Nonuniqueness for the Transport Equation with Sobolev Vector Fields
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Abstract
We construct a large class of examples of nonuniqueness for the linear transport equation and the transportdiffusion equation with divergencefree vector fields in Sobolev spaces \(W^{1, p}\).
Keywords
Transport equation Renormalized solutions Convex integration Nonuniqueness hprincipleIntroduction
Both theorems can be generalized as follows: we can construct vector fields with arbitrary large regularity \(u \in W^{{\tilde{m}},\tilde{p}}\), \({\tilde{m}} \in \mathbb {N}\), for which uniqueness of solutions to (1)–(2) or (7)–(2) fails, in the class of densities \(\rho \in W^{m, p}\), with arbitrary large \(m \in \mathbb {N}\); moreover, we can do that even when on the r.h.s. of (7) there is a higher order diffusion operator (see Theorems 1.6 and 1.10).
Before stating the precise statements of these results, we present a brief (and far from complete) overview of the main wellposedness achievements present in the literature. We start with the analysis of the wellposedness for the transport equation in class of bounded densities, then we pass to the analysis of wellposedness for the transport equation in the class of \(L^p\)integrable densities, with the statement of our Theorems 1.2 and 1.6 and finally we discuss the transportdiffusion equation, with the statements of our Theorems 1.9 and 1.10. The last part of this introduction is devoted to a brief overview of the main techniques used in our proofs.
The Case of Bounded Densities
The literature about rough vector fields mainly concerns the wellposedness of (1)–(2) in the class of bounded densities, \(\rho \in L^\infty \). The reason for that can be found in the fact that the scientific community has been mainly interested in the wellposedness of ODE (5) and has used the PDE as a tool to attack the ODE problem: the general strategy is that a wellposedness result for the transport equation in the class of bounded densities yields a unique solution to the PDE (6) and thus one tries to prove that the flow \(X(t) := \Phi (t)^{1}\) is the unique meaningful solution, in the sense of regular Lagrangian flow, to the ODE (5). We refer to [6] for a precise definition of the notion of regular Lagrangian flow and for a detailed discussion about the link between the Eulerian and the Lagrangian approach.
The first result in this direction dates back to DiPerna and Lions [26], when they proved uniqueness, in the class of bounded densities, for vector fields \(u \in L^1_t W^{1,1}_x\) with bounded divergence. This result was extended in 2004 by Ambrosio [5] to vector fields \(u \in L^1_t BV_x \cap L^\infty \) and with bounded divergence (see also [16, 17]) and very recently by Bianchini and Bonicatto [8] for vector fields \(u \in L^1_t BV_x\) which are merely nearly incompressible (see, for instance, [6] for a definition of nearly incompressibility).
The proofs of these results are very subtle and involves several deep ideas and sophisticated techniques. We could however try to summarize the heuristic behind all of them as follows: (very) roughly speaking, a Sobolev or BV vector field u is Lipschitzlike (i.e. Du is bounded) on a large set and there is just a small “bad” set, where Du is very large. On the big set where u is “Lipschitzlike”, the classical uniqueness theory applies. Nonuniqueness phenomena could thus occur only on the small “bad” set. Uniqueness of solutions in the class of bounded densities is then a consequence of the fact that a bounded density \(\rho \) can not “see” this bad set, or, in other words, cannot concentrate on this bad set.
As we mentioned, uniqueness at the PDE level in the class of bounded densities implies, in all the cases considered above, uniqueness at the ODE level (again in the sense of the regular Lagrangian flow). On the other hand, based on a selfsimilar mixing example of Aizenmann [1], Depauw [24] constructed an example of nonuniqueness for weak solutions with \(\rho \in L^{\infty }((0,T)\times \mathbb {T}^d)\) and \(u\in L^1(\varepsilon ,1;BV(\mathbb {T}^d))\) for any \(\varepsilon >0\) but \(u\notin L^1(0,1;BV(\mathbb {T}^d))\). This example has been revisited in [3, 4, 17, 37]. It should be observed, though, that the phenomenon of nonuniqueness in such “mixing” examples is Lagrangian in the sense that it is a consequence of the degeneration of the flow map X(t, x) as \(t\rightarrow 0\); in particular, once again, the link between (1) and (5) is crucial.
