{k})^c\), that are smaller than \(W_{(\xi )}^k\) satisfying \(\mathrm{{div}}\,(W_{(\xi )}^k + (W_{(\xi )}^{k})^c)=0\) and \(\partial _t (Q_{(\xi )}^{k})^c = \mathrm{{div}}\,(W_{(\xi )}^k \otimes W_{(\xi )}^k)\). The vector fields \(W_i^c\) and \(Q_i\) are defined bundling together \((W_{(\xi )}^{k})^c\) and \((Q_{(\xi )}^{k})^c\) as we did in (16).
Another important property we need is that \(W_i^p \otimes W_j^p =0\) when \(i\ne j\). It is ensured by (v) in Proposition 4.1, which builds upon a delicate combinatorial lemma presented in Sect. 4.6.
We finally explain the role of the matrix \(A_i\) in our construction. Let us begin by noticing that the principal term \(W_i^p\) has big time derivative, being fast translating in time. Hence, the term \(\partial _t W_i^p\) cannot be treated as an error. To overcome this difficulty we impose an extra structure on \(W^p_i\) and \(W^c_i\). We construct them in order to have the identity \(\partial _t(W_i^p + W_i^c) = \mathrm{{div}}\,(A_i)\), for some symmetric matrix \(A_i\) which has small \(L^1\)-norm. The latter can be added to the new error term \(R_1\).
4.1 General notation
Given a velocity field \(u:=(u_1, u_2): {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2\) we write
Let us fix \(r_\perp \ll r_\parallel \ll 1\) and \(k\in {\mathbb {N}}\), \(k\ge 1\). We adopt the following convention: given any \(\rho : \mathbb {R} \rightarrow \mathbb {R}\) supported in \((-1,1)\) we write
Notice that \(\textrm{supp}(\rho _{r_\perp }^k) \subset ( 3 \cdot 2^{-k} r_\perp , 5\cdot 2^{-k}r_\perp )\), in particular
and
With a slight abuse of notation we keep denoting by \(\rho _{r_\perp }^k, \rho _{r_\parallel }^k: {{\mathbb {T}}}\rightarrow {\mathbb {R}}\) their periodized version.
4.2 Construction of the principal block
We consider \(\Phi , \psi : {\mathbb {R}}\rightarrow {\mathbb {R}}\) supported in \((-1,1)\), we set \(\phi := -\Phi '''\) and assume \(\int \psi ^2 = \int \phi ^2 =1\). Given \(r_\perp \ll r_\parallel \ll 1\) and \(k\in {\mathbb {N}}\) we have
and
We periodize \((\Phi ')_{r_\perp }^k\), \((\Phi '')_{r_\perp }^k\), \(\phi _{r_\perp }^k\), \(\psi _{r_\parallel }^k\) keeping the same notation.
Given a vector \(\xi \in \mathbb {Q}^2\), and parameters \(\lambda , \mu \gg 1\) we set
We finally fix \(K, n_0 \in {\mathbb {N}}\), and define the principal block
where \(t_k\) are time translations that will be chosen later. The following fundamental identity holds
4.3 Correction of the divergence
Observe that
Setting
and using the identity \( 2^{-k} r_\perp \partial _{x_1} (\Phi '')_{r_{\perp }}^k = -\phi _{r_\perp }^k\) we get \(\mathrm{{div}}\,(W_{(\xi )} + W^c_{(\xi )})=0\).
To correct the divergence of \( W_{\xi , K,n_0}\) we introduce
and set
4.4 Time correction
Let us now set
and observe that
Hence
The time corrector is defined as
4.5 Estimates on building blocks
In this section we collect the relevant estimates on the building blocks. Given \(N\ge 0\) we write \(D^N = D^{\left\lfloor {N}\right\rfloor } \Delta ^{s/2}\) where \(\left\lfloor {N}\right\rfloor \) is the integer part of N, \(s:= N - \left\lfloor {N}\right\rfloor \) and \(D^{\left\lfloor {N}\right\rfloor }\) is the standard derivative operator.
Lemma 4.2
For any \(N\in [0,\infty )\), \(M\ge 0\) integer and \(p\in [1,\infty ]\) there exists \(C=C(N,M,p,|\xi |,\Phi ,\psi )>0\) such that the following hold.
