Non-uniqueness of the transport equation at high spacial integrability

In the smooth setting, the method of characteristics solves first-order PDE by converting the PDE into an appropriate system of ODE [37, Sec.3.2]. In particular, for Lipschitz vector fields $u\in C_{t}W^{1,\infty}_{x}$, the solution to (1.1) can be given by the Lagrangian representation $\rho(t,x)=\rho_{0}(X(0,t,x))$ with the flow map $X$ solving the ODEs

$$\left\{\begin{split}\partial_{t}X(t,\tau,x)&=\boldsymbol{u}(t,X(t,\tau,x)),\\ X(\tau,\tau,x)&=x.\end{split}\right.$$

(1.2)

This link results in the existence and uniqueness of the PDE (1.1) by the Cauchy-Lipschitz (Picard-Linderöf) theory for the ODE (1.2) (The Lipschitz condition on the vector fields can be replaced by one-side Lipschitz, $\log$-Lipschitz or Osgood condition, we refer to [1] for the elaborate surveys).

For non-Lipschitz vector fields, the link is less obvious and the uniqueness issue of (1.1) becomes subtler. The first breakthrough result traces back to the celebrated work of DiPerna and Lions [35], they proved that for all $p\in[1,\infty]$, the Lagrangian representation and uniqueness hold in the class $L^{\infty}_{t}L^{p}_{x}$ for any given vector field $\boldsymbol{u}\in L^{1}_{t}W^{1,p^{\prime}}_{x}$ with $\operatorname{div}\boldsymbol{u}\in L^{\infty}$, where $p^{\prime}$ is the Hölder conjugate exponent of $p$. This result was extended to the case $\rho\in L^{\infty}_{t,x},\boldsymbol{u}\in L^{1}_{t}BV$ by Ambrosio [6] with deep tools from geometric measure theory. Whereafter, there are abundant researches for the theory of non-smooth vector fields and their applications on non-linear PDEs. For instance, the analogous results of [35, 6] have been established, for vector fields with gradient given by a singular integral in [13] and for nearly incompressible $BV$ vector fields in [11], and the results in [13] has been adapted to obtain the Lagrangian solutions for the Vlasov-Poisson system with $L^{1}$ data [12], see also [8]. We refer to the works [22, 23, 38, 25, 18, 19] and the surveys [9, 28, 24, 7] for other important progresses and related results in this direction.

For the non-uniqueness issue of (1.1), two examples were given by DiPerna and Lions [35], for an autonomous vector field $\boldsymbol{u}\in W^{1,p^{\prime}}$ but $\operatorname{div}\boldsymbol{u}\notin L^{\infty}$ and for an autonomous divergence-free vector field $\boldsymbol{u}\in\cap_{0\leq s<1}W^{s,1}$ but $\boldsymbol{u}\notin W^{1,1}$. Much later, based on the work [2], Depauw [33] constructed an example of a divergence-free vector field $\boldsymbol{u}\in L^{1}_{\rm loc}((0,T];BV(\mathbb{R}^{2}))$ but $\boldsymbol{u}\notin L^{1}(0,T;BV(\mathbb{R}^{2}))$, in the class of bounded densities. See also [10, 42, 5]. In [3, 4], Alberti, Bianchini and Crippa showed an example of an autonomous divergence-free vector field $\boldsymbol{u}\in\cap_{0\leq\alpha<1}C^{0,\alpha}(\mathbb{R}^{2})$ but $\boldsymbol{u}\notin C^{0,1}(\mathbb{R}^{2})$, in the class of bounded densities. More recently, based on anomalous dissipation and mixing, Drivas, Elgindi, Iyer and Jeong proved non-uniqueness in the class $\rho\in L^{\infty}_{t}L^{2}_{x}$ for $\boldsymbol{u}\in L^{1}_{t}C^{1-}$ [36]. Roughly speaking, these results are proved by showing the non-uniqueness of the flow maps (the ODE level), with the violation of either the boundedness of $\operatorname{div}\boldsymbol{u}$ or the spacial $W^{1,1}_{\rm loc}$ regularity of $\boldsymbol{u}$.

