Abstract
We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential \(W(x) \sim \log |x|\) for \(|x| \gg 1\) appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the \(C^\alpha \) regularity with a novel approach for this family of equations using modulus of continuity techniques.
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1 Introduction
In this work, we analyse the long time asymptotics for probability density solutions to the general aggregation-diffusion equation of the form
with \(W:{{\mathbb {R}}^n}\rightarrow {\mathbb {R}}\) being the interaction potential which is assumed to be symmetric \(W(x) = W(-x)\). The assumption of unit mass is not restrictive up to a change of variables due to the (formal) conservation of mass. This work is devoted to identify sharp conditions on the interaction potential W such that the intermediate asymptotic behaviour of the solutions to (P) is given by the heat kernel,
More precisely, our goal is to find the best possible conditions on W such that
and if possible, recover the optimal decay rates of the heat equation. Aggregation-diffusion equations of the form (P) with linear or nonlinear diffusion are ubiquitous in the literature due to the large number of applications in mathematical biology and mathematical physics, we refer to [16] and the references therein for a recent survey of related results. In the case of linear diffusion, (P) is usually referred as the McKean–Vlasov equation associated to a nonlinear SDE process via the mean field limit [12, 27].
The asymptotic simplification of (P) can be understood as the case in which the long-time asymptotics of McKean–Vlasov equations is dominated by the linear diffusion term leading to self-similar diffusive behavior for large times. Notice that this result is not true for instance for singular attractive potentials as the Keller–Segel model for chemotaxis or its variants in the diffusion dominated regime [7, 9, 15, 18, 19] or for McKean–Vlasov equations where the potentials may lead to phase transitions as in [3, 40]. In these cases, there are non-trivial stationary states of the equation (P) that attract the long-time dynamics for certain initial data.
Therefore, finding the sharpest conditions on W such that the asymptotic simplification occurs is a challenging question. Notice that even for bounded interaction potentials W, the mere time decay of \(L^p\) norms, \(1<p\leqq \infty \), was not known for general initial data. We also extend previous results of [14] in which strong integrability assumptions on W and \(\nabla W\) were imposed, as well as smallness conditions on \(\rho _0\). In [28] the authors study the case \(n = 1\) with \(\nabla W \in L^1\) showing (1.1) without rate for general initial datum.
There is a long literature devoted to the intermediate asymptotics of convection-diffusion equations. Results for the heat equation (\(W=0\)) can be recovered directly from the heat kernel representation (see, e.g. [41]). Better decay rates can be deduced by cancellation of higher order moments, as presented in [24]. In [2] the authors introduce entropy dissipation arguments through the logarithmic Sobolev inequality that work for a large array of diffusion problems (see also [39]). In [1] this method is applied to the heat equation, to recover improved decay rates. This technique was also used in [4] to recover similar results for (P) to recover decay rates, even when the linear diffusion \(\Delta \rho \) is replaced by the nonlinear diffusion \(\Delta A(\rho )\). In [25] the authors study the case where the convection is of the type \(a \cdot \nabla (|u|^{q-1} u)\).
As mentioned above, when W has certain growth at infinity there is no decay, and, in fact, \(\rho (t)\) converges to an stationary solution which can be recovered by minimisation of free-energy functional (see [17]). The key example, as we will discuss below, is \(W(x) = \chi \ln |x|\) (\(\chi >0\)), known as the Keller–Segel problem in \(n=2\). In [8] the authors discuss the case where \(\chi \) is smaller than a critical value, and prove there exists an asymptotic profile different from the Gaussian. This result also holds for any other n (see [6]). A variation of this problem is studied [36] also for small \(\chi \). When \(\chi \) is larger than the critical value, solutions may produce a Dirac delta in finite time.
To analyse the intermediate asymptotics of (P), one classically works in rescaled variables [2, 21, 39]. Following the parabolic scaling of the heat kernel, we introduce the new variables \(\tau :=\log \sqrt{2t+1}\) and \(y:=\frac{x}{\sqrt{2t+1}}\), and consider a rescaled density \({{\widetilde{\rho }}}\) given by
The first key element is the existence of a PDE for \({{\widetilde{\rho }}}\). It was shown in [14] that we can write
Notice that, even though \(\Vert {{\widetilde{W}}}(\tau ,\cdot ) \Vert _{L^1} = e^{-n\tau } \Vert W \Vert _{L^1}\) is decaying in \(\tau \) if \(W\in L^1({{\mathbb {R}}^n})\), the norms of \(\nabla {{{\widetilde{W}}}},\Delta {{{\widetilde{W}}}}\) can exponentially grow in \(L^p ({{\mathbb {R}}^n})\) depending on n and p. We still expect, under some assumptions, this last term to vanish asymptotically to recover the steady-state of the usual Fokker–Planck equation. However, one cannot directly use classical energy estimates to prove uniform-in-time \(L^p\) bounds of \({{\widetilde{\rho }}}\) without any additional smallness assumptions on W or \(\rho _0\).