The Case of Unbounded Densities
There are important mathematical models, related, for instance, to the Boltzmann equation (see [25]), incompressible 2D Euler [19], or to the compressible Euler equations, in which the density under consideration is not bounded, but it belongs just to some \(L^\infty _t L^p_x\) space. It is thus an important question to understand the wellposedness of the Cauchy problem (1)–(2) in such larger functional spaces.
As a first step, we observe that for a density \(\rho \in L^\infty _t L^p_x\) and a field \(u \in L^1_t L^1_x\), the product \(\rho u\) is not well defined in \(L^1\) and thus the notion of weak solution as in (8) has to be modified. There are several possibilities to overcome this issue. We mention two of them: either we require that \(u \in L^1_t L^{p'}_x\), where \(p'\) is the dual Hölder exponent to p, or we consider a notion of solution which cut off the regions where \(\rho \) is unbounded. Indeed, this second possibility is encoded in the notion of renormalized solution (9).
The wellposedness theory provided by (9) for bounded densities is sufficient for the existence of a regular Lagrangian flow, which in turn leads to existence also for unbounded densities. For the uniqueness, an additional integrability condition is required:
Theorem 1.1
As we have already observed for the case of bounded densities, also in this more general setting existence of weak and renormalized solutions is not a difficult problem. It is as well not hard to show uniqueness in the class of renormalized solutions, using the fact that renormalized solutions in \(L^\infty _t L^p_x\) have constant in time \(L^p\) norm (it suffices to choose \(\beta \) as a bounded smooth approximation of \(\tau \mapsto \tau ^p\)).
The crucial point in Theorem 1.1 concerns the uniqueness of the renormalized solution among all the weak solutions in \(L^\infty _t L^p_x\), provided (11) is satisfied. The reason why such uniqueness holds can be explained by the same heuristic as in the case of bounded densities: a vector field in \(W^{1,{\tilde{p}}}\) is “Lipschitzlike” except on a small bad set, which can not “be seen” by a density in \(L^p\), if (11) holds, i.e. if p, although it is less than \(\infty \), is sufficiently large w.r.t. \({\tilde{p}}\). On the more technical side, the integrability condition (11) is necessary in the proof in [26] to show convergence of the commutator (10) in \(L^1_{loc}\).
The following question is therefore left open: does uniqueness of weak solutions hold in the class of densities \(\rho \in L^\infty _t L^p_x\) for a vector field in \(L^1_t L^{p'}_x \cap L^1_t W^{1, \tilde{p}}_x\), when (11) fails?
In a recent note Caravenna and Crippa [15], addressed this issue for the case \(p=1\) and \({\tilde{p}}>1\), announcing the result that uniqueness holds under the additional assumption that u is continuous.
Previously, examples of such Eulerian nonuniqueness have been constructed, for instance, in [18], based on the method of convex integration from [21], yielding merely bounded velocity u and density \(\rho \). However, such examples do not satisfy the differentiability condition \(u \in W^{1, {\tilde{p}}}\) for any \({\tilde{p}}\ge 1\) and therefore do not possess an associated Lagrangian flow.
Here is the statement of our first and main result.
Theorem 1.2
 (a)
\(\rho \in C\bigl ([0,T]; L^p (\mathbb {T}^d)\bigr )\), \(u \in C\bigl ([0,T]; W^{1,{\tilde{p}}} (\mathbb {T}^d)\cap L^{p'} (\mathbb {T}^d)\bigr )\);
 (b)
 (c)at initial and final time \(\rho \) coincides with \({\bar{\rho }}\), i.e.$$\begin{aligned} \rho (0,\cdot ) = {\bar{\rho }}(0, \cdot ), \quad \rho (T, \cdot ) = \bar{\rho }(T, \cdot ); \end{aligned}$$
 (d)\(\rho \) is \(\varepsilon \)close to \({\bar{\rho }}\) i.e.$$\begin{aligned} \begin{aligned} \sup _{t \in [0,T]} \big \Vert \rho (t,\cdot )  {\bar{\rho }}(t, \cdot )\big \Vert _{L^p(\mathbb {T}^d)}&\le \varepsilon . \end{aligned} \end{aligned}$$
Our theorem has the following immediate consequences.
Corollary 1.3
Proof
Let \(\chi : [0,T] \rightarrow \mathbb {R}\) such that \(\chi \equiv 0\) on [0, T / 4], \(\chi \equiv 1\) on [3T / 4, T]. Apply Theorem 1.2 with \({\bar{\rho }}(t,x) := \chi (t) {\bar{\rho }}(x)\). \(\square \)
Corollary 1.4
Proof
Remark 1.5
 1.