Another approach is based on the convex integration technique (for complete summaries on this enormous theory, we refer to the surveys articles [30, 31, 32, 15, 17]), which gives counterexamples of uniqueness directly at the PDE level. The first non-uniqueness result with this technique was obtained by Crippa et al in [26] using the framework of [29], but for vector field $\boldsymbol{u}$ was merely bounded. Later, a breakthrough result for the Sobolev vector field was obtained by Modena and Székelyhidi [40]. Based on their work, a series of works for the non-uniqueness to (1.1) have been done recently, we list the main functional setting of them below:

1. (I).

   [40](Modena and Székelyhidi): $\rho\in C_{t}L^{p}_{x}$, $\boldsymbol{u}\in C_{t}(W^{1,q}_{x}\cap L^{p^{\prime}}_{x})$ for $\frac{1}{p}+\frac{1}{q}>1+\frac{1}{d-1}$, $p>1$ and $d\geq 3$.

2. (II).

   [41](Modena and Székelyhidi): $\rho\in C_{t}L^{1}_{x}$, $\boldsymbol{u}\in C_{t}W^{1,q}_{x}\cap C_{t,x}$ for $q<d-1$ and $d\geq 3$.(The case $p=1$ of [40])

3. (III).

   [39](Modena and Sattig): $\rho\in C_{t}L^{p}_{x}$, $\boldsymbol{u}\in C_{t}(W^{1,q}_{x}\cap L^{p^{\prime}}_{x})$ for $\frac{1}{p}+\frac{1}{q}>1+\frac{1}{d}$ and $d\geq 2$.

4. (IV).

   [14](Bruè, Colombo and De Lellis): $\rho>0$, $\rho\in C_{t}L^{p}_{x}$, $\boldsymbol{u}\in C_{t}(W^{1,q}_{x}\cap L^{p^{\prime}}_{x})$ for $\frac{1}{p}+\frac{1}{q}>1+\frac{1}{d}$, $p>1$ and $d\geq 2$.

5. (V).

   [20](Cheskidov and Luo): $\rho\in L^{1}_{t}L^{p}_{x}$, $\boldsymbol{u}\in L^{1}_{t}W^{1,q}_{x}\cap L^{\infty}_{t}L^{p^{\prime}}_{x}$ for $\frac{1}{p}+\frac{1}{q}>1$, $p>1$ and $d\geq 3$.

Comments:

Very recently, [19] proves that the uniqueness result holds for $\boldsymbol{u}\in L^{\infty}_{t}W^{1,q}_{x}$ with $q>d$ in the class $\rho\in L^{1}_{t,x}$ under the assumption that the so called forward-backward integral curves of $\boldsymbol{u}$ are trivial. Whereas, the non-uniqueness result of [39] holds for $\boldsymbol{u}\in L^{\infty}_{t}W^{1,q}_{x}$ with $q<d$ in the class $\rho\in C_{t}L^{1}_{x}$. Hence, roughly speaking when $p=1$, the regularity condition $\boldsymbol{u}\in L^{\infty}_{t}W^{1,q}_{x}$ with $q>d$ is optimal for the uniqueness of the weak solution in the class $L^{1}_{t,x}$.

When $p>1$, the uniqueness is unclear in the range $1<\frac{1}{p}+\frac{1}{q}\leq 1+\frac{1}{d}$ for $\rho\in C_{t}L^{p}_{x}$, $\boldsymbol{u}\in C_{t}(W^{1,q}_{x}\cap L^{p^{\prime}}_{x})$. However, [14, Theorem 1.5] states that the uniqueness result holds for positive density $\rho\in L^{\infty}_{t}L^{p}_{x}$ and $\boldsymbol{u}\in L^{1}_{t}W^{1,q}_{x}$ with $\frac{1}{p}+\frac{1}{q}<1+\frac{q-1}{(d-1)q}$, which goes beyond the DiPerna-Lions range.

In [20], the non-uniqueness with the sharp spacial integrability ($1/p+1/q>1$) has been reached but at the expensive of time regularity. They proved the result using the convex integration scheme of [40] combined with temporal intermittency and oscillations in the spirit of [16]. We refer to [16, 20, 17] for a more comprehensive discussion of the temporal intermittency.

A few months after we finished this paper, we learn that Cheskidov and Luo post a extremely nice result [21] on arXiv, in which they show the non-uniqueness result for $u\in L_{t}^{1}W_{x}^{1,p}$, $\forall p<\infty$, $\rho\in\cap_{p<\infty,k\in\mathbb{N}}L_{t}^{p}C^{k}$, it also implies that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. In view of this surprising result, we have reconsidered the implications of our study, see Remark 1.3.