First, we obtain a result of global existence and instant regularisation by standard techniques. We then introduce a new estimate on the variational structure of the equation to prove uniform-in-time bounds of natural quantities for the problem such as the second moment, the energy and the entropy. As a consequence, we also show uniform-in-time propagation of the \(L^2\), \(H^1\) and \(C^\alpha \) norms.
Theorem 1.1
Let \(W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\) and \(\rho _0 \in L^1_+ ({\mathbb {R}}^n)\) with unit mass. Then, there exists a unique mild solution of (P) such that \(\rho \in C([0,\infty ) ; L^1_+ ({{\mathbb {R}}^n})) \cap C((0,+\infty ); W^{k,p} ({{\mathbb {R}}^n}))\) for all \(p \in [1,\infty ], k \in {\mathbb {R}}\) with \(\rho (t)\) also of unit mass for \(t\geqq 0\). Assume, furthermore, \(W(x) = W(-x)\) and the initial datum has bounded second order moment and entropy
Then the rescaled density \({{\widetilde{\rho }}}\) satisfies
Moreover,
-
1.
If \(\nabla W \in L^n ({{\mathbb {R}}^n})\) then
$$\begin{aligned} \sup _{\tau \geqq 1} \Vert {{\widetilde{\rho }}}(\tau , \cdot ) \Vert _{H^1 } < \infty . \end{aligned}$$ -
2.
If \(n \geqq 2\), \(\nabla W \in L^n ({{\mathbb {R}}^n})\) and \(\Delta W \in L^{\frac{n}{2}} ({{\mathbb {R}}^n})\) then
$$\begin{aligned} \sup _{\tau \geqq 1} \Vert {{\widetilde{\rho }}}(\tau , \cdot ) \Vert _{C^\alpha } < \infty , \qquad \forall \alpha \in (0,1). \end{aligned}$$
The well-posedness and instant regularisation parts of the proof of Theorem 1.1 is based on the study of the Duhamel formula for
Existence and uniqueness are proven by a fixed-point argument. Regularity is achieved by a bootstrap argument in fractional Sobolev spaces. The details are presented in Sect. 2. To this end, we develop a new Young inequality for fractional spaces that we present in “Appendix A”.
In order to recover the propagation of regularity, we take advantage of a second key fact: a sharp decay of the free energy, that leads to a uniform-in-time entropy bound in rescaled variables (1.2). This is the objective of Sect. 3. More precisely, since \(W(x) = W(-x)\), problem (P) is the 2-Wassertein flow associated to the free energy
When \(W \in L^\infty ({{\mathbb {R}}^n})\), we prove the sharp decay of the energy \(E[\rho ]\) in Lemma 3.1. Thus implies that
which tends to \(-\infty \) with the same rate \(-\frac{n}{2}\ln t\) as for the heat equation. We next show that there is a suitable free-energy-like quantity that is bounded below in rescaled variables (1.2), and hence we will be able to estimate the second moment. Through the second moment and the free energy, we are able to show uniform-in-time equi-integrability in the form
The uniform-in-time propagation of regularity is analysed in Sect. 4, where (1.6) is used in a crucial way. For the uniform-in-time bounds of the \(H^1\) norm we apply a standard energy estimate. For the propagation of the \(C^\alpha \) norm we apply a modulus of continuity argument, which to our knowledge is new for (P) but has been applied successfully for other equations (see, e.g., [30, 31]).
Equipped with all these uniform-in-time estimates, we can finally characterise, in Sect. 5, the intermediate asymptotics of the solutions of (P).