Condition (12) implies that \(d \ge 3\). In fact it is not clear if a similar statement could hold for \(d=2\)  see for instance [2] for the case of autonomous vector fields.
 2.
Our theorem shows the optimality of the condition of DiPerna–Lions in (11), at least for sufficiently high dimension \(d\ge 3\).
 3.
The requirement that \({\bar{\rho }}\) has constant (in time) spatial mean value is necessary because weak solutions to (1), (3) preserve the spatial mean.
 4.
The condition (12) implies that the \(L^{p'}\)integrability of the velocity u does not follow from the Sobolev embedding theorem.
 5.We expect that the statement of Theorem 1.2 remains valid if (12) is replaced byIt would be interesting to see if this condition is sharp in the sense that uniqueness holds provided$$\begin{aligned} \frac{1}{p}+\frac{1}{{\tilde{p}}}>1+\frac{1}{d}. \end{aligned}$$(13)In this regard we note that (13) implies \(\tilde{p}<d\). Conversely, if \(u\in W^{1,{\tilde{p}}}\) with \({\tilde{p}}>d\), the Sobolev embedding implies that u is continuous so that the uniqueness statement in [15] applies.$$\begin{aligned} \frac{1}{p} + \frac{1}{{\tilde{p}}} \le 1 + \frac{1}{d}\,. \end{aligned}$$
 6.
The given function \({\bar{\rho }}\) could be less regular than \(C^\infty \), but we are not interested in following this direction here.
 7.
It can be shown that the dependence of \(\rho ,u\) on time is actually \(C^\infty _t\), not just continuous, since we treat time just as a parameter.
Inspired by the heuristic described above, the proof of our theorem is based on the construction of densities \(\rho \) and vector fields u so that \(\rho \) is, in some sense, concentrated on the “bad” set of u, provided (12) holds. To construct such densities and fields, we treat the linear transport equation (1) as a nonlinear PDE, whose unknowns are both \(\rho \) and u: this allows us to control the interplay between density and field. More precisely, we must deal with two opposite needs: on one side, to produce “anomalous” solutions, we need to highly concentrate \(\rho \) and u; on the other side, too highly concentrated functions fail to be Sobolev or even \(L^p\)integrable. The balance between these two needs is expressed by (12).
It is therefore possible to guess that, under a more restrictive assumption than (12), one could produce anomalous solutions enjoying much more regularity than just \(\rho \in L^p\) and \(u \in W^{1, {\tilde{p}}}\). Indeed, we can produce anomalous solutions as regular as we like, as shown in the next theorem, where (12) is replaced by (14).
Theorem 1.6
 (a)
\(\rho \in C([0,T], W^{m,p} (\mathbb {T}^d))\), \(u \in C([0,T]; W^{{\tilde{m}},{\tilde{p}}} (\mathbb {T}^d))\), \(\rho u \in C([0,1]; L^1 (\mathbb {T}^d))\);
 (b)
 (c)at initial and final time \(\rho \) coincides with \(\bar{\rho }\), i.e.$$\begin{aligned} \rho (0,\cdot ) = {\bar{\rho }}(0, \cdot ), \quad \rho (T, \cdot ) = \bar{\rho }(T, \cdot ); \end{aligned}$$
 (d)\(\rho \) is \(\varepsilon \)close to \({\bar{\rho }}\) i.e.$$\begin{aligned} \begin{aligned} \sup _{t \in [0,T]} \big \Vert \rho (t,\cdot )  {\bar{\rho }}(t, \cdot )\big \Vert _{W^{m,p}(\mathbb {T}^d)}&\le \varepsilon . \\ \end{aligned} \end{aligned}$$
Remark 1.7
The analogues of Corollaries 1.3 and 1.4 continue to hold in Theorems 1.6. Observe also that (14) reduces to (12) if we choose \(m=0\) and \({\tilde{m}} =1\).
Remark 1.8
Contrary to Theorem 1.2, here we do not show that \(u \in C([0,T], L^{p'}(\mathbb {T}^d))\). Here we prove that \(\rho u \in C([0,T], L^1(\mathbb {T}^d))\) by showing that \(\rho \in C([0,T]; L^s(\mathbb {T}^d))\) and \(u \in C([0,T]; L^{s'}(\mathbb {T}^d))\) for some suitably chosen \(s,s' \in (1, \infty )\). This is also the reason why in Theorem 1.6 we allow the case \(p = 1\). Indeed, Theorem 1.2, for any given p, produces a vector field \(u \in C_t L^{p'}_x\); on the contrary, Theorem 1.6 just produces a field \(u \in C_t L^{s'}_x\), for some \(s' < p'\).