Based on the frameworks of [39, 20], we obtain the following main result in this paper, which indicates that the non-uniqueness happens even in the range $1/p+1/q>1-\frac{p-1}{4(p+1)p}$, as long as $1\leq s<\infty$, $p>1$.

Notice $1-\frac{p-1}{4(p+1)p}<1$ as long as $p>1$. Hence, combine the uniqueness result in [35, Corollary II.1], we obtain immediately from Theorem 1.1 that at least in the following sense, the $L^{\infty}$ in time of the density $\rho$ is critical for the uniqueness of weak solutions to (1.1).

We identify $[0,T]$ with an 1-dimensional torus and the time-periodic function $f$ on $[0,T]$ means $f(t+nT)=f(t)$ for all $n\in\mathbb{Z}$. Theorem 1.1 follows immediately from the following theorem.

Indeed, assume $\|\rho(t)\|_{L^{p}}\equiv C$ for some $C>0$. On the one hand, due to $\|\rho-\tilde{\rho}\|_{L^{s}_{t}L^{p}_{x}}\leq\frac{1}{4}(\frac{T}{4})^{1/s}$, we have

$$\frac{T}{2}|C-1|\leq\int_{\frac{T}{4}}^{\frac{3T}{4}}\|\rho(t)-\tilde{\rho}(t)\|_{L^{p}_{x}}\,dt\leq\big{(}\frac{T}{2}\big{)}^{1/s^{\prime}}\|\rho-\tilde{\rho}\|_{L^{s}_{t}L^{p}_{x}}\leq\frac{T}{8},$$

hence $|C-1|\leq 1/4$, which implies $C>3/4$.

On the other hand,

$$\frac{T}{8}\geq\big{(}\frac{T}{4}\big{)}^{1/s^{\prime}}\|\rho-\tilde{\rho}\|_{L^{s}_{t}L^{p}_{x}}\geq\int_{[0,\frac{T}{8}]\cup[\frac{7T}{8},T]}\|\rho(t)-\tilde{\rho}(t)\|_{L^{p}_{x}}\,dt=\frac{T}{4}C,$$

hence $C\leq 1/2$, in contradiction with $C>3/4$, we prove the claim. ∎

The norms of $L^{p}(\mathbb{T}^{d})$, $L^{s}(0,T)$, $L^{s}(0,T;L^{p}(\mathbb{T}^{d}))$ will be denoted standardly as $\|\cdot\|_{L^{p}_{x}}$, $\|\cdot\|_{L^{s}_{t}}$, $\|\cdot\|_{L^{s}_{t}L^{p}_{x}}$ or just $\|\cdot\|_{L^{p}}$ when there is no confusion. The norm of $C^{k}([0,T]\times\mathbb{T}^{d})$ will be denoted as $\|\cdot\|_{C^{k}}$.

For any $f\in L^{1}(\mathbb{T}^{d})$, its spacial mean is $\fint_{\mathbb{T}^{d}}f\,dx=\int_{\mathbb{T}^{d}}f\,dx$ and denote simply as $\fint f$. We denote $C_{0}^{\infty}(\mathbb{T}^{d})$ as the space of smooth periodic functions with zero mean.

Denote the standard partial differential operators with multiindex ${\boldsymbol{k}}\in\mathbb{N}^{d}$ as $\partial^{\boldsymbol{k}}:=\partial_{x_{1}}^{k_{1}}\cdots\partial_{x_{d}}^{k_{d}}$. For any $f\in C^{\infty}(\mathbb{T}^{d})$, the obvious facts are $\|\partial^{\boldsymbol{k}}(f(\sigma\cdot))\|_{L^{p}}=\sigma^{|\boldsymbol{k}|}\|\partial^{\boldsymbol{k}}f\|_{L^{p}}$ for any $\forall p\in[1,\infty)$, $\forall\sigma\in\mathbb{N}_{+}$ and of course $\partial^{\boldsymbol{k}}f\in C_{0}^{\infty}(\mathbb{T}^{d})$ as long as ${\boldsymbol{k}}\neq 0$.

$a\lesssim b$ will be denoted as $a\leq Cb$ with some inessential constant $C$. If the constant depends on some quantities, for instance $r$, it will be denoted as $C_{r}$, and $a\lesssim_{r}b$ means $a\leq C_{r}b$.

