Uniqueness for active scalar equations in a Zygmund space

Abstract

We consider a class of active scalar equations which includes, for example, the 2D Euler equations, the 2D Navier–Stokes equations, and various aggregation equations including the Keller–Segel model. For this class of equations, we establish uniqueness of solutions in the Zygmund space $C_{*}^{0}$ . This result improves upon that in (Trans. Amer. Math. Soc. 367 (2015) 3095–3118), where the authors show uniqueness of solutions in $BMO$ . As a corollary of our methods, we establish the uniform in space vanishing viscosity limit of Hölder continuous solutions to the aggregation equation with Newtonian potential.

Keywords

Active scalars inviscid limit

1. Introduction

In this paper we investigate uniqueness of solutions for a class of active scalar equations of the form $\begin{matrix} (1.1) & \partial_{t} ρ + \nabla \cdot (ρ V ρ) = ν Δ ρ, \end{matrix}$ where $ν ⩾ 0$ is a fixed constant, V is, broadly speakly, a linear smoothing operator of order one, and $ρ : R^{d} \times [0, T] \to R$ is the unknown scalar-valued function of space and time variables. Examples of PDE in this class include the two-dimensional Euler equations or Navier–Stokes equations, where ρ denotes the scalar vorticity, and some aggregation equations, including the Keller–Segel model [3,5] and the inviscid aggregation equation with Newtonian potential [4,8], where ρ typically denotes the density of some population.

Our goal is to show uniqueness of weak solutions to (1.1) in the Zygmund space $C_{*}^{0} (R^{d})$ , with $d = 2$ or 3. This is, to our knowledge, the strongest uniqueness result for this class of equations, improving upon a recent result of Azzam and Bedrossian [1], who establish uniqueness in the space $BMO (R^{d})$ . To prove our result, we use Littlewood–Paley theory and Bony’s paraproduct decomposition to estimate the difference of two solutions to (1.1) in a homogeneous Besov space ${\dot{B}}_{q, \infty}^{- 1}$ , with $q < \infty$ . Our methods are similar to those in [1] in that our uniqueness proof utilizes energy methods in a homogeneous space (in [1] the authors use the space ${\dot{H}}^{- 1}$ ). Unlike [1], however, we make use of Littlewood–Paley operators to prove estimates which are localized in Fourier space, allowing for a sharper result. Our techniques are motivated by those in [11], where Vishik applies Littlewood–Paley methods to prove uniqueness of solutions to the Euler equations in a Besov type space which contains $C_{*}^{0}$ .

As a corollary of our uniqueness theorem, in Section 4 we establish the uniform-in-space vanishing viscosity limit of the aggregation equation with Newtonian potential for weak solutions in a Hölder space $C^{α}$ , $α > 0$ . This result is an improvement of a result in [8], in which the authors establish the vanishing viscosity limit for strong solutions in $C^{α}$ with $α > 1$ . Our strategy is to first apply methods from the uniqueness proof to establish a ${\dot{B}}_{q, \infty}^{- 1}$ estimate for the difference of the solutions of the viscous and inviscid equations, respectively. We then use interpolation and Hölder regularity of the solutions to derive an estimate in the $L^{\infty}$ -norm.

The paper is organized as follows: in Section 2, we introduce the Littlewood–Paley operators and useful function spaces. We also state a few useful lemmas. We then introduce properties of solutions to the aggregation equation with Newtonian potential. In Section 3, we prove uniqueness of solutions to (1.1) in $C_{*}^{0}$ . In Section 4, we apply estimates from Section 3 to establish the vanishing viscosity limit of solutions to the aggregation equation with Newtonian potential.

2. Background and preliminary lemmas

2.1. Littlewood–Paley operators and function spaces

We first define the Littlewood–Paley operators. We let $φ \in S (R^{d})$ satisfy $supp φ \subset {ξ \in R^{d} : \frac{3}{4} ⩽ | ξ | ⩽ \frac{8}{3}}$ , and for every $j \in Z$ we let $φ_{j} (ξ) = φ (2^{- j} ξ)$ (so ${\overset{ˇ}{φ}}_{j} (x) = 2^{j d} \overset{ˇ}{φ} (2^{j} x)$ ). Observe that, if $| j - j^{'} | ⩾ 2$ , then $supp φ_{j} \cap supp φ_{j^{'}} = \emptyset$ . We define $ψ_{n} \in S (R^{d})$ by the equality $\begin{matrix} ψ_{n} (ξ) = 1 - \sum_{j > n} φ_{j} (ξ) \end{matrix}$ for all $ξ \in R^{d}$ . For $f \in S^{'} (R^{d})$ and $j \in Z$ , we define the inhomogeneous Littlewood–Paley operators $Δ_{j}$ by $\begin{array}{l} Δ_{j} f = \{\begin{matrix} 0, & j < - 1, \\ {\overset{ˇ}{ψ}}_{- 1} * f, & j = - 1, \\ {\overset{ˇ}{φ}}_{j} * f, & j > - 1, \end{matrix} \end{array}$ and for all $j \in Z$ , we define the homogeneous Littlewood–Paley operators ${\dot{Δ}}_{j}$ by $\begin{matrix} {\dot{Δ}}_{j} f = {\overset{ˇ}{φ}}_{j} * f . \end{matrix}$ Note that ${\dot{Δ}}_{j} f = Δ_{j} f$ when $j ⩾ 0$ .

Finally, for $f \in S^{'} (R^{d})$ we define the operator $S_{n} f$ by $\begin{matrix} S_{n} f = {\overset{ˇ}{ψ}}_{n} * f = \sum_{j = - \infty}^{n} Δ_{j} f . \end{matrix}$ It is well known that for all $f \in S^{'} (R^{d})$ , $S_{n} f$ converges to f in the sense of distributions (see, for example, [7]).

In the proof of the main theorem we use the paraproduct decomposition introduced by J.-M. Bony in [6]. We recall the definition of the paraproduct and remainder used in this decomposition.

Definition 2.1.
Define the paraproduct of two functions f and g by $\begin{matrix} T_{f} g = \sum_{\overset{i, j}{i ⩽ j - 2}} Δ_{i} f Δ_{j} g = \sum_{j = 1}^{\infty} S_{j - 2} f Δ_{j} g . \end{matrix}$ We use $R (f, g)$ to denote the remainder. $R (f, g)$ is given by the following bilinear operator: $\begin{matrix} R (f, g) = \sum_{\overset{i, j}{| i - j | ⩽ 1}} Δ_{i} f Δ_{j} g . \end{matrix}$

Bony’s decomposition then gives $\begin{matrix} f g = T_{f} g + T_{g} f + R (f, g) . \end{matrix}$ We now define the homogeneous and inhomogeneous Besov spaces.
Definition 2.2.
Let $s \in R$ , $(p, q) \in [1, \infty] \times [1, \infty)$ . The inhomogeneous Besov space $B_{p, q}^{s} (R^{d})$ is defined to be the space of tempered distributions f on $R^{d}$ such that $\begin{matrix} ‖ f ‖_{B_{p, q}^{s}} : = {(\sum_{j = - 1}^{\infty} 2^{j q s} {‖ Δ_{j} f ‖}_{L^{p}}^{q})}^{\frac{1}{q}} < \infty . \end{matrix}$ When $q = \infty$ , write $\begin{matrix} ‖ f ‖_{B_{p, \infty}^{s}} : = sup_{j ⩾ - 1} 2^{j s} {‖ Δ_{j} f ‖}_{L^{p}} . \end{matrix}$ Let $s \in R$ , $(p, q) \in [1, \infty] \times [1, \infty)$ . The homogeneous Besov space ${\dot{B}}_{p, q}^{s} (R^{d})$ is defined to be the space of tempered distributions f on $R^{d}$ such that $\begin{matrix} ‖ f ‖_{{\dot{B}}_{p, q}^{s}} : = {(\sum_{j = - \infty}^{\infty} 2^{j q s} {‖ {\dot{Δ}}_{j} f ‖}_{L^{p}}^{q})}^{\frac{1}{q}} < \infty . \end{matrix}$ When $q = \infty$ , write $\begin{matrix} ‖ f ‖_{{\dot{B}}_{p, \infty}^{s}} : = sup_{j \in Z} 2^{j s} {‖ {\dot{Δ}}_{j} f ‖}_{L^{p}} . \end{matrix}$

We also define the Zygmund spaces.
Definition 2.3.
Let $s \in R$ . The Zygmund space $C_{}^{s} (R^{d})$ is the set of all tempered distributions f on $R^{d}$ such that $\begin{matrix} ‖ f ‖_{C_{}^{s}} : = sup_{j ⩾ - 1} 2^{j s} ‖ Δ_{j} f ‖_{L^{\infty}} < \infty . \end{matrix}$

It is well-known that $C_{*}^{s} (R^{d})$ coincides with the classical Hölder space $C^{s} (R^{d})$ when s is not an integer and $s > 0$ .
2.2. Useful lemmas

We will make frequent use of Bernstein’s Lemma. We refer the reader to [7], chapter 2, for a proof of the lemma.

Lemma 2.4 (Bernstein’s Lemma).

Let $r_{1}$ and $r_{2}$ satisfy $0 < r_{1} < r_{2} < \infty$ , and let p and q satisfy $1 ⩽ p ⩽ q ⩽ \infty$ . There exists a positive constant C such that for every integer k, if u belongs to $L^{p} (R^{d})$ , and $supp \hat{u} \subset B (0, r_{1} λ)$ , then $\begin{matrix} (2.1) & sup_{| α | = k} {‖ \partial^{α} u ‖}_{L^{q}} ⩽ C^{k} λ^{k + d (\frac{1}{p} - \frac{1}{q})} ‖ u ‖_{L^{p}} . \end{matrix}$ Furthermore, if $supp \hat{u} \subset C (0, r_{1} λ, r_{2} λ)$ , then $\begin{matrix} (2.2) & C^{- k} λ^{k} ‖ u ‖_{L^{p}} ⩽ sup_{| α | = k} {‖ \partial^{α} u ‖}_{L^{p}} ⩽ C^{k} λ^{k} ‖ u ‖_{L^{p}} . \end{matrix}$

We also make use of the following positivity lemma. A proof of the lemma can be found in [9].