Extension to the TransportDiffusion Equation
The mechanism of concentrating the density in the same set where the field is concentrated, used to construct anomalous solutions to the transport equation, can be used as well to prove nonuniqueness for the transportdiffusion equation (7).
The diffusion term \(\Delta \rho \) “dissipates the energy” and therefore, heuristically, it helps for uniqueness. Nonuniqueness can thus be caused only by the transport term \(\text {div }(\rho u)= u \cdot \nabla \rho \). Therefore, as a general principle, whenever a uniqueness result is available for the transport equation, the same result applies to the transportdiffusion equation (see, for instance, [19, 32, 34]). Moreover, the diffusion term \(\Delta \rho \) is so strong that minimal assumptions on u are enough to have uniqueness: this is the case, for instance, if u is just bounded, or even \(u \in L^r_t L^q_x\), with \(2/r + d/q \le 1\) (see [31] and also [9], where this relation between r, q, d is proven to be sharp). Essentially, in this regime the transport term can be treated as a lower order perturbation of the heat equation.
Theorem 1.9
 (a)
\(\rho \in C\bigl ([0,T]; L^p (\mathbb {T}^d)\bigr )\), \(u \in C\bigl ([0,T]; W^{1,{\tilde{p}}} (\mathbb {T}^d)\cap L^{p'} (\mathbb {T}^d)\bigr )\);
 (b)
 (c)at initial and final time \(\rho \) coincides with \({\bar{\rho }}\), i.e.$$\begin{aligned} \rho (0,\cdot ) = {\bar{\rho }}(0, \cdot ), \quad \rho (T, \cdot ) = \bar{\rho }(T, \cdot ); \end{aligned}$$
 (d)\(\rho \) is \(\varepsilon \)close to \({\bar{\rho }}\) i.e.$$\begin{aligned} \begin{aligned} \sup _{t \in [0,T]} \big \Vert \rho (t,\cdot )  {\bar{\rho }}(t, \cdot )\big \Vert _{L^p(\mathbb {T}^d)}&\le \varepsilon . \end{aligned} \end{aligned}$$
Theorem 1.10
 (a)
\(\rho \in C\bigl ([0,T]; W^{m,p} (\mathbb {T}^d)\bigr )\), \(u \in C\bigl ([0,T]; W^{{\tilde{m}},{\tilde{p}}} (\mathbb {T}^d)\bigr )\), \(\rho u \in C([0,1]; L^1(\mathbb {T}^d))\);
 (b)
 (c)at initial and final time \(\rho \) coincides with \({\bar{\rho }}\), i.e.$$\begin{aligned} \rho (0,\cdot ) = {\bar{\rho }}(0, \cdot ), \quad \rho (T, \cdot ) = \bar{\rho }(T, \cdot ); \end{aligned}$$
 (d)\(\rho \) is \(\varepsilon \)close to \({\bar{\rho }}\) i.e.$$\begin{aligned} \begin{aligned} \sup _{t \in [0,T]} \big \Vert \rho (t,\cdot )  {\bar{\rho }}(t, \cdot )\big \Vert _{W^{m,p}(\mathbb {T}^d)}&\le \varepsilon . \end{aligned} \end{aligned}$$
Remark 1.11
The analogues of Corollaries 1.3 and 1.4 continue to hold in Theorems 1.9 and 1.10. Remark 1.8 applies also to the statement of Theorem 1.10.
Observe also that, if we choose \(m=0\), \({\tilde{m}}=1\), \(k=2\), the first condition in (15) reduces to the first condition in (17), nevertheless (15) is not equivalent to (17). Indeed, (15) implies (17), but the viceversa is not true, in general. This can be explained by the fact that Theorem 1.9, for any given p, produces a vector field \(u \in C_t L^{p'}_x\), while Theorem 1.10 just produces a field \(u \in C_t L^{s'}_x\) for some \(s' < p'\).