Theorem 1.2
Let \(W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n}) \cap L^1 ({{\mathbb {R}}^n})\) such that \(W(x) = W(-x)\), \(\nabla W \in L^n ({{\mathbb {R}}^n})\) and, if \(n \geqq 2\), also that \(\Delta W \in L^{\frac{n}{2}} ({{\mathbb {R}}^n})\). Assume that \(\rho _0 \in L^1_+ ({{\mathbb {R}}^n})\) is such that it satisfies (1.4). Then we have
This result is obtained by a classical entropy-entropy dissipation argument. Following the idea in [2, 14, 21, 39], we measure the distance between the rescaled version of the solution \({{\widetilde{\rho }}}\) and the Gaussian in the \(L^1\) relative entropy defined by
As in the case of the heat equation, this functional can be differentiated in \(\tau \). We apply the logarithmic Sobolev and Csiszar–Kullback inequalities to reduce ourselves to estimate the remainder terms with respect the classical heat equation due to \({{{\widetilde{W}}}}\). We also discuss the \(L^2\) relative entropy and the related \(L^2\) intermediate asymptotics (see Sect. 5.2).
For \(n \geqq 3\) the decay rate coincides with that of the heat equation under our assumptions, and hence it seems sharp as a generic rate. Notice that it is a simple computation that
Better decay rates for the heat equation can be obtained by correctly matching higher moments, as shown in [24] (see also the survey [41] for a clear explanation). For \(n \leqq 2\), we do not expect better rates with our technique, as explained in Sect. 5.1. It is an open problem to improve these rates in \(n=1,2\) possibly under stronger assumptions on W. One reason the results in dimension \(n \leqq 2\) are not as sharp as for \(n\geqq 3\) is that the limit case \(W(x) = \pm \log |x|\) gets more singular in the Sobolev scale as the dimension gets smaller.
We also answer in Sect. 5 the question on minimal assumptions on W such that the asymptotic simplification of the system happens with arbitrarily slow rate.
Theorem 1.3
Let \(n\geqq 2\), \( W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\) such that \(W(x) = W(-x)\), \(\nabla W \in L^{n-\varepsilon } ({{\mathbb {R}}^n})\), \(\Delta W \in L^{\frac{n}{2}} ({{\mathbb {R}}^n})\) (and also \(\Delta W \in L^{\frac{n}{2} - \varepsilon } ({{\mathbb {R}}^n})\) if \(n\geqq 3\)) for some \(\varepsilon > 0\), and that \(\rho _0 \in L^1_+ ({{\mathbb {R}}^n})\) is such that it satisfies (1.4). Then \( \Vert \rho (t, \cdot ) - K(t, \cdot ) \Vert _{L^1} \rightarrow 0\) as \(t\rightarrow \infty \).
This theorem also works for \(n = 1\) under suitable assumptions on W (see Theorem 5.4). Lastly, let us discuss the assumptions \(\nabla W \in L^n\) (and \(\Delta W \in L^{\frac{n}{2}}\) if \(n\geqq 2\)). A borderline case outside these assumptions is the key example alluded above, \(W(x) = \chi \ln |x|\). The rescaling leads to \({{{\widetilde{W}}}}= \chi \ln |y| + \chi \tau \), so \(\nabla {{{\widetilde{W}}}}\) does not evolve in time. It is easy to see that any solution of
for some constant C, is a stationary solution for the Fokker–Planck equation (1.3) with \(\nabla {{{\widetilde{W}}}}= \nabla W\), and so the corresponding \(\rho \) in original variables is of self-similar form with profile \({{\widetilde{\rho }}}\). The existence and uniqueness of these self-similar solutions for subcritical values of \(\chi \) was proven in [6, 9] by variational methods, moreover they are the intermediate asymptotics for subcritical \(\chi \). This explains to some extent how the hypotheses on W are almost sharp for the asymptotic simplification towards the heat kernel profile.
2 Well-Posedness and Regularity
We make use of the classical approach using Duhamel’s formula to obtain sharp well-posedness global in time results, under the assumptions specified below on the potential W (see similar results in [14]). In this section, we use a sub-index t to denote the time variable. Using the variation of constants formula we can re-write the problem (P) as a fixed-point problem of the form
or, equivalently, moving derivatives in the convolution
For two vector fields F and \({\overline{F}}\), we denote the component-wise convolution \( F*{\overline{F}} = \sum _{i=1}^n F_i * {\overline{F}}_i . \) The corresponding formula for is
Below we collect several estimates for the solution u in (2.2).