Lemma 2.5.
Assume f satisfies $f, Δ f \in L^{p} (R^{d})$ for some $p \in [2, \infty)$ . Then $\begin{matrix} - \int_{R^{d}} | f |^{p - 2} f Δ f d x ⩾ 0 . \end{matrix}$

Finally, Osgood’s Lemma will be useful in what follows. A proof of the lemma can be found in [7]. Lemma 2.6 (Osgood’s Lemma).

Let ρ be a positive Borel function, let γ be a locally integrable positive function, and let μ be a continuous, increasing function. Assume that for some number $β ⩾ 0$ , the function ρ satisfies $\begin{matrix} ρ (t) ⩽ β + \int_{t_{0}}^{t} γ (s) μ (ρ (s)) d s . \end{matrix}$ If $β > 0$ , then $\begin{matrix} - ϕ (ρ (t)) + ϕ (β) ⩽ \int_{t_{0}}^{t} γ (s) d s, \end{matrix}$ where $ϕ (x) = \int_{x}^{1} \frac{1}{μ (r)} d r$ . If $β = 0$ , and μ satisfies $\begin{matrix} \int_{0}^{1} \frac{d r}{μ (r)} = + \infty, \end{matrix}$ then ρ is identically zero.

2.3. Properties of the smoothing operator V

In what follows, we assume that V is a linear operator satisfying the following properties:

For each $p \in (1, \infty)$ and $f \in L^{p} (R^{d})$ , $‖ \nabla V f ‖_{L^{p}} ⩽ C (p) ‖ f ‖_{L^{p}}$ .

V commutes with the Littlewood–Paley operators, and for all $f \in S^{'} (R^{d})$ , there exists $C > 0$ such that for all $j \in Z$ , $‖ \nabla V {\dot{Δ}}_{j} f ‖_{L^{\infty}} ⩽ C ‖ {\dot{Δ}}_{j} f ‖_{L^{\infty}}$ .

Given p and q with $1 < p < q < \infty$ and $1 + \frac{1}{q} = \frac{d - 1}{d} + \frac{1}{p}$ , for all $f \in L^{p} (R^{d})$ , there exists $C > 0$ such that $‖ V f ‖_{L^{q}} ⩽ C ‖ f ‖_{L^{p}}$ .

Note that P1 is motivated by the boundedness of Calderón-Zygmund operators on $L^{p}$ , $p \in (1, \infty)$ , while P3 is motivated by the Hardy-Littlewood-Sobolev inequality.

2.4. Definition of weak solution to (1.1)

We will use the following definition of a weak solution to (1.1) in Section 3.

Definition 2.7.
We say that $ρ : [0, T] \times R^{d} \to R$ is a weak solution to (1.1) on $[0, T]$ if ρ belongs to $L^{2} (0, T; L^{2} (R^{d}))$ and if for all $ϕ \in C_{c}^{\infty} ([0, T] \times R^{d})$ , $\begin{matrix} \int_{0}^{T} \int_{R^{d}} ρ (ϕ_{t} + V ρ \cdot \nabla ϕ + ν Δ ϕ) d x d t . \end{matrix}$ Equivalently, to say that ρ is a weak solution to (1.1) means that (1.1) holds in the sense of distributions.

2.5. The aggregation equation with Newtonian potential

The aggregation equation with Newtonian potential is given by $\begin{array}{l} ({AG}_{ν}) \{\begin{matrix} \partial_{t} ρ^{ν} + \nabla \cdot (ρ^{ν} v^{ν}) = ν Δ ρ^{ν}, \\ v^{ν} = - \nabla Φ * ρ^{ν}, \\ ρ^{ν} (0) = ρ_{0} . \end{matrix} \end{array}$ Here Φ denotes the Newtonian potential, $ρ^{ν}$ is the density, and $v^{ν}$ is the velocity. The system ( ${AG}_{1}$ ) represents a limiting case of the Keller–Segel equation modeling chemotaxis. Note also that ( ${AG}_{ν}$ ) is a special case of ( $ASE$ ) with $V ρ^{ν} = - \nabla Φ * ρ^{ν}$ .

In Section 4, we establish the vanishing viscosity limit of $C^{α}$ solutions to ( ${AG}_{ν}$ ) for any fixed $α > 0$ . The vanishing viscosity limit for ( ${AG}_{ν}$ ) is addressed in [8], where the authors establish the limit under the assumption that the solutions belong to $C^{α}$ for $α > 1$ .

The solutions under consideration in Section 4 are weak solutions, satisfying Definition 2.7 above. We will assume in addition that the limiting solution $ρ^{0}$ to ( ${AG}_{0}$ ) is a so-called Lagrangian solution. Properties of Lagrangian solutions to ( ${AG}_{0}$ ) are discussed at length in Section 5 of [8]. We define Lagrangian solutions here and state a few properties that will be useful in Section 4. We refer the reader to [4] and Section 5 of [8] for further details.

Definition 2.8.
Fix $T > 0$ . Let $X : [0, T] \times R^{d} \to R^{d}$ with $X (t, \cdot)$ a homeomorphism for all $t \in [0, T]$ and let $ρ_{0} \in L^{\infty} (R^{d})$ . Define $ρ^{0} : [0, T] \times R^{d} \to R$ by $\begin{array}{l} (2.3) & ρ^{0} (t, x) = \frac{ρ_{0} (X^{- 1} (t, x))}{1 - t ρ_{0} (X^{- 1} (t, x))} \end{array}$ and let $v^{0} : = - \nabla Φ * ρ^{0}$ . Here, $X^{- 1}$ is defined by $X^{- 1} (t, X (t, x)) = x$ for all $(t, x) \in [0, T] \times R^{d}$ . Then $(X, ρ^{0}, v^{0})$ (or more simply $ρ^{0}$ ) is a Lagrangian solution to ( ${AG}_{0}$ ) with initial density $ρ_{0}$ if X is the flow map for $v^{0}$ ; that is, if $\begin{array}{l} X (t, x) = x + \int_{0}^{t} v^{0} (s, X (s, x)) d s \end{array}$ for all $t ⩾ 0$ , $x \in R^{d}$ .

We remark that a Lagrangian solution $ρ^{0}$ to ( ${AG}_{0}$ ) is also a weak solution to ( ${AG}_{0}$ ); see, for example, Theorem 5.4 of [8] and its proof.

The short-time existence of $C^{α}$ Lagrangian solutions to ( ${AG}_{0}$ ) in dimensions 2 and 3 with compactly supported initial density was first established in [4], with an alternate proof given in [8]. It follows from the equality (2.3) that, for each $t \in [0, \frac{1}{‖ ρ_{0} ‖_{L^{\infty}}})$ , Lagrangian solutions satisfy $\begin{matrix} (2.4) & {‖ ρ^{0} (t) ‖}_{L^{\infty}} ⩽ \frac{‖ ρ_{0} ‖_{L^{\infty}}}{1 - ‖ ρ_{0} ‖_{L^{\infty}} t}, \end{matrix}$ and for all $q \in [1, \infty)$ , $\begin{matrix} (2.5) & {‖ ρ^{0} (t) ‖}_{L^{q}} ⩽ ‖ ρ_{0} ‖_{L^{q}} {(1 - ‖ ρ_{0} ‖_{L^{\infty}} t)}^{\frac{1}{q} - 1} . \end{matrix}$ Moreover, if the support of the initial density $ρ_{0}$ is contained in a ball of radius $R_{0}$ , then for each fixed $t \in [0, T)$ , $ρ^{0} (t)$ has compactly support in $R^{d}$ , and the support of $ρ^{0} (t)$ is contained in a ball of radius $\begin{matrix} (2.6) & R (t) = R_{0} + \frac{C}{1 - T ‖ ρ_{0} ‖_{L^{\infty}}} (‖ ρ_{0} ‖_{L^{1}} + ‖ ρ_{0} ‖_{L^{\infty}}) t . \end{matrix}$ Finally, for $C^{α}$ Lagrangian solutions to ( ${AG}_{0}$ ), $α \in [0, 1)$ , the following inequality holds for each $t \in [0, T)$ : $\begin{matrix} (2.7) & \begin{matrix} {‖ \nabla X (t) ‖}_{L^{\infty}}, {‖ \nabla X^{- 1} (t) ‖}_{L^{\infty}} ⩽ C (T, ‖ ρ_{0} ‖_{L^{1}}, ‖ ρ_{0} ‖_{C^{α}}) . \end{matrix} \end{matrix}$ If, in addition, $ρ_{0}$ is in $C^{1} (R^{d})$ , then by the quotient rule, for each $t \in [0, T)$ , $\begin{matrix} (2.8) & \begin{matrix} {‖ \nabla ρ^{0} (t) ‖}_{L^{\infty}} ⩽ (1 + C t ‖ ρ_{0} ‖_{L^{\infty}}) {‖ {(1 - t ρ_{0})}^{- 1} ‖}_{L^{\infty}}^{2} ‖ \nabla ρ_{0} ‖_{L^{\infty}} {‖ \nabla X^{- 1} (t) ‖}_{L^{\infty}} . \end{matrix} \end{matrix}$
3. Statement and proof of uniqueness

In this section we prove the following theorem.