The proof of Lemma 4.2 is a simple computation, so we omit it. Let us draw some useful consequence. Summing on k and reminding that then terms in the sum in (18) have disjoint support, we get
(in particular, this says that the principal part is much smaller than the corrector),
and
Moreover, for \(s\in (0,1)\) and \(p<\frac{2}{1 + s}\) it holds
where \(\gamma := -s - 1 + 2/p >0\). In particular
Lemma 4.3
(Lorentz estimates) There exists \(C=C(|\xi |, \Phi ,\psi )>0\) such that
Proof
Observe that
and similarly
where for \(i=1,2\)
Let us now fix \(s\ge 1\) and \(k_*\) the smallest integer satisfying \(k_*\ge n_0+1\) and \(C2^{2k_*} \ge s K^{1/2} r_\perp r_\parallel \). It holds
From (26) and the choice of \(k_*\) we get
hence
the estimate on \(\Vert D W^c_{\xi ,K,n_0} \Vert _{L^{1,\infty }}\) can be obtained following the same strategy. An analogous argument gives
yielding
\(\square \)
Remark 4.4
It is not hard to prove the following extension of Lemma 4.3. For any \(q\ge 1\) it holds
Lemma 4.5
There exists a smooth \(\lambda \)-periodic function \(A_{\xi , K, n_0}: {{\mathbb {T}}}^2 \rightarrow \textrm{Sym}_2\) such that
Proof
Setting
it holds
and similarly
Hence (27) is satisfied. Defining
and arguing as in Lemma 4.2, we obtain that
which yields (28). \(\square \)
4.6 Combinatorial lemma
The following proposition shows that, up to a suitable (time) translation of each element in the bundle, the building blocks associated to different directions can be taken disjoint.
Proposition 4.6
Let \(\xi _1=e_1\), \(\xi _2 = e_2\), \(\xi _3= e_1+e_2\) and \(\xi _4= e_1-e_2\). Then for \(n_0= 5K\) the functions in the family \(\{ W_{(\xi _{i+1})}^k(x,t+ i \mu ^{-1} 2^{-5K})\}_{k=n_0,\ldots ,n_0+K;\; i=0,1,2,3}\) have all supports mutually disjoint in space-time.
Proof
We apply Lemma 4.7 below to the families \(\{ W_{(\xi _2)}^k(x,t+ i \mu ^{-1} 2^{-5K})\}_{k=n_0,...,n_0+K}\) and \(\{ W_{(\xi _2)}^k(x,t+ j \mu ^{-1} 2^{-5K})\}_{k=n_0,...,n_0+K}\); up to shifting the time axis, we can assume that \(i=0\) and that \(j\in \{1,2,3\}\) and conclude the proof.\(\square \)
Lemma 4.7
Let \(\xi _1, \xi _2 \in \{ e_1,e_2, e_1+e_2, e_1-e_2\} \) be two different vector fields. Let us consider two families \(\{ W_{(\xi _1)}^k(x,t)\}_{k=n_0,\ldots ,n_0+K}\) and \(\{ W_{(\xi _2)}^k(x,t+t_0)\}_{k=n_0,\ldots ,n_0+K}\) for some \(t_0 \in [\mu ^{-1} 2^{-7K}, \mu ^{-1} 2^{-7K+2}] \) and for \(n_0=5K\). Then the supports of all these functions are disjoint in space-time, namely
Proof
The family \(\{ W_{(\xi _1)}^k(x,t)\}_{k=n_0,\ldots ,n_0+K}\) is supported by (17) in space in a tube along \(\xi _1\) of size \(r_\parallel 2^{-n_0}\) and similarly the family \(\{ W_{(\xi _2)}^k(x,t+t_0)\}_{k=n_0,\ldots ,n_0+K}\) is supported in the tube along \(\xi _2\) of size \(r_\parallel 2^{-n_0}\). Since these two thin tubes intersect only in a neighborhood of the origin, we deduce that the supports of \(W_{(\xi _1)}^k(x,t)\) and \(W_{(\xi _2)}^h(x,t)\), where \(h, k \in \{n_0,\ldots , n_0+K\}\), can intersect for some time \(t>0\) only if they both belong to \(B_{R}(0)\), where \(R:= r_\parallel 2^{-n_0 + 1}\).
We claim the following: suppose that for a certain \(t>0\) and \(k \in \{n_0,\ldots , n_0+K\}\) we have \(\textrm{supp}W_{(\xi _1)}^k (\cdot , t) \cap B_{r_\parallel 2^{-n_0+1}} \ne \emptyset \). Then \(\textrm{supp}W_{(\xi _2)}^h(\cdot , t+t_0) \cap B_R = \emptyset \) for every \(h \in \{n_0,\ldots , n_0+K\}\).
The previous claim excludes the simultaneous presence at any \(t>0\) of the support of \(W_{(\xi _1)}^k (\cdot , t)\) and the support of \(W_{(\xi _2)}^h(\cdot , t+t_0)\) in \(B_R(0)\), thereby concluding the proof of the lemma.