$C_{*}$ represents a positive constant that might depend on the old solution $(\rho,\boldsymbol{u},\boldsymbol{R})$ and the constant $N$ but never on $\nu,\delta$ given in Proposition 2.1. $C_{*}$ may change from line to line.

By the classical Fourier analysis, for any $f\in C^{\infty}(\mathbb{T}^{d})$, a unique solution in $C_{0}^{\infty}(\mathbb{T}^{d})$ of the Poisson equation

$$\Delta u=f-\fint f$$

is given by $u(x)=\sum_{k\in\mathbb{Z}^{d}\setminus\{0\}}(4\pi|k|^{2})^{-1}e^{2\pi ik\cdot x}\hat{f}(k)$. Hence the standard anti-divergence operator $\mathcal{R}\colon C^{\infty}(\mathbb{T}^{d})\to C^{\infty}_{0}(\mathbb{T}^{d};\mathbb{R}^{d})$ can be defined as

$$\mathcal{R}f:=\Delta^{-1}\nabla f,$$

which satisfies

$$\operatorname{div}(\mathcal{R}f)=f-\fint f.$$

Obviously, for every $\sigma\in\mathbb{N}_{+}$ and $f\in C_{0}^{\infty}(\mathbb{T}^{d})$ there holds

$$\mathcal{R}[f(\sigma\cdot)](x)=\sigma^{-1}(\mathcal{R}f)(\sigma x).$$

Further, the first order bilinear anti-divergence operator $\mathcal{B}\colon C^{\infty}(\mathbb{T}^{d})\times C^{\infty}(\mathbb{T}^{d})\to C^{\infty}(\mathbb{T}^{d};\mathbb{R}^{d})$ can be defined as

$$\mathcal{B}(a,f):=a\mathcal{R}f-\mathcal{R}(\nabla a\cdot\mathcal{R}f),$$

which satisfies

$$\operatorname{div}(\mathcal{B}(a,f))=af-\fint af\text{ provided that }f\in C^{\infty}_{0}(\mathbb{T}^{d}).$$

The bilinear anti-divergence operator $\mathcal{B}$ has the additional advantage of gaining derivative from $f$ when $f$ has zero mean and a very small period. See also higher order variants in [39].

Now we define the Mikado flow $(\Phi_{j}^{\mu},\boldsymbol{W}_{j}^{\mu})$ given in [20], which are a family of periodic stationary solutions to the transport equation (1.1). Where $\Phi_{j}^{\mu}$ is the Mikado density, $\boldsymbol{W}_{j}^{\mu}$ is the Mikado filed.

Let $d\geq 3$. Fix $\boldsymbol{\Omega}\in C_{c}^{\infty}(\mathbb{R}^{d-1};\mathbb{R}^{d-1})$ satisfying

$$\operatorname{supp}\boldsymbol{\Omega}\subset(0,1)^{d-1},\quad\int_{\mathbb{R}^{d-1}}(\operatorname{div}\boldsymbol{\Omega})^{2}\,dx=1.$$

Denote $\phi=\operatorname{div}\boldsymbol{\Omega}$ and $\boldsymbol{\Omega}^{\mu}(x)=\boldsymbol{\Omega}(\mu x)$, $\phi^{\mu}(x)=\phi(\mu x)$. We can define a family of non-periodic stationary solutions to (1.1) as

	$\displaystyle\tilde{\Phi}_{j}^{\mu}(x)$	$\displaystyle=\mu^{\frac{d-1}{p}}\phi^{\mu}(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{d}),$
	$\displaystyle\tilde{\boldsymbol{W}}_{j}^{\mu}(x)$	$\displaystyle=\mu^{\frac{d-1}{p^{\prime}}}\phi^{\mu}(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{d})\boldsymbol{e}_{j}.$

The potential is defined as

$$\tilde{\boldsymbol{\Omega}}_{j}^{\mu}(x)=\mu^{-1+\frac{d-1}{p}}\boldsymbol{\Omega}^{\mu}(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{d}).$$

The periodic solutions $(\Phi_{j}^{\mu},\boldsymbol{W}_{j}^{\mu})$ with mutually disjoint supports can be constructed by translation and periodization of the non-periodic flow $(\tilde{\Phi}_{j}^{\mu},\tilde{\boldsymbol{W}}_{j}^{\mu})$. This is based on the geometrical fact that along any two directions in $\mathbb{R}^{d}$ for $d\geq 3$, there exist two disjoint lines.