Theorem 3.1.
Let $ν ⩾ 0$ and $T > 0$ , and assume $p_{0}$ and $p_{1}$ satisfy $1 < p_{0} < d < p_{1} < \infty$ . Suppose $ρ^{1}$ and $ρ^{2}$ are weak solutions of ( 1.1 ) in $C ([0, T]; L^{p_{0}} \cap L^{p_{1}} \cap C_{}^{0} (R^{d}))$ with* $ρ_{0}^{1} = ρ_{0}^{2}$ . Then $ρ^{1} (t) = ρ^{2} (t)$ for every $t \in [0, T]$ .
Proof.
To prove Theorem 3.1, we estimate the difference between $ρ^{1}$ and $ρ^{2}$ in the homogeneous Besov space ${\dot{B}}_{q, \infty}^{- 1}$ for q sufficiently large.

Assume that $ρ^{1}$ and $ρ^{2}$ solve (1.1). Let $\bar{ρ} = ρ^{1} - ρ^{2}$ . Then $\bar{ρ}$ satisfies $\begin{matrix} (3.1) & \partial_{t} \bar{ρ} + \nabla \cdot (\bar{ρ} V ρ_{1}) + \nabla \cdot (ρ_{2} V \bar{ρ}) = ν Δ \bar{ρ} . \end{matrix}$ For fixed $l \in Z$ , we apply the ${\dot{Δ}}_{l}$ operator to (3.1). This gives $\begin{matrix} (3.2) & \partial_{t} {\dot{Δ}}_{l} \bar{ρ} - ν Δ {\dot{Δ}}_{l} \bar{ρ} = - ({\dot{Δ}}_{l} \nabla \cdot (\bar{ρ} V ρ_{1}) + {\dot{Δ}}_{l} \nabla \cdot (ρ_{2} V \bar{ρ})) : = - (A_{l} + B_{l}) . \end{matrix}$ We must manipulate $A_{l} = {\dot{Δ}}_{l} \nabla \cdot (\bar{ρ} V ρ_{1})$ further. By the product rule, $\begin{matrix} A_{l} & = {\dot{Δ}}_{l} \nabla \cdot (\bar{ρ} V ρ_{1}) \\ = \nabla \cdot [{\dot{Δ}}_{l} (V ρ_{1} \bar{ρ}) - S_{l - 2} V ρ_{1} {\dot{Δ}}_{l} \bar{ρ}] + \nabla \cdot (S_{l - 2} V ρ_{1} {\dot{Δ}}_{l} \bar{ρ}) \\ = R_{l} (V ρ_{1}, \bar{ρ}) + \nabla \cdot (S_{l - 2} V ρ_{1} {\dot{Δ}}_{l} \bar{ρ}) \\ = R_{l} (V ρ_{1}, \bar{ρ}) + \nabla {\dot{Δ}}_{l} \bar{ρ} \cdot S_{l - 2} V ρ_{1} + {\dot{Δ}}_{l} \bar{ρ} \nabla \cdot S_{l - 2} V ρ_{1} \\ = R_{l} (V ρ_{1}, \bar{ρ}) + A_{l}^{1} + A_{l}^{2}, \end{matrix}$ where $\begin{matrix} R_{l} (V ρ_{1}, \bar{ρ}) = \nabla \cdot [{\dot{Δ}}_{l} (V ρ_{1} \bar{ρ}) - S_{l - 2} V ρ_{1} {\dot{Δ}}_{l} \bar{ρ}], \\ A_{l}^{1} = \nabla {\dot{Δ}}_{l} \bar{ρ} \cdot S_{l - 2} V ρ_{1}, and \\ A_{l}^{2} = {\dot{Δ}}_{l} \bar{ρ} \nabla \cdot S_{l - 2} V ρ_{1} . \end{matrix}$ In order to utilize Property P3 of V in what follows, we assume s satisfies $1 + \frac{1}{s} = \frac{d - 1}{d} + \frac{1}{p_{0}}$ , and we fix $q \in [2, \infty)$ sufficiently large to justify the calculations. We multiply (3.2) by ${\dot{Δ}}_{l} \bar{ρ} | {\dot{Δ}}_{l} \bar{ρ} |^{q - 2}$ and integrate to obtain $\begin{matrix} (3.3) & \begin{matrix} \frac{1}{q} \frac{d}{d t} \int_{R^{n}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q} - ν \int_{R^{d}} {\dot{Δ}}_{l} \bar{ρ} | {\dot{Δ}}_{l} \bar{ρ} |^{q - 2} Δ {\dot{Δ}}_{l} \bar{ρ} \\ = - \int_{R^{d}} {\dot{Δ}}_{l} \bar{ρ} | {\dot{Δ}}_{l} \bar{ρ} |^{q - 2} (R_{l} (V ρ_{1}, \bar{ρ}) + A_{l}^{1} + A_{l}^{2} + B_{l}) . \end{matrix} \end{matrix}$ It follows from Lemma 2.5 that $\begin{matrix} - ν \int_{R^{d}} {\dot{Δ}}_{l} \bar{ρ} | {\dot{Δ}}_{l} \bar{ρ} |^{q - 2} Δ {\dot{Δ}}_{l} \bar{ρ} ⩾ 0 . \end{matrix}$ Moreover, it follows from integration by parts that $\begin{matrix} \int_{R^{d}} {\dot{Δ}}_{l} \bar{ρ} | {\dot{Δ}}_{l} \bar{ρ} |^{q - 2} A_{l}^{1} d x ⩽ C \int_{R^{d}} | S_{l - 2} \nabla V ρ_{1} | | {\dot{Δ}}_{l} \bar{ρ} |^{q} d x ⩽ C ‖ S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q} . \end{matrix}$ Then (3.3) reduces to $\begin{matrix} (3.4) & \begin{matrix} \frac{d}{d t} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q} ⩽ & q (‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q - 1} {‖ R_{l} (V ρ_{1}, \bar{ρ}) ‖}_{L^{q}} + ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q - 1} {‖ A_{l}^{2} ‖}_{L^{q}} \\ + ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q - 1} ‖ B_{l} ‖_{L^{q}} + C ‖ S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q}) . \end{matrix} \end{matrix}$ After taking the time derivative of the left hand side and dividing through by $q ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}^{q - 1}$ , we conclude that $\begin{matrix} (3.5) & \frac{d}{d t} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} ⩽ {‖ R_{l} (V ρ_{1}, \bar{ρ}) ‖}_{L^{q}} + {‖ A_{l}^{2} ‖}_{L^{q}} + ‖ B_{l} ‖_{L^{q}} + C ‖ S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} . \end{matrix}$ We now estimate the terms on the right hand side. In what follows, we assume $p ⩾ 2$ is fixed (to be chosen later).