We now prove the claim. Let us fix a time t such that \(\textrm{supp}W_{(\xi _1)}^k(\cdot ,t) \cap B_{R} \ne \emptyset \). Since \(\textrm{supp}W_{(\xi _1)}^k(\cdot ,t) \) is moving at constant speed \(\mu 2^k\) along the tube on the torus, there exists \({\bar{t}}\) such that \(|t -{\bar{t}} | \le R \mu ^{-1} 2^{-k}\) and \(\textrm{supp}W_{(\xi _1)}^k(\cdot ,{\bar{t}}) = \textrm{supp}W_{(\xi _1)}^k(x,0)\). At time \({\bar{t}}\) we have information about the position of \(\textrm{supp}W_{(\xi _2)}^h(\cdot ,{\bar{t}}+t_0)\); more precisely, we have that
because the ratio between the (constant) velocity of \(\textrm{supp}W_{(\xi _1)}^k(\cdot ,t)\) and the velocity of \(\textrm{supp}W_{(\xi _2)}^k(\cdot ,t)\) is of the form \(2^j\) for some \(j \in \{-K,..., K\}\).
In the union in the right-hand side of (29), thanks to the upper bound on \(t_0\), the choice \(n=0\) identifies the ball of the (finite) union at minimal distance from the origin for every k. By the lower bound on \(t_0\) and the fact that the minimal velocity is \(\mu 2^{n_0}\), we get that this distance is greater than \(2^{n_0-7K}\). At time t the distance between \(\textrm{supp}W_{(\xi _2)}^h(\cdot , t+t_0)\) and \(B_R(0)\) is therefore bigger than
This concludes the proof of the claim. \(\square \)
4.7 Proof of Proposition 4.1
Let \(n_0=5K\) and \(\{ W_{(\xi _{i+1})}^k(x,t+ i \mu ^{-1} 2^{-5K})\}_{k=n_0,\ldots ,n_0+K;\; i=0,1,2,3}\) be as in Proposition 4.6. Since \(\textrm{supp}W_{\xi _{i+1}}^k = \textrm{supp}(W_{\xi _{i+1}}^k)^c = \textrm{supp}Q_{\xi _{i+1}}^k\), by translating in time \((W_{\xi _{i+1}}^k)^c\) and \(Q_{\xi _{i+1}}^k\) with \(t_{k,i}:= i \mu ^{-1} 2^{-5K}\) we deduce that \(W_{i+1}^p:= W^p_{\xi _{i+1},K,n_0}\), \(W_{i+1}^c:= W^c_{\xi _{i+1},K,n_0}\), \(Q_{i+1}:= Q_{\xi _{i+1},K,n_0}\) and \(A_{i+1}:=A_{\xi ,K,n_0}\) satisfy (v) in Lemma 4.1. We refer the reader to Lemma 4.5 for the construction of \(A_{\xi ,K,n_0}\). Properties (i) and (ii) in Lemma 4.1 are now immediate from (19), (20) and Lemma 4.5. We are left with the proof of (iii) and (iv) in Lemma 4.1. To do so we have to choose appropriately the parameters \(\lambda , \mu , K, r_\perp \) and \(r_\parallel \). Let \(\delta < 1/2\) to be chosen later in terms of \({\varepsilon }>0\), we set
leaving \(r_\perp \ll r_\parallel \ll 1\) free. From Lemma 4.3, Lemma 4.5, (21), (22) and (23) we deduce
Moreover, from (24) and (25) we deduce
for any \(s\in (0,1)\) and \(p\in \left( 1, \frac{2}{1+s}\right) \). The conclusions (iii) and (iv) in Lemma 4.1 follow by choosing first \(\delta \) small enough so that \(C\delta \le {\varepsilon }\), and after \(r_\perp \ll r_\parallel \ll 1\) so that \(\frac{r_\perp }{r_\parallel } \le {\varepsilon }\), \(\lambda = \delta ^4 r_\parallel ^{1/2} r_\perp ^{-1/2} \ge {\varepsilon }^{-1}\) and \( C\exp \left\{ -C(p,s) \delta ^4 \left( \frac{r_\perp }{r_\parallel }\right) ^{-2} \right\} \le {\varepsilon }^{c(p,s)}\).
5 Definition of the Perturbations
Let us begin by observing that there exist \(\Gamma _i \in C^\infty (\textrm{Sym}_2, \mathbb {R})\), \(i=1,\ldots , 4\) such that
where \(e_1:=(1,0)\), \(e_2:=(0,1)\), \(e_3:= (1/\sqrt{2}, 1/\sqrt{2})\) and \(e_4:= (1/\sqrt{2},-1/\sqrt{2})\).