Strategy of the Proof
Our strategy is based on the technique of convex integration that has been developed in the past years for the incompressible Euler equations in connection with Onsager’s conjecture, see [10, 11, 12, 13, 22, 23, 29] and in particular inspired by the recent extension of the techniques to weak solutions of the Navier–Stokes equations in [14]. Whilst the techniques that led to progress and eventual resolution of Onsager’s conjecture in [29] are suitable for producing examples with Hölder continuous velocity (with small exponent) [30], being able to ensure that the velocity is in a Sobolev space \(W^{1,{\tilde{p}}}\), i.e. with one full derivative, requires new ideas.
A similar issue appears when one wants to control the dissipative term \(\Delta u\) in the Navier–Stokes equations. Inspired by the theory of intermittency in hydrodynamic turbulence, Buckmaster and Vicol [14] introduced “intermittent Beltrami flows”, which are spatially inhomogeneous versions of the classical Beltrami flows used in [10, 11, 12, 22, 23]. In contrast to the homogeneous case, these have different scaling for different \(L^q\) norms at the expense of a diffuse Fourier support. In particular, one can ensure small \(L^q\) norm for small \(q>1\), which in turn leads to control of the dissipative term.
In this paper we introduce concentrations to the convex integration scheme in a different way, closer in spirit to the \(\beta \)model, introduced by Frisch et al. [27, 28] as a simple model for intermittency in turbulent flows. In addition to a large parameter \(\lambda \) that controls the frequency of oscillations, we introduce a second large parameter \(\mu \) aimed at controlling concentrations. Rather than working in Fourier space, we work entirely in xspace and use “Mikado flows”, introduced in [20] and used in [13, 29] as the basic building blocks. These building blocks consist of pairwise disjoint (periodic) pipes in which the divergencefree velocity and, in our case, the density are supported. In particular, our construction only works for dimensions \(d\ge 3\). The oscillation parameter \(\lambda \) controls the frequency of the periodic arrangement  the pipes are arranged periodically with period \(1{/}\lambda \). The concentration parameter \(\mu \) controls the relative (to \(1{/}\lambda \)) radius of the pipes and the size of the velocity and density. Thus, for large \(\mu \) our building blocks consist of a \(1{/}\lambda \)periodic arrangement of very thin pipes of total volume fraction \(1{/}\mu ^{d1}\) where the velocity and density are concentrated—see Proposition 4.1 and Remark 4.2 below.
We prove in details only Theorem 1.2, in Sects. 2–6. The proofs of Theorems 1.6, 1.9, 1.10 can be obtained from the one of Theorem 1.2 with minor changes. A sketch is provided in Sect. 7.
Technical Tools

\(\mathbb {T}^d = \mathbb {R}^d / \mathbb {Z}^d\) is the ddimensional flat torus.

For \(p \in [1,\infty ]\) we will always denote by \(p'\) its dual exponent.
 If f(t, x) is a smooth function of \(t \in [0,T]\) and \(x \in \mathbb {T}^d\), we denote by

\(\Vert f\Vert _{C^k}\) the sup norm of f together with the sup norm of all its derivatives in time and space up to order k;

\(\Vert f(t, \cdot )\Vert _{C^k(\mathbb {T}^d)}\) the sup norm of f together with the sup norm of all its spatial derivatives up to order k at fixed time t;

\(\Vert f(t,\cdot )\Vert _{L^p(\mathbb {T}^d)}\) the \(L^p\) norm of f in the spatial derivatives, at fixed time t. Since we will take always \(L^p\) norms in the spatial variable (and never in the time variable), we will also use the shorter notation \(\Vert f(t, \cdot )\Vert _{L^p} = \Vert f(t)\Vert _{L^p}\) to denote the \(L^p\) norm of f in the spatial variable.


\(C^\infty _0(\mathbb {T}^d)\) is the set of smooth functions on the torus with zero mean value.

\(\mathbb {N}= \{0,1,2, \dots \}\).

We will use the notation \(C(A_1, \dots , A_n)\) to denote a constant which depends only on the numbers \(A_1, \dots , A_n\).
Improved Hölder Inequality
We start with the statement of the improved Hölder inequality, inspired by Lemma 3.7 in [14].
Lemma 2.1
Proof
Antidivergence Operators
For \(f \in C^\infty _0(\mathbb {T}^d)\) there exists a unique \(u \in C^\infty _0(\mathbb {T}^d)\) such that \(\Delta u = f\). The operator \(\Delta ^{1}: C^\infty _0(\mathbb {T}^d) \rightarrow C^\infty _0(\mathbb {T}^d)\) is thus well defined. We define the standard antidivergence operator as \(\nabla \Delta ^{1}: C^\infty _0(\mathbb {T}^d) \rightarrow C^\infty (\mathbb {T}^d; \mathbb {R}^d)\). It clearly satisfies \(\text {div }(\nabla \Delta ^{1} f) = f\).