We begin with $‖ S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}}$ . First note that $S_{l - 2} \nabla V ρ_{1} = 0$ when $l ⩽ 0$ , so we may assume that $l ⩾ 1$ . For the case $l ⩾ 1$ , Bernstein’s Lemma and property P2 of V imply that $\begin{matrix} (3.6) & \begin{matrix} ‖ S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} & ⩽ ‖ Δ_{- 1} \nabla V ρ_{1} ‖_{L^{\infty}} + \sum_{k = 0}^{l - 2} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}} \\ ⩽ C ‖ Δ_{- 1} \nabla V ρ_{1} ‖_{L^{p_{1}}} + \sum_{k = 0}^{l - 2} ‖ Δ_{k} ρ_{1} ‖_{L^{\infty}} \\ ⩽ C ‖ ρ_{1} ‖_{L^{p_{1}}} + (l - 1) ‖ ρ_{1} ‖_{C_{}^{0}} . \end{matrix} \end{matrix}$ This gives $\begin{matrix} (3.7) & \begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} ‖ S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} & ⩽ C (‖ ρ_{1} ‖_{L^{p_{1}}} + (p - 1) ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} \\ ⩽ C (p - 1) (‖ ρ_{1} ‖_{L^{p_{1}}} + ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix} \end{matrix}$ We now estimate $‖ B_{l} ‖_{L^{q}} = ‖ {\dot{Δ}}_{l} \nabla \cdot (ρ_{2} V \bar{ρ}) ‖_{L^{q}}$ . Following [10], we define $\begin{matrix} J^{l, k} : = \partial_{k} {\dot{Δ}}_{l} ({(V \bar{ρ})}^{k} ρ_{2}), \end{matrix}$ and we use Bony’s paraproduct decomposition to write $\begin{matrix} J^{l, k} = & \partial_{k} {\dot{Δ}}_{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} S_{j - 2} {(V \bar{ρ})}^{k} Δ_{j} ρ_{2} + \partial_{k} {\dot{Δ}}_{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} Δ_{j} {(V \bar{ρ})}^{k} S_{j - 2} ρ_{2} \\ + \partial_{k} {\dot{Δ}}_{l} \sum_{| j - j^{'} | ⩽ 1, max {j, j^{'}} ⩾ l - 3} Δ_{j} {(V \bar{ρ})}^{k} Δ_{j^{'}} ρ_{2} \\ : = & J_{1}^{l, k} + J_{2}^{l, k} + J_{3}^{l, k} . \end{matrix}$ We first estimate $J_{1}^{l, k}$ in the $L^{q}$ -norm. By Bernstein’s Lemma and properties P1 and P2 of V, $\begin{matrix} {‖ J_{1}^{l, k} ‖}_{L^{q}} & ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} ‖ S_{j - 2} V \bar{ρ} ‖_{L^{q}} ‖ Δ_{j} ρ_{2} ‖_{L^{\infty}} \\ ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} ‖ ρ_{2} ‖_{C_{}^{0}} \sum_{k ⩽ j - 2} ‖ Δ_{k} V \bar{ρ} ‖_{L^{q}} \\ ⩽ C 2^{l} ‖ ρ_{2} ‖_{C_{}^{0}} \sum_{k ⩽ l + 1} ‖ Δ_{k} V \bar{ρ} ‖_{L^{q}} \\ ⩽ C 2^{l} ‖ ρ_{2} ‖_{C_{}^{0}} (‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + \sum_{k = 0}^{l + 1} ‖ Δ_{k} V \bar{ρ} ‖_{L^{q}}) \\ ⩽ C 2^{l} ‖ ρ_{2} ‖_{C_{}^{0}} (‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + \sum_{k = 0}^{l + 1} 2^{- k} ‖ Δ_{k} \bar{ρ} ‖_{L^{q}}) \\ ⩽ C 2^{l} ‖ ρ_{2} ‖_{C_{}^{0}} (‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + (l + 2) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}}) . \end{matrix}$ Multiplying by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ J_{1}^{l, k} ‖}_{L^{q}} ⩽ C ‖ ρ_{2} ‖_{C_{}^{0}} (‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + (p + 2) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}}) . \end{matrix}$ We now estimate $J_{2}^{l, k}$ in the $L^{q}$ -norm. Again by Bernstein’s Lemma and properties P1 and P2 of V, $\begin{matrix} {‖ J_{2}^{l, k} ‖}_{L^{q}} & ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} ‖ S_{j - 2} ρ_{2} ‖_{L^{\infty}} ‖ Δ_{j} V \bar{ρ} ‖_{L^{q}} \\ ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} j ‖ ρ_{2} ‖_{C_{}^{0}} ‖ Δ_{j} V \bar{ρ} ‖_{L^{q}} \\ ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} j ‖ ρ_{2} ‖_{C_{}^{0}} 2^{- j} ‖ Δ_{j} \nabla V \bar{ρ} ‖_{L^{q}} \\ ⩽ C 2^{l} max {l + 3, 1} ‖ ρ_{2} ‖_{C_{}^{0}} ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix}$ Multiplying by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ J_{2}^{l, k} ‖}_{L^{q}} ⩽ C ‖ ρ_{2} ‖_{C_{}^{0}} (p + 3) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix}$ Finally, we estimate $J_{3}^{l, k}$ . By Bernstein’s Lemma and properties of Littlewood–Paley operators, $\begin{matrix} (3.8) & \begin{matrix} {‖ J_{3}^{l, k} ‖}_{L^{q}} ⩽ C 2^{l} \sum_{j ⩾ max {l - 3, - 1}} ‖ Δ_{j} ρ_{2} ‖_{L^{\infty}} ‖ Δ_{j} V \bar{ρ} ‖_{L^{q}} ⩽ C 2^{l} ‖ ρ_{2} ‖_{C_{}^{0}} ‖ V \bar{ρ} ‖_{B_{q, 1}^{0}} . \end{matrix} \end{matrix}$ By Bernstein’s Lemma and property P1 of V, $\begin{matrix} ‖ V \bar{ρ} ‖_{B_{q, 1}^{0}} & ⩽ ‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + \sum_{j = 0}^{p - 1} ‖ Δ_{j} V \bar{ρ} ‖_{L^{q}} + \sum_{j ⩾ p} ‖ Δ_{j} V \bar{ρ} ‖_{L^{q}} \\ ⩽ ‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + \sum_{j = 0}^{p - 1} 2^{- j} ‖ Δ_{j} \nabla V \bar{ρ} ‖_{L^{q}} + \sum_{j ⩾ p} 2^{j d (\frac{1}{p_{1}} - \frac{1}{q})} 2^{- j} ‖ Δ_{j} \nabla V \bar{ρ} ‖_{L^{p_{1}}} \\ ⩽ ‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}, \end{matrix}$ where we used that $p_{1} > d$ to get the last inequality. Substituting this estimate into (3.8), multiplying by $2^{- l}$ , and taking the supremum over $l ⩽ p$ gives $\begin{matrix} (3.9) & sup_{- \infty < l ⩽ p} 2^{- l} {‖ J_{3}^{l, k} ‖}_{L^{q}} ⩽ C ‖ ρ_{2} ‖_{C_{}^{0}} (‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}) . \end{matrix}$ Combining the estimates for $J_{1}^{l, k}$ , $J_{2}^{l, k}$ , and $J_{3}^{l, k}$ , we conclude that $\begin{matrix} (3.10) & \begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} ‖ B_{l} ‖_{L^{q}} & = sup_{- \infty < l ⩽ p} 2^{- l} {‖ {\dot{Δ}}_{l} \nabla \cdot (ρ_{2} V \bar{ρ}) ‖}_{L^{q}} \\ ⩽ sup_{- \infty < l ⩽ p} \sum_{k = 1}^{d} 2^{- l} ({‖ J_{1}^{l, k} ‖}_{L^{q}} + {‖ J_{2}^{l, k} ‖}_{L^{q}} + {‖ J_{3}^{l, k} ‖}_{L^{q}}) \\ ⩽ C ‖ ρ_{2} ‖_{C_{}^{0}} (‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} + (p + 3) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}) . \end{matrix} \end{matrix}$

We now estimate $R_{l} (V ρ_{1}, \bar{ρ}) = \nabla \cdot ({\dot{Δ}}_{l} (V ρ_{1} \bar{ρ}) - S_{l - 2} V ρ_{1} {\dot{Δ}}_{l} \bar{ρ})$ . We follow techniques used in [11] and [2]. Specifically, we write $\begin{matrix} (3.11) & R_{l} = R_{l}^{1} + R_{l}^{2} + R_{l}^{3} + R_{l}^{4}, \end{matrix}$ where $\begin{matrix} (3.12) & \begin{matrix} R_{l}^{1} = \nabla \cdot ({\dot{Δ}}_{l} T_{\bar{ρ}} V ρ_{1}), \\ R_{l}^{2} = - \nabla \cdot [T_{V ρ_{1}}, {\dot{Δ}}_{l}] \bar{ρ}, \\ R_{l}^{3} = \nabla \cdot (T_{V ρ_{1} - S_{l - 2} V ρ_{1}} {\dot{Δ}}_{l}) \bar{ρ}, \\ R_{l}^{4} = \nabla \cdot {{\dot{Δ}}_{l} R (V ρ_{1}, \bar{ρ}) - R (S_{l - 2} V ρ_{1}, {\dot{Δ}}_{l} \bar{ρ})} . \end{matrix} \end{matrix}$ By estimating each of the four terms individually, we will show that $\begin{matrix} (3.13) & \begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} ‖ R_{l} ‖_{L^{q}} ⩽ & C (‖ ρ_{1} ‖_{C_{}^{0}} + ‖ ρ_{1} ‖_{L^{p_{1}}} + ‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}}) \\ \times (‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + (p + 3) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}) . \end{matrix} \end{matrix}$

We begin with $R_{l}^{1}$ . By Bernstein’s Lemma and property P2 of V, $\begin{matrix} (3.14) & \begin{matrix} {‖ R_{l}^{1} ‖}_{L^{q}} & ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} ‖ S_{j - 2} \bar{ρ} ‖_{L^{q}} ‖ Δ_{j} V ρ_{1} ‖_{L^{\infty}} \\ ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} \sum_{k ⩽ j - 2} 2^{k} 2^{- k} ‖ Δ_{k} \bar{ρ} ‖_{L^{q}} 2^{- j} ‖ \nabla Δ_{j} V ρ_{1} ‖_{L^{\infty}} \\ ⩽ 2^{l} \sum_{| j - l | ⩽ 3, j ⩾ 1} 2^{j} \sum_{k ⩽ j - 2} 2^{- k} ‖ Δ_{k} \bar{ρ} ‖_{L^{q}} 2^{- j} ‖ Δ_{j} ρ_{1} ‖_{L^{\infty}} \\ ⩽ C 2^{l} ‖ ρ_{1} ‖_{C_{}^{0}} (‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + max {1, l + 2} ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}}) . \end{matrix} \end{matrix}$ Multiplying by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} (3.15) & sup_{- \infty < l ⩽ p} 2^{- l} {‖ R_{l}^{1} ‖}_{L^{2}} ⩽ C ‖ ρ_{1} ‖_{C_{}^{0}} (‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + (p + 2) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}}) . \end{matrix}$ We now estimate $R_{l}^{2}$ . Writing out the commutator and using properties of Littlewood–Paley operators gives $\begin{matrix} R_{l}^{2} & = - \nabla \cdot [T_{V ρ_{1}}, {\dot{Δ}}_{l}] \bar{ρ} \\ = - \nabla \cdot (T_{V ρ_{1}} {\dot{Δ}}_{l} \bar{ρ} - {\dot{Δ}}_{l} (T_{V ρ_{1}} \bar{ρ})) \\ = - \nabla \cdot (\sum_{j ⩾ 1, | j - l | ⩽ 3} S_{j - 2} V ρ_{1} Δ_{j} {\dot{Δ}}_{l} \bar{ρ} - {\dot{Δ}}_{l} \sum_{j ⩾ 1, | j - l | ⩽ 3} S_{j - 2} V ρ_{1} Δ_{j} \bar{ρ}) . \end{matrix}$ Applying the $L^{q}$ -norm, Bernstein’s Lemma, and property P2 of V gives $\begin{matrix} {‖ R_{l}^{2} ‖}_{L^{q}} & ⩽ 2^{l} {‖ \sum_{j ⩾ 1, | j - l | ⩽ 3} (S_{j - 2} V ρ_{1} Δ_{j} {\dot{Δ}}_{l} \bar{ρ} - {\dot{Δ}}_{l} (S_{j - 2} V ρ_{1} Δ_{j} \bar{ρ})) ‖}_{L^{q}} \\ = 2^{l} {‖ \sum_{j ⩾ 1, | j - l | ⩽ 3} \int_{R^{d}} {\overset{ˇ}{φ}}_{l} (y) (S_{j - 2} V ρ_{1} (x - y) - S_{j - 2} V ρ_{1} (x)) Δ_{j} \bar{ρ} (x - y) d y ‖}_{L_{x}^{q}} \\ ⩽ 2^{l} \sum_{j ⩾ 1, | j - l | ⩽ 3} \int_{R^{d}} {‖ {\overset{ˇ}{φ}}_{l} (y) (S_{j - 2} V ρ_{1} (x - y) - S_{j - 2} V ρ_{1} (x)) Δ_{j} \bar{ρ} (x - y) ‖}_{L_{x}^{q}} d y \\ = 2^{l} \sum_{j ⩾ 1, | j - l | ⩽ 3} ‖ S_{j - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}} \int_{R^{d}} | 2^{l d} \overset{ˇ}{φ} (2^{l} y) | | y | d y \\ ⩽ C \sum_{j ⩾ 1, | j - l | ⩽ 3} ‖ S_{j - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}}, \\ ⩽ C (‖ Δ_{- 1} \nabla V ρ_{1} ‖_{L^{\infty}} + max {1, l + 2} ‖ ρ_{1} ‖_{C_{0}^{}}) sup_{j ⩾ 1, | j - l | ⩽ 3} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}}, \end{matrix}$ where we used the mean value theorem to get the last equality, and we applied a change of variables to get the second-to-last inequality. Multiplying the resulting estimate by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ R_{l}^{2} ‖}_{L^{q}} ⩽ C (‖ Δ_{- 1} \nabla V ρ_{1} ‖_{L^{\infty}} + (p + 2) ‖ ρ_{1} ‖_{C_{0}^{}}) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix}$