We can define, for instance,
It is immediate to show the identity \(R= \sum _{i=1}^4 \Gamma _i(R)^2 e_i \otimes e_i\). Moreover, using that \(|R-I|<1/8\), we deduce
which implies that \(\Gamma _i\) are smooth functions.
We define
where \(\chi \in C^\infty ({\mathbb {R}})\) satisfies \(0\le \chi \le 1\), \(\chi = 0\) on \([0,t_0/2]\), and \(\chi = 1\) on \([t_0, \infty )\). Our choice leads to
where \(\xi _1=(1,0)\), \(\xi _2=(0,2)\), \(\xi _3=(1,1)\) and \(\xi _4=(1,-1)\). The latter implies that
for some smooth pressure function P.
We observe that the coefficient \(a_i\) is a “slow function”, namely its derivatives are estimated only in terms of the smoothness of \(R_0\)
For \({\varepsilon }>0\) to be chosen later, we consider the functions \(W_i^p\), \(W_i^c\), \(Q_i\), \(A_i\) from Proposition 4.1. We define the new velocity field as the sum of the previous one, a principal perturbation, a divergence corrector and a temporal corrector
where
where \(\mathbb {P} = \nabla ^\perp \Delta ^{-1}\mathrm{{div}}\,: C^\infty ({{\mathbb {T}}}^2;{\mathbb {R}}^2) \rightarrow C^\infty ({{\mathbb {T}}}^2;{\mathbb {R}}^2)\) is the Leray projector.
We refer the reader to Remark 3.2 for the definition of \(\mathcal {R}\). From now on, in order to simplify our notation, for any function space X and any map f which depends on t and x, we will write \(\Vert f \Vert _X\) meaning \(\Vert f \Vert _{L^\infty (X)}\).
5.1 Estimate on \(\Vert u_{1} - u_0 \Vert _{L^2}\) and on \(\Vert u_{1} - u_0 \Vert _{L^1}\)
By the triangular inequality,
and we estimate the right-hand side separately as
where in the second line we used the improved Holder inequality (8) and (iii) in Proposition 4.1.
From Remark 3.2 we deduce
Finally we employ (iv) in Proposition 4.1 to get
Analogously
5.2 Estimate on \(\Vert {{\,\textrm{curl}\,}}(u_{1} - u_0 )\Vert _{L^{1,\infty }}\) and \(\Vert D^s (u_1 - u_0)\Vert _{L^p}\)
By triangular inequality,
we estimate the right-hand side separately as
where we employed (iv) in Proposition 4.1. Using that \(D\mathcal {R}\) is a Calderon-Zygmund operator we deduce
Let us now fix \(s\in (0,1)\) and \(p\in (1,\frac{2}{1+s})\). Arguing as above employing (15) we get
6 New Error
We define \(R_1\) in such a way that
which, by subtracting the equation for \(u_0\), is equivalent to
We are going to define
where the various addends are defined in the following paragraphs, and show that
The proof of Proposition 2.1 will follow by choosing \({\varepsilon }\) small enough.
6.1 Linear error
Let us set
thanks to (31) it holds
6.2 Temporal error
Let us set
Using that
for some pressure term P, it is immediate to verify the identity
Since \({\mathcal {R}}\) and \(\mathcal {R}_0\) send \(L^1\) to \(L^1\) (cf. Lemma 3.1 and Remark 3.2), we have that
From (iv) in Proposition 4.1 we get
By employing (14) we bound
6.3 Quadratic error terms
Let us set
and show that (32) holds. In view of (33), (30) and (34) it amounts to check that
The latter easily follows by noticing that, as a consequence of (ii) in Proposition 4.1, one has
Let us finally prove that \(\Vert R_1^{(q)}\Vert _{L^1} \le {\varepsilon }C(t_0, \Vert R_0\Vert _{C^2})\). We begin by observing that
From (iv) in Proposition 4.1, the estimates in Sect. 5.1 on \( \Vert u_{1}^{(c)}\Vert _{L^2},\Vert u_{1}^{(t)} \Vert _{L^2}, \Vert u_1-u_0\Vert _{L^2}\) and Lemma 3.1 we deduce
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Acknowledgements
EB was supported by the Giorgio and Elena Petronio Fellowship at the Institute for Advanced Study. MC was supported by the SNSF Grant 182565. The authors wish to thank Camillo De Lellis for interesting discussions on the theme of the paper.
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Brué, E., Colombo, M. Nonuniqueness of Solutions to the Euler Equations with Vorticity in a Lorentz Space. Commun. Math. Phys. 403, 1171–1192 (2023). https://doi.org/10.1007/s00220-023-04816-4
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DOI: https://doi.org/10.1007/s00220-023-04816-4