Lemma 2.2
Proof
With the help of the standard antidivergence operator, we now define an improved antidivergence operator, which lets us gain a factor \(\lambda ^{1}\) when applied to functions of the form \(f(x) g(\lambda x)\).
Lemma 2.3
Remark 2.4
The same result holds if f, g are vector fields and we want to solve the equation \(\text {div }u = f \cdot g_\lambda \), where \(\cdot \) denotes the scalar product.
Proof
Remark 2.5
Mean Value and Fast Oscillations
Lemma 2.6
Proof
Statement of the Main Proposition and Proof of Theorem 1.2
Proposition 3.1
Proof of Theorem 1.2 assuming Proposition 3.1
Let M be the constant in Proposition 3.1. Let \(\tilde{\varepsilon }>0\) and \(\eta >0\) (their precise value will be fixed later, with \(\eta \) depending on \({{\tilde{\varepsilon }}}\)). Let \(\sigma _q = \delta _q := 2^{q}\) and \(I_q := I_{\sigma _q} = (\sigma _q, 1  \sigma _q)\).
Lemma 3.2
Proof
 if \(t \in I_q\),$$\begin{aligned} \Vert \rho _{q+1}(t)  \rho _q(t)\Vert _{L^p} \le M \eta \Vert R_q(t)\Vert _{L^1}^{1/p}, \le M \eta \delta _{q+1}^{1/p}; \end{aligned}$$
 if \(t \in I_{q+1} {{\setminus }} I_q\),$$\begin{aligned} \Vert \rho _{q+1}(t)  \rho _q(t)\Vert _{L^p} \le M \eta \Vert R_q(t)\Vert _{L^1}^{1/p} \le M \eta \Big [R_0(t)\Vert _{L^1} + \delta _{q+1} \Big ]^{1/p}; \end{aligned}$$
 if \(t \in I_{q+2} {{\setminus }} I_{q+1}\),$$\begin{aligned} \Vert \rho _{q+1}(t)  \rho _q(t)\Vert _{L^p} \le M \eta \Vert R_q(t)\Vert _{L^1}^{1/p} \le M \eta \Vert R_0(t)\Vert _{L^1}^{1/p}, \end{aligned}$$
 if \(t \in I_{q+2} {{\setminus }} I_{q+1}\),$$\begin{aligned} \Vert R_{q+1}(t)\Vert _{L^1} \le \Vert R_q(t)\Vert _{L^1} + \delta _{q+2} \le \Vert R_0(t)\Vert _{L^1} + \delta _{q+2}; \end{aligned}$$
 if \(t \in [0,1] {{\setminus }} I_{q+2}\),$$\begin{aligned} \Vert R_{q+1}(t)\Vert _{L^1} \le \Vert R_q(t)\Vert _{L^1} \le \Vert R_0(t)\Vert _{L^1}, \end{aligned}$$
The Perturbations
In this and the next two sections we prove Proposition 3.1. In particular in this section we fix the constant M in the statement of the proposition, we define the functions \(\rho _1\) and \(u_1\) and we prove some estimates on them. In Sect. 5 we define \(R_1\) and we prove some estimates on it. In Sect. 6 we conclude the proof of Proposition 3.1, by proving estimates (26a)–(26d).
Mikado Fields and Mikado Densities
The first step towards the definition of \(\rho _1, u_1\) is the construction of Mikado fields and Mikado densities.
Proposition 4.1
 (a)It holds where \(\{e_j\}_{j=1,\dots ,d}\) is the standard basis in \(\mathbb {R}^d\).
 (b)For every \(k \in \mathbb {N}\) and \(r \in [1,\infty ]\)$$\begin{aligned} \begin{aligned} \Vert D^k \Theta _\mu ^j\Vert _{L^r(\mathbb {T}^d)}&\le \Vert \Phi \Vert _{L^r(\mathbb {R}^{d1})} \mu ^{a + k  (d1) /r}, \\ \Vert D^k W_\mu ^j\Vert _{L^{r}(\mathbb {T}^d)}&\le \Vert \Phi \Vert _{L^r(\mathbb {R}^{d1})} \mu ^{b + k  (d1)/r}, \\ \end{aligned} \end{aligned}$$(35)
 (c)
For \(j \ne k\), \(\mathrm {supp} \ \Theta _\mu ^j = \mathrm {supp} \ W_\mu ^j\) and \(\mathrm {supp} \ \Theta _\mu ^j \cap \mathrm {supp} \ W_{\mu }^k = \emptyset \).