We now estimate $R_{l}^{3}$ . First note that by the product rule and properties of Littlewood–Paley operators, $\begin{matrix} (3.16) & \begin{matrix} {‖ R_{l}^{3} ‖}_{L^{q}} & = {‖ \nabla \cdot \sum_{| j - l | ⩽ 3, j ⩾ 1} S_{j - 2} (V ρ_{1} - S_{l - 2} V ρ_{1}) Δ_{j} {\dot{Δ}}_{l} \bar{ρ} ‖}_{L^{q}} \\ ⩽ \sum_{| k - l | ⩽ 3} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + \sum_{| k - l | ⩽ 3} ‖ Δ_{k} V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \nabla \bar{ρ} ‖_{L^{q}} . \end{matrix} \end{matrix}$ We consider three cases separately: $l < - 2$ , $- 2 ⩽ l ⩽ 2$ , and $l > 2$ . For the first case, $R_{l}^{3}$ is identically zero. For the second case, Bernstein’s Lemma and properties P1 and P2 of V give $\begin{array}{l} {‖ R_{l}^{3} ‖}_{L^{q}} ⩽ & (‖ Δ_{- 1} \nabla V ρ_{1} ‖_{L^{\infty}} + \sum_{| k - l | ⩽ 3, k ⩾ 0} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}}) ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ + (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + \sum_{| k - l | ⩽ 3, k ⩾ 0} ‖ Δ_{k} V ρ_{1} ‖_{L^{\infty}}) ‖ {\dot{Δ}}_{l} \nabla \bar{ρ} ‖_{L^{q}} \\ ⩽ & (‖ Δ_{- 1} \nabla V ρ_{1} ‖_{L^{\infty}} + C ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ + C (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + \sum_{| k - l | ⩽ 3, k ⩾ 0} 2^{- k} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}}) 2^{l} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ ⩽ & C (‖ ρ_{1} ‖_{L^{p_{1}}} + ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + C (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} . \end{array}$ Finally, for the third case, we can again use Bernstein’s Lemma and property P2 of V to write $\begin{matrix} {‖ R_{l}^{3} ‖}_{L^{q}} & ⩽ \sum_{| k - l | ⩽ 3, k ⩾ 0} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + \sum_{| k - l | ⩽ 3, k ⩾ 0} ‖ Δ_{k} V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \nabla \bar{ρ} ‖_{L^{q}} \\ ⩽ \sum_{| k - l | ⩽ 3, k ⩾ 0} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + \sum_{| k - l | ⩽ 3, k ⩾ 0} 2^{l - k} ‖ Δ_{k} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ ⩽ C ‖ ρ_{1} ‖_{C_{}^{0}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} . \end{matrix}$

Combining the three cases above, multiplying by $2^{- l}$ , and taking the supremum over $l ⩽ p$ gives $\begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ R_{l}^{3} ‖}_{L^{q}} ⩽ C (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + ‖ ρ_{1} ‖_{L^{p_{1}}} + ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix}$ Finally, we estimate $R_{l}^{4}$ . We do this without utilizing the difference. We first write $\begin{matrix} {‖ R_{l}^{4} ‖}_{L^{q}} ⩽ {‖ \nabla \cdot {\dot{Δ}}_{l} R (\bar{ρ}, V ρ_{1}) ‖}_{L^{q}} + {‖ \nabla \cdot R (S_{l - 2} V ρ_{1}, {\dot{Δ}}_{l} \bar{ρ}) ‖}_{L^{q}} : = {‖ R_{l}^{4, 1} ‖}_{L^{q}} + {‖ R_{l}^{4, 2} ‖}_{L^{q}} . \end{matrix}$ Then using an argument similar to that in [10], we apply Bernstein’s Lemma and property P2 of V to deduce that $\begin{matrix} {‖ R_{l}^{4, 1} ‖}_{L^{q}} & ⩽ C 2^{l} \sum_{j ⩾ max {l - 3, - 1}} ‖ Δ_{j} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}} \\ ⩽ C 2^{l} (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + \sum_{j ⩾ 0} 2^{- j} ‖ \nabla Δ_{j} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}}) \\ ⩽ C 2^{l} (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + \sum_{j ⩾ 0} 2^{- j} ‖ Δ_{j} ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}}) \\ ⩽ C 2^{l} (‖ ρ_{1} ‖_{C_{}^{0}} + ‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}}) ‖ \bar{ρ} ‖_{B_{q, 1}^{- 1}} . \end{matrix}$ Multiplying by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} (3.17) & sup_{- \infty < l ⩽ p} 2^{- l} {‖ R_{l}^{4, 1} ‖}_{L^{q}} ⩽ C (‖ ρ_{1} ‖_{C_{}^{0}} + ‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}}) ‖ \bar{ρ} ‖_{B_{q, 1}^{- 1}} . \end{matrix}$ By the definition of ${\dot{B}}_{q, \infty}^{- 1}$ and Bernstein’s Lemma, $\begin{matrix} ‖ \bar{ρ} ‖_{B_{q, 1}^{- 1}} & ⩽ C ‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + \sum_{0 ⩽ j < p} 2^{- j} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}} + \sum_{j ⩾ p} 2^{- j} ‖ Δ_{j} \bar{ρ} ‖_{L^{q}} \\ ⩽ C ‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + \sum_{j ⩾ p} 2^{- j} 2^{j d (\frac{1}{p_{1}} - \frac{1}{q})} ‖ \bar{ρ} ‖_{L^{p_{1}}} \\ ⩽ C ‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}, \end{matrix}$ where we used that $p_{1} > d$ to get the last inequality. After substituting this estimate into (3.17), we conclude that $\begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ R_{l}^{4, 1} ‖}_{L^{q}} ⩽ C (‖ ρ_{1} ‖_{C_{}^{0}} + ‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}}) (‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}) . \end{matrix}$ We now estimate $‖ R_{l}^{4, 2} ‖_{L^{q}}$ . We apply the definition of the remainder term, the product rule, properties of Littlewood–Paley operators, Bernstein’s Lemma, and property P2 of V to write $\begin{matrix} {‖ R_{l}^{4, 2} ‖}_{L^{q}} ⩽ & C \sum_{| j - l | ⩽ 1} ‖ Δ_{j} S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + C \sum_{| j - l | ⩽ 1} ‖ Δ_{j} S_{l - 2} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} {\dot{Δ}}_{l} \nabla \bar{ρ} ‖_{L^{q}} \\ ⩽ & C ‖ Δ_{- 1} S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{- 1} {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + C ‖ Δ_{- 1} S_{l - 2} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{- 1} {\dot{Δ}}_{l} \nabla \bar{ρ} ‖_{L^{q}} \\ + C \sum_{| j - l | ⩽ 1, j ⩾ 0} (‖ Δ_{j} S_{l - 2} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + ‖ Δ_{j} S_{l - 2} V ρ_{1} ‖_{L^{\infty}} ‖ Δ_{j} {\dot{Δ}}_{l} \nabla \bar{ρ} ‖_{L^{q}}) \\ ⩽ & C ‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + C \sum_{| j - l | ⩽ 1, j ⩾ 0} ‖ Δ_{j} \nabla V ρ_{1} ‖_{L^{\infty}} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ ⩽ & C (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + \sum_{| j - l | ⩽ 1, j ⩾ 0} ‖ Δ_{j} ρ_{1} ‖_{L^{\infty}}) ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} . \end{matrix}$ Multiplying by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ R_{l}^{4, 2} ‖}_{L^{q}} ⩽ C (‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + ‖ ρ_{1} ‖_{C_{}^{0}}) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix}$ Combining the estimates for $R_{l}^{1}$ through $R_{l}^{4}$ , we conclude that $\begin{matrix} (3.18) & \begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} ‖ R_{l} ‖_{L^{q}} ⩽ & C (‖ ρ_{1} ‖_{C_{}^{0}} + ‖ ρ_{1} ‖_{L^{p_{1}}} + ‖ Δ_{- 1} V ρ_{1} ‖_{L^{\infty}}) \\ \times (‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} + (p + 3) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}}) . \end{matrix} \end{matrix}$

We now estimate $A_{l}^{2} = {\dot{Δ}}_{l} \bar{ρ} \nabla \cdot S_{l - 2} V ρ_{1}$ . Note that $A_{l}^{2}$ is identically zero when $l ⩽ 0$ . For $l ⩾ 1$ , write $\begin{matrix} ‖ {\dot{Δ}}_{l} \bar{ρ} \nabla \cdot S_{l - 2} V ρ_{1} ‖_{L^{q}} & ⩽ ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} ‖ \nabla \cdot S_{l - 2} V ρ_{1} ‖_{L^{\infty}} \\ ⩽ ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \sum_{k = - 1}^{l - 2} ‖ \nabla \cdot Δ_{k} V ρ_{1} ‖_{L^{\infty}} \\ ⩽ ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} (‖ \nabla \cdot Δ_{- 1} V ρ_{1} ‖_{L^{\infty}} + \sum_{k = 0}^{max {0, l - 2}} ‖ \nabla \cdot Δ_{k} V ρ_{1} ‖_{L^{\infty}}) \\ ⩽ ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} (C ‖ ρ_{1} ‖_{L^{p_{1}}} + (l - 1) ‖ ρ_{1} ‖_{C_{}^{0}}), \end{matrix}$ where we used Bernstein’s Lemma and properties P1 and P2 of V to get the last inequality. Multiplying by $2^{- l}$ and taking the supremum over $l ⩽ p$ gives $\begin{matrix} (3.19) & \begin{matrix} sup_{- \infty < l ⩽ p} 2^{- l} {‖ A_{l}^{2} ‖}_{L^{q}} & ⩽ C ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} (‖ ρ_{1} ‖_{L^{p_{1}}} + (p - 1) ‖ ρ_{1} ‖_{C_{}^{0}}) \\ ⩽ C (p - 1) ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} (‖ ρ_{1} ‖_{L^{p_{1}}} + ‖ ρ_{1} ‖_{C_{}^{0}}) . \end{matrix} \end{matrix}$

We now estimate the high frequencies. By Bernstein’s Lemma, $\begin{matrix} (3.20) & sup_{l ⩾ p} 2^{- l} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} ⩽ sup_{l ⩾ p} 2^{- l} 2^{l d (\frac{1}{p_{1}} - \frac{1}{q})} ‖ \bar{ρ} ‖_{L^{p_{1}}} ⩽ sup_{l ⩾ p} 2^{- \frac{l d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}} ⩽ 2^{- \frac{p d}{q}} ‖ \bar{ρ} ‖_{L^{p_{1}}} \end{matrix}$ since $p_{1} > d$ by assumption.

We integrate (3.5) in time, multiply by $2^{- l}$ , take the supremum over $l ⩽ p$ , and apply the estimates (3.7), (3.10), (3.18) and (3.19). Combining the resulting estimate with (3.20) gives $\begin{matrix} (3.21) & \begin{matrix} {‖ \bar{ρ} (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ & 2^{- \frac{p d}{q}} {‖ \bar{ρ} (t) ‖}_{L^{p_{1}}} \\ + C \int_{0}^{t} ({‖ ρ_{1} (s) ‖}_{C_{}^{0}} + {‖ ρ_{2} (s) ‖}_{C_{}^{0}} + {‖ ρ_{1} (s) ‖}_{L^{p_{1}}} + {‖ Δ_{- 1} V ρ_{1} (s) ‖}_{L^{\infty}}) \\ \times ({‖ Δ_{- 1} \bar{ρ} (s) ‖}_{L^{q}} + {‖ Δ_{- 1} V \bar{ρ} (s) ‖}_{L^{q}} + p {‖ \bar{ρ} (s) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- \frac{p d}{q}} {‖ \bar{ρ} (s) ‖}_{L^{p_{1}}}) d s . \end{matrix} \end{matrix}$ Before we apply Osgood’s Lemma, we must estimate several of the terms under the time integral in (3.21). We first estimate $‖ Δ_{- 1} V \bar{ρ} (s) ‖_{L^{q}}$ . Observe that, by Bernstein’s Lemma and properties P1, P2, and P3 of V, $\begin{matrix} (3.22) & \begin{matrix} ‖ Δ_{- 1} V \bar{ρ} ‖_{L^{q}} & ⩽ ‖ {\overset{ˇ}{ψ}}_{- p} * V \bar{ρ} ‖_{L^{q}} + ‖ Δ_{- 1} V \bar{ρ} - {\overset{ˇ}{ψ}}_{- p} * V \bar{ρ} ‖_{L^{q}} \\ ⩽ ‖ {\overset{ˇ}{ψ}}_{- p} * V \bar{ρ} ‖_{L^{q}} + \sum_{k = - p}^{- 1} ‖ {\dot{Δ}}_{k} V \bar{ρ} ‖_{L^{q}} \\ ⩽ 2^{- p d (\frac{1}{s} - \frac{1}{q})} ‖ {\overset{ˇ}{ψ}}_{- p} * V \bar{ρ} ‖_{L^{s}} + C p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} \\ ⩽ 2^{- p d (\frac{1}{s} - \frac{1}{q})} ‖ \bar{ρ} ‖_{L^{p_{0}}} + C p ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix} \end{matrix}$

We must also estimate $‖ Δ_{- 1} V ρ_{1} (s) ‖_{L^{\infty}}$ . By Bernstein’s Lemma, $\begin{matrix} (3.23) & {‖ Δ_{- 1} V ρ_{1} (s) ‖}_{L^{\infty}} ⩽ C {‖ V ρ_{1} (s) ‖}_{L^{s}} ⩽ C {‖ ρ_{1} (s) ‖}_{L^{p_{0}}}, \end{matrix}$ where we again used property P3 of V.

To estimate $‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}}$ , we again apply Bernstein’s Lemma and our choice of $s > p_{0}$ to write $\begin{matrix} (3.24) & \begin{matrix} ‖ Δ_{- 1} \bar{ρ} ‖_{L^{q}} & ⩽ ‖ {\overset{ˇ}{ψ}}_{- p} * \bar{ρ} ‖_{L^{q}} + ‖ Δ_{- 1} \bar{ρ} - {\overset{ˇ}{ψ}}_{- p} * \bar{ρ} ‖_{L^{q}} \\ ⩽ 2^{- p d (\frac{1}{p_{0}} - \frac{1}{q})} ‖ {\overset{ˇ}{ψ}}_{- p} * \bar{ρ} ‖_{L^{p_{0}}} + \sum_{j = - p}^{- 1} 2^{j} 2^{- j} ‖ {\dot{Δ}}_{j} \bar{ρ} ‖_{L^{q}} \\ ⩽ 2^{- p d (\frac{1}{p_{0}} - \frac{1}{q})} ‖ \bar{ρ} ‖_{L^{p_{0}}} + C ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ 2^{- p d (\frac{1}{s} - \frac{1}{q})} ‖ \bar{ρ} ‖_{L^{p_{0}}} + C ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix} \end{matrix}$ Assume that q is sufficiently large to ensure that $ε_{0} = min {d (\frac{1}{s} - \frac{1}{q}), \frac{d}{q}} > 0$ . Substituting (3.22), (3.23), and (3.24) into (3.21) gives $\begin{matrix} (3.25) & \begin{matrix} {‖ \bar{ρ} (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ & 2^{- ε_{0} p} {‖ \bar{ρ} (t) ‖}_{L^{p_{1}}} \\ + C \int_{0}^{t} ({‖ ρ_{1} (s) ‖}_{C_{}^{0}} + {‖ ρ_{2} (s) ‖}_{C_{}^{0}} + {‖ ρ_{1} (s) ‖}_{L^{p_{1}}} + {‖ ρ_{1} (s) ‖}_{L^{p_{0}}}) \\ \times (p {‖ \bar{ρ} (s) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- ε_{0} p} {‖ \bar{ρ} (s) ‖}_{L^{p_{0}} \cap L^{p_{1}}}) d s . \end{matrix} \end{matrix}$

Observe that, for q sufficiently large, Bernstein’s Lemma gives $\begin{matrix} ‖ \bar{ρ} ‖_{{\dot{B}}_{q, \infty}^{- 1}} & = sup_{l \in Z} 2^{- l} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ ⩽ sup_{l ⩽ 0} 2^{- l} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} + sup_{l > 0} 2^{- l} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{q}} \\ ⩽ sup_{l ⩽ 0} 2^{- l} 2^{l d (\frac{1}{p_{0}} - \frac{1}{q})} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{p_{0}}} + sup_{l > 0} 2^{- l} 2^{l d (\frac{1}{p_{1}} - \frac{1}{q})} ‖ {\dot{Δ}}_{l} \bar{ρ} ‖_{L^{p_{1}}} \\ ⩽ ‖ \bar{ρ} ‖_{L^{p_{0}}} + ‖ \bar{ρ} ‖_{L^{p_{1}}} ⩽ ‖ ρ_{1} ‖_{L^{p_{0}} \cap L^{p_{1}}} + ‖ ρ_{2} ‖_{L^{p_{0}} \cap L^{p_{1}}} . \end{matrix}$ Set $\begin{matrix} M = sup_{t \in [0, T]} ({‖ ρ_{1} (t) ‖}_{L^{p_{0}} \cap L^{p_{1}}} + {‖ ρ_{2} (t) ‖}_{L^{p_{0}} \cap L^{p_{1}}}), \end{matrix}$ and for each $t \in [0, T]$ , set $\begin{matrix} δ (t) = \frac{\int_{0}^{t} ‖ \bar{ρ} (s) ‖_{{\dot{B}}_{q, \infty}^{- 1}} d s}{M T} ⩽ 1 . \end{matrix}$ Let $p = \frac{1}{ε_{0}} (2 - log δ (t))$ . Substituting this value of p into (3.25) gives $\begin{matrix} {‖ \bar{ρ} (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ (\frac{C M^{2} (T + 1)}{ε_{0}}) δ (t) (2 - log δ (t)) = C (M, T, ε_{0}) δ (t) (2 - log δ (t)) . \end{matrix}$ Integrating both sides from 0 to t and dividing both sides by $M T$ gives $\begin{matrix} (3.26) & δ (t) ⩽ C (M, T, ε_{0}) \int_{0}^{t} δ (s) (2 - log δ (s)) d s . \end{matrix}$ We apply Osgood’s Lemma with $ρ (t) = δ (t)$ , $μ (r) = r (2 - log r)$ , and $γ (t) = C (M, T, ε_{0})$ for each $t \in [0, T]$ . This proves Theorem 3.1. □

4. The vanishing viscosity limit for ( ${AG}_{ν}$ )

In this section we establish the vanishing viscosity limit for Hölder continuous solutions to the aggregation equation with Newtonian potential. These solutions and their properties are discussed in Section 2. We prove the following theorem.