Remark 4.2
Proof of Proposition 4.1
Step 3 Finally notice that conditions (c) in the statement are not verified by \(\Theta _\mu ^j\) and \(W_\mu ^j\) defined in Step 2. However we can achieve (c), using that \(\mu > 2d\) and redefining \(\Theta _\mu ^j, W_\mu ^j\) after a suitable translation of the independent variable \(x \in \mathbb {T}^d\) for each \(j=1,\dots , d\). \(\square \)
Definition of the Perturbations
We are now in a position to define \(\rho _1\), \(u_1\). The constant M has already been fixed in (36). Let thus \(\eta , \delta , \sigma >0\) and \((\rho _0, u_0, R_0)\) be a smooth solution to the continuitydefect equation (25).
Estimates on the Perturbation
In this section we provide some estimates on \(\vartheta \), \(\vartheta _c\), w, \(w_c\).
Lemma 4.3
Proof
Lemma 4.4
Proof
Lemma 4.5
Proof
The proof is completely analogous to the proof of Lemma 4.3, with \(\eta ^{1}\) instead of \(\eta \) and \(\Vert W_\mu ^j\Vert _{L^{p'}}\) instead of \(\Vert \Theta _\mu ^j\Vert _{L^p}\), and thus it is omitted. \(\square \)
Lemma 4.6
Proof
Lemma 4.7
Proof
Lemma 4.8
Proof
The New Defect Field
In this section we continue the proof of Proposition 3.1, defining the new defect field \(R_1\) and proving some estimates on it.
Definition of the New Defect Field
Estimates on the Defect Field
We now prove some estimates on the different terms which define \(R_1\).
Lemma 5.1
Proof
Lemma 5.2
Proof
Lemma 5.3
Proof
The proof follows immediately from the definition of \(R^\psi \) in (46) and the definition of the cutoff \(\psi \). \(\square \)
Lemma 5.4
Proof
Lemma 5.5
Proof
We estimate separately each term in the definition (45) of \(R^{\mathrm{corr}}\).
Remark 5.6
Proof of Proposition 3.1
2. Estimate (26b). The estimate uses Lemmas 4.5 and 4.7 and it is completely similar to what we just did for (26a).
Sketch of the Proofs of Theorems 1.6, 1.9, 1.10
Theorems 1.6, 1.9, 1.10 can be proven in a very similar way to Theorem 1.2 and thus we present just a sketch of their proofs.
The proof of Theorem 1.2 follows from Proposition 3.1: similarly, for each one of Theorems 1.6, 1.9, 1.10 there is a corresponding main proposition, from which the proof the theorem follows.
Sketch of the proof of Theorem 1.9
Sketch of the proof of Proposition 7.1
Sketch of the proof of Theorem 1.6
Also for Theorem 1.6 there is a main proposition, analog to Proposition 3.1.
Proposition 7.2
Theorem 1.6 can be deduced from Proposition 7.2 exactly in the same way as Theorem 1.2 was deduced from Proposition 3.1. The only difference here is the following. In general, it is not true that \(\rho (t) \in L^p(\mathbb {T}^d)\), \(u(t) \in L^{p'} (\mathbb {T}^d)\). Therefore the fact that \(\rho u \in C((0,T); L^1(\mathbb {T}^d))\) is proven by showing that \(\rho \in C((0,T); L^s(\mathbb {T}^d))\) (thanks to (52a)) and \(u \in C((0,T); L^{s'} (\mathbb {T}^d))\) (thanks to (52b)).
Sketch of the proof of Proposition 7.2
Proposition 7.2 can now be proven exactly as we proved Proposition 3.1 in Sects. 4–6, this time using (53)–(55) instead of (37)–(38). \(\square \)
Sketch of the proof of Theorem 1.10
Sketch of the proof of Proposition 7.3
Notes
Acknowledgements
The authors would like to thank Gianluca Crippa for several very useful comments. This research was supported by the ERC Grant Agreement No. 724298.
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