Theorem 4.1.
Let $T > 0$ and $α > 0$ be fixed, and let $d = 2$ or 3. Let $ρ^{ν}$ and $ρ^{0}$ be solutions to ( ${AG}_{ν}$ ) and ( ${AG}_{0}$ ), respectively, in $C ([0, T]; C^{α} (R^{d}))$ , generated from the same compactly supported initial data $ρ_{0} \in C^{α} (R^{d})$ . There exists $C > 0$ such that for ν sufficiently small and for any $t < T$ and $β < α$ , $\begin{matrix} {‖ (ρ^{ν} - ρ^{0}) (t) ‖}_{L^{\infty}} ⩽ C ν^{β C e^{- C t}} . \end{matrix}$
Proof.
We first apply Theorem 3.1 to show that $ρ^{ν}$ converges to $ρ^{0}$ as $ν \to 0$ in the ${\dot{B}}_{q, \infty}^{- 1}$ -norm for q sufficiently large. Specifically, we show that under the assumptions of Theorem 4.1, $\begin{matrix} (4.1) & {‖ (ρ^{ν} - ρ^{0}) (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ C ν^{β e^{- C t}} . \end{matrix}$ We then use interpolation and spatial regularity of $ρ^{ν}$ and $ρ^{0}$ to complete the proof of Theorem 4.1.

Let $χ \in C_{c}^{\infty} (R^{d})$ be a smooth bump function with $χ (x) = 1$ for all $x \in B_{1} (0)$ , and $χ (x) = 0$ for all $x \in {B_{2} (0)}^{c}$ . Set $χ_{n} (x) = χ (\frac{x}{n})$ for each n and each $x \in R^{d}$ , and set $ρ_{0, n} = χ_{n} S_{n} ρ_{0}$ . Assume $ρ_{n}^{ν}$ and $ρ_{n}^{0}$ are solutions to ( ${AG}_{ν}$ ) and ( ${AG}_{0}$ ), respectively, with initial data $ρ_{0, n}$ . For fixed $t \in [0, T]$ , write $\begin{matrix} {‖ (ρ^{ν} - ρ^{0}) (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} & ⩽ {‖ (ρ^{ν} - ρ_{n}^{ν}) (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} + ‖ (ρ_{n}^{ν} - ρ_{n}^{0}) (t) ‖_{{\dot{B}}_{q, \infty}^{- 1}} + ‖ (ρ_{n}^{0} - ρ^{0}) (t) ‖_{{\dot{B}}_{q, \infty}^{- 1}} \\ : = δ_{1} (t) + δ_{2} (t) + δ_{3} (t) . \end{matrix}$ We apply the proof of Theorem 3.1 to $δ_{1}$ and $δ_{3}$ , keeping in mind that we are now considering the difference of two solutions generated from two distinct initial densities. The resulting estimates on $δ_{1}$ and $δ_{3}$ , while similar to (3.26), will thus have an extra term on the right hand side involving $ρ_{0}$ and $ρ_{0, n}$ . Specifically, one can conclude that, for $j = 1$ , 3, $\begin{matrix} (4.2) & \begin{matrix} δ_{j} (t) ⩽ \frac{‖ ρ_{0} - ρ_{0, n} ‖_{{\dot{B}}_{q, \infty}^{- 1}}}{M} + C (M, T, ε_{0}) \int_{0}^{t} δ_{j} (s) (2 - log δ_{j} (s)) d s, \end{matrix} \end{matrix}$ We must estimate $‖ ρ_{0} - ρ_{0, n} ‖_{{\dot{B}}_{q, \infty}^{- 1}}$ . Write $\begin{matrix} ‖ ρ_{0} - ρ_{0, n} ‖_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ ‖ ρ_{0} - χ_{n} ρ_{0} ‖_{{\dot{B}}_{q, \infty}^{- 1}} + ‖ χ_{n} ρ_{0} - χ_{n} S_{n} ρ_{0} ‖_{{\dot{B}}_{q, \infty}^{- 1}} . \end{matrix}$ Since $ρ_{0}$ is compactly supported, for sufficiently large n, $‖ ρ_{0} - χ_{n} ρ_{0} ‖_{{\dot{B}}_{q, \infty}^{- 1}} = 0$ . It remains to estimate $‖ χ_{n} ρ_{0} - χ_{n} S_{n} ρ_{0} ‖_{{\dot{B}}_{q, \infty}^{- 1}}$ . By Bernstein’s Lemma, for q sufficiently large, $\begin{matrix} ‖ χ_{n} ρ_{0} - χ_{n} S_{n} ρ_{0} ‖_{{\dot{B}}_{q, \infty}^{- 1}} & = sup_{j \in Z} 2^{- j} {‖ {\dot{Δ}}_{j} (χ_{n} (ρ_{0} - S_{n} ρ_{0})) ‖}_{L^{q}} \\ ⩽ sup_{j < 0} 2^{- j} {‖ {\dot{Δ}}_{j} (χ_{n} (ρ_{0} - S_{n} ρ_{0})) ‖}_{L^{q}} + sup_{j ⩾ 0} 2^{- j} {‖ {\dot{Δ}}_{j} (χ_{n} (ρ_{0} - S_{n} ρ_{0})) ‖}_{L^{q}} \\ ⩽ sup_{j < 0} 2^{- j} 2^{j d (1 - 1 / q)} {‖ {\dot{Δ}}_{j} (χ_{n} (ρ_{0} - S_{n} ρ_{0})) ‖}_{L^{1}} + ‖ χ_{n} ‖_{L^{q}} ‖ ρ_{0} - S_{n} ρ_{0} ‖_{L^{\infty}} \\ ⩽ ‖ χ_{n} ‖_{L^{1}} ‖ ρ_{0} - S_{n} ρ_{0} ‖_{L^{\infty}} + ‖ χ_{n} ‖_{L^{q}} ‖ ρ_{0} - S_{n} ρ_{0} ‖_{L^{\infty}} \\ ⩽ (‖ χ_{n} ‖_{L^{1}} + ‖ χ_{n} ‖_{L^{q}}) \sum_{k ⩾ n} 2^{- k α} 2^{k α} ‖ Δ_{k} ρ_{0} ‖_{L^{\infty}} \\ ⩽ C n^{d} 2^{- n α} ⩽ C 2^{- n β} \end{matrix}$ for all $β < α$ , where we performed a change of variables on $χ_{n}$ to get the second to last inequality, and where C depends on the initial data.

Combining these estimates gives, for sufficiently large n and q, $\begin{matrix} ‖ ρ_{0} - ρ_{0, n} ‖_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ C 2^{- n β} \end{matrix}$ for any $β < α$ . Substituting this estimate into (4.2) gives, for any $β < α$ , and for $j = 1$ , 3, $\begin{matrix} δ_{j} (t) ⩽ \frac{C 2^{- n β}}{M} + C (M, T, ε_{0}) \int_{0}^{t} δ_{j} (s) (2 - log δ_{j} (s)) d s . \end{matrix}$ By Osgood’s Lemma, for $j = 1$ , 3, $\begin{matrix} - log (2 - log δ_{j} (t)) + log (2 - log (\frac{C 2^{- n β}}{M})) ⩽ C (M, T, ε_{0}) t . \end{matrix}$ Taking the exponential twice gives, for $j = 1$ , 3, $\begin{matrix} (4.3) & δ_{j} (t) ⩽ e^{2 - 2 e^{- C (M, T, ε_{0}) t}} {(\frac{C 2^{- n β}}{M})}^{e^{- C (M, T, ε_{0}) t}} . \end{matrix}$

Now consider the term $δ_{2} (t) = ‖ (ρ_{n}^{ν} - ρ_{n}^{0}) (t) ‖_{{\dot{B}}_{q, \infty}^{- 1}}$ . In this case, the two solutions in the difference are generated from the same initial data. However, as $ρ_{n}^{0}$ satisfies the inviscid equation, when taking the difference of $ρ_{n}^{ν}$ and $ρ_{n}^{0}$ , we see that an equation analogous to (3.1) holds, but with the extra term $ν Δ ρ_{n}^{0}$ on the right hand side. Applying the proof of Theorem 3.1 to this slightly modified equation results in the estimate $\begin{matrix} (4.4) & δ_{2} (t) ⩽ \frac{C ν T ‖ Δ ρ_{n}^{0} ‖_{L^{\infty} (0, T; {\dot{B}}_{q, \infty}^{- 1})}}{M} + C (M, T, ε_{0}) \int_{0}^{t} δ_{2} (s) (2 - log δ_{2} (s)) d s . \end{matrix}$ To estimate $‖ Δ ρ_{n}^{0} ‖_{{\dot{B}}_{q, \infty}^{- 1}}$ , first observe that, by Bernstein’s Lemma, $\begin{matrix} {‖ Δ ρ_{n}^{0} ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ {‖ ρ_{n}^{0} ‖}_{{\dot{B}}_{q, \infty}^{1}} . \end{matrix}$ Using the compact support of $ρ_{n}^{0}$ and Bernstein’s Lemma, we can write $\begin{matrix} {‖ ρ_{n}^{0} ‖}_{{\dot{B}}_{q, \infty}^{1}} & ⩽ sup_{j ⩽ 0} {‖ {\dot{Δ}}_{j} ρ_{n}^{0} ‖}_{L^{q}} + sup_{j > 0} 2^{- j} 2^{j} {‖ {\dot{Δ}}_{j} \nabla ρ_{n}^{0} ‖}_{L^{q}} \\ ⩽ {‖ ρ_{n}^{0} ‖}_{L^{q}} + {‖ \nabla ρ_{n}^{0} ‖}_{L^{q}} ⩽ C {(m (B_{n} (t)))}^{1 / q} {‖ ρ_{n}^{0} ‖}_{C^{1}}, \end{matrix}$ where the support of $ρ_{n}^{0} (t)$ is contained in $B_{n} (t)$ , a ball with radius $R_{n} (t)$ . By (2.6), it follows that $R_{n} (t)$ satisfies $\begin{matrix} R_{n} (t) = R_{n} (0) + \frac{C t}{1 - T ‖ ρ_{0, n} ‖_{L^{\infty}}} ‖ ρ_{0, n} ‖_{L^{1} \cap L^{\infty}} . \end{matrix}$ Since $R_{n} (0) = 2 n$ , and $‖ ρ_{0, n} ‖_{L^{1} \cap L^{\infty}} ⩽ ‖ ρ_{0} ‖_{L^{1} \cap L^{\infty}}$ , for sufficiently large n, $m (B_{n} (t))$ can be bounded above by $C (T, ρ_{0}) n^{d}$ . Therefore, $\begin{matrix} (4.5) & {‖ Δ ρ_{n}^{0} ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ {‖ ρ_{n}^{0} ‖}_{{\dot{B}}_{q, \infty}^{1}} ⩽ C (T, ρ_{0}) n^{d / q} {‖ ρ_{n}^{0} ‖}_{C^{1}} . \end{matrix}$ Moreover, by (2.4), (2.7), (2.8), and the estimates $\begin{matrix} ‖ ρ_{n, 0} ‖_{L^{\infty}} ⩽ ‖ ρ_{0} ‖_{L^{\infty}}, \\ ‖ ρ_{n, 0} ‖_{C^{α}} ⩽ ‖ χ_{n} ‖_{C^{α}} ‖ S_{n} ρ_{0} ‖_{C^{α}} ⩽ C ‖ ρ_{0} ‖_{C^{α}}, \end{matrix}$ it follows that $\begin{matrix} {‖ ρ_{n}^{0} ‖}_{C^{1}} & ⩽ {‖ ρ_{n}^{0} ‖}_{L^{\infty}} + {‖ \nabla ρ_{n}^{0} ‖}_{L^{\infty}} \\ ⩽ \frac{‖ ρ_{n, 0} ‖_{L^{\infty}}}{1 - ‖ ρ_{n, 0} ‖_{L^{\infty}} t} + (1 + C t ‖ ρ_{n, 0} ‖_{L^{\infty}}) {‖ {(1 - t ρ_{n, 0})}^{- 1} ‖}_{L^{\infty}}^{2} ‖ \nabla ρ_{n, 0} ‖_{L^{\infty}} {‖ \nabla X_{n}^{- 1} (t) ‖}_{L^{\infty}} \\ ⩽ C (T, ‖ ρ_{0} ‖_{L^{\infty}}) (1 + ‖ \nabla ρ_{n, 0} ‖_{L^{\infty}} {‖ \nabla X_{n}^{- 1} (t) ‖}_{L^{\infty}}) \\ ⩽ C (T, ‖ ρ_{0} ‖_{L^{1}}, ‖ ρ_{0} ‖_{C^{α}}) (1 + ‖ ρ_{n, 0} ‖_{C^{1}}) \\ ⩽ C (T, ‖ ρ_{0} ‖_{L^{1}}, ‖ ρ_{0} ‖_{C^{α}}) (1 + ‖ χ_{n} ‖_{C^{1}} ‖ S_{n} ρ_{0} ‖_{C^{1}}) . \end{matrix}$ Note that $‖ χ_{n} ‖_{C^{1}} ⩽ \frac{C}{n} ⩽ C$ , and $\begin{matrix} ‖ S_{n} ρ_{0} ‖_{C^{1}} ⩽ \sum_{j = - 1}^{n} (‖ Δ_{j} ρ_{0} ‖_{L^{\infty}} + ‖ Δ_{j} \nabla ρ_{0} ‖_{L^{\infty}}) ⩽ C ‖ ρ_{0} ‖_{C^{α}} \sum_{j = - 1}^{n} 2^{j (1 - α)} ⩽ C 2^{n (1 - α)} . \end{matrix}$ Thus, $\begin{matrix} {‖ ρ_{n}^{0} ‖}_{C^{1}} ⩽ C (T, ‖ ρ_{0} ‖_{L^{1}}, ‖ ρ_{0} ‖_{C^{α}}) 2^{n (1 - α)} . \end{matrix}$ Substituting this estimate into (4.5) gives, for n sufficiently large, $\begin{matrix} {‖ Δ ρ_{n}^{0} ‖}_{{\dot{B}}_{q, \infty}^{- 1}} ⩽ C (T, ρ_{0}) n^{d / q} 2^{n (1 - α)} ⩽ C (T, ρ_{0}) 2^{n (1 - β)} \end{matrix}$ for any $β < α$ . Setting $ν = 2^{- n}$ in (4.4) and applying the above estimate gives $\begin{matrix} δ_{2} (t) ⩽ \frac{C 2^{- n β}}{M} + C (M, T, ε_{0}) \int_{0}^{t} δ_{2} (s) (2 - log δ_{2} (s)) d s . \end{matrix}$ Again applying Osgood’s Lemma and taking the exponential twice gives $\begin{matrix} (4.6) & δ_{2} (t) ⩽ e^{2 - 2 e^{- C (M, T, ε_{0}) t}} {(\frac{C 2^{- n β}}{M})}^{e^{- C (M, T, ε_{0}) t}} . \end{matrix}$ Combining the estimates for $δ_{1}$ , $δ_{2}$ , and $δ_{3}$ and using that $ν = 2^{- n}$ gives, for all $β < α$ and for ν sufficiently small, $\begin{matrix} {‖ (ρ^{ν} - ρ^{0}) (t) ‖}_{{\dot{B}}_{q, \infty}^{- 1}} & ⩽ δ_{1} (t) + δ_{2} (t) + δ_{3} (t) \\ ⩽ C e^{2 - 2 e^{- C (M, T, ε_{0}) t}} {(\frac{C ν^{β}}{M})}^{e^{- C (M, T, ε_{0}) t}} \\ ⩽ C (M, T, ε_{0}) ν^{β e^{- C (M, T, ε_{0}) t}}, \end{matrix}$ establishing (4.1).

4.1. Convergence in $L^{\infty}$

We now apply an interpolation argument to show that the vanishing viscosity limit actually holds in the $L^{\infty}$ -norm. By Bernstein’s Lemma, for any fixed $N ⩾ 1$ , $\begin{array}{l} {‖ ρ^{ν} - ρ^{0} ‖}_{L^{\infty}} \\ ⩽ {‖ {\overset{ˇ}{ψ}}_{- N} * (ρ^{ν} - ρ^{0}) ‖}_{L^{\infty}} + \sum_{j = - N}^{N} {‖ {\dot{Δ}}_{j} (ρ^{ν} - ρ^{0}) ‖}_{L^{\infty}} + \sum_{j = N + 1}^{\infty} {‖ {\dot{Δ}}_{j} (ρ^{ν} - ρ^{0}) ‖}_{L^{\infty}} \\ ⩽ C 2^{- N \frac{d}{p_{0}}} {‖ {\overset{ˇ}{ψ}}_{- N} * (ρ^{ν} - ρ^{0}) ‖}_{L^{p_{0}}} + \sum_{j = - N}^{N} 2^{j (d / q + 1)} {‖ ρ^{ν} - ρ^{0} ‖}_{{\dot{B}}_{q, \infty}^{- 1}} + \sum_{j = N + 1}^{\infty} {‖ {\dot{Δ}}_{j} (ρ^{ν} - ρ^{0}) ‖}_{L^{\infty}} \\ ⩽ C 2^{- N \frac{d}{p_{0}}} {‖ ρ^{ν} - ρ^{0} ‖}_{L^{p_{0}}} + (C + 2^{N (d / q + 1)}) {‖ ρ^{ν} - ρ^{0} ‖}_{{\dot{B}}_{q, \infty}^{- 1}} + 2^{- N α} ({‖ ρ^{ν} ‖}_{C^{α}} + {‖ ρ^{0} ‖}_{C^{α}}) \\ ⩽ C 2^{- N α} ({‖ ρ^{ν} ‖}_{L^{p_{0}}} + {‖ ρ^{0} ‖}_{L^{p_{0}}} + {‖ ρ^{ν} ‖}_{C^{α}} + {‖ ρ^{0} ‖}_{C^{α}}) + C 2^{N (d / q + 1)} {‖ ρ^{ν} - ρ^{0} ‖}_{{\dot{B}}_{q, \infty}^{- 1}} \\ ⩽ C (M, T, ε_{0}) (2^{- N α} + 2^{N (d / q + 1)} ν^{β e^{- C (M, T, ε_{0}) t}}), \end{array}$ where we used that $d > p_{0}$ to get the fourth inequality, and where we applied (4.1) to get the last inequality. To optimize the rate of convergence, we choose N such that $2^{- N α} = 2^{N (d / q + 1)} ν^{β e^{- C (M, T, ε_{0}) t}}$ . This gives $\begin{matrix} N = \frac{- 1}{1 + α + d / q} log (ν^{β e^{- C (M, T, ε_{0}) t}}) . \end{matrix}$ After substituting this value of N into the above calculation, we conclude that $\begin{matrix} {‖ (ρ_{ν} - ρ) (t) ‖}_{L^{\infty}} ⩽ C (M, T, ε_{0}) ν^{β C (α, d, q) e^{- C (M, T, ε_{0}) t}} . \end{matrix}$ This completes the proof of Theorem 4.1. □

Footnotes

Acknowledgements

The author was supported by Simons Foundation Grant No. 429578.

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