\(L^1\) estimates for u(t; F). Let us start by obtaining direct basic \(L^1\) estimates for u(t; F). We begin by recalling some properties of the heat kernel \(G_t\). Clearly \(\Vert G_t \Vert _{L^1} = 1\). For integer derivatives
Then, \(\Vert D^k G_t\Vert _{L^p}\) is integrable in time for t near zero as long as \(p < \frac{n}{n+k-2}\). A similar scaling holds in the range of fractional Sobolev spaces \({\mathcal {W}}^{s,p}\) (defined in “Appendix A”) by applying the classical computations presented in “Appendix A.1”. In particular,
With these estimates, we can directly recover \(L^1\) estimates for u(t; F) by using Young’s inequality
Continuous dependence with respect to F. Similarly, we can also state a result of continuous dependence with respect to F
Computing the supremum, we recover
Modulus of continuity in time We claim that, if F has a modulus of continuity in \(C([0,T];L^1({{\mathbb {R}}^n}))\), it is preserved for u(t; F). We already know that, for \(t > s\),
This last element is a modulus of continuity, by the classical result of strong convergence of convolutions. For the continuity of the second term in (2.2), we can write
On the one hand, we can compute that
On the other hand, letting \(\omega _{F,T}\) be the modulus of continuity of F on [0, T] to \(L^1\) we have that
Hence, Duhamel’s formula preserves the continuity, in the sense that
\(L^p\) estimates for u(t; F). The final result that we need is about the regularisation between \(L^p\) spaces. Following a similar procedure as above, we can write
where \((t - s)^{\frac{n}{2p} - \frac{n+1}{2}}\) is locally integrable in t if \(p < \frac{n}{n-1}\). Thus
Analogously, we have
Thus for \(q \in (1, n)\) we have
Now we can obtain our first result of existence and uniqueness for (P), generalising the results of [14], and fitting our current purpose.
Theorem 2.1
(Local in time well-posedness) Given \(\rho _0 \in L^1_+ ({\mathbb {R}}^n)\) and \(\nabla W \in L^\infty \) there exists a unique solution \(\rho (t)\) in \(C([0,T]; L^1 ({{\mathbb {R}}^n}))\) for some \(T > 0\) of (P) in the sense that it satisfies (2.1). The solution has a maximal existence time \(T^*\). If \(T^* < \infty \), then
Furthermore, let \(\rho \) and \({\overline{\rho }}\) solutions of (2.1) corresponding to initial data \(\rho _0\) and \({\overline{\rho }}_0\) respectively. We have that
Proof
We apply Banach’s fixed-point theorem in \(X = C([0,T]; Y)\), where \(Y = \{ \rho \in L^1 ({{\mathbb {R}}^n}) : \Vert \rho \Vert _{L^1} \leqq \Vert \rho _0 \Vert _{L^1} + 1 \}\), to the solutions of \(u_t - \Delta u = \nabla \cdot F\) and \(F = \rho \nabla W * \rho \). Hence, we define an operator \({\mathcal {F}}\) through the right-hand side of (2.1), i.e.
We first point out that, by Young’s convolution inequality
This means, on the one hand that it does not reduce the modulus of continuity in time of \(\rho \), since
We check that \({\mathcal {F}}: X \rightarrow X\) by joining (2.10) with (2.5) and (2.3) for T small enough. Let us now show that \({\mathcal {F}}\) is contractive for T small enough. Pick \(\rho , {\overline{\rho }} \in X\). Due to (2.9) and (2.4)
We can select \(T > 0\) small so that there is a contraction. Lastly, let us show the continuous dependence. With a similar argument as above we obtain that
Hence, for T small enough that \(C T ^{\frac{1}{2} } \Vert \nabla W \Vert _{L^\infty } < 1\),
Since C does not depend on \(\rho \), this argument can be applied iteratively to deduce the result. \(\square \)
A similar argument provides continuous dependence on \(\nabla W\). The next theorem is our main result in this section. We will apply a bootstrap argument to show the solution \(\rho \) in Theorem 2.1 is in \(C((0,T^*);{\mathcal {W}}^{s,p}({\mathbb {R}}^n))\) during its existence for any \(s>0\) and \(p\in [1,\infty )\), and in fact we have \(T^*=\infty \), i.e. the solution is global in time. Once the regularity in space is shown, we immediately obtain the regularity of \(\rho \) in time by passing it through the equation (P), thus \(\rho \) is a classical solution of (P).
Theorem 2.2
(Global in time solutions and instant regularisation) Let \(W \in {\mathcal {W}}^{1,\infty } ({\mathbb {R}}^n)\). Then the solution constructed in Theorem 2.1 is defined for all \(T > 0\) and it satisfies
and, \(\rho \in C ((0, T]; {\mathcal {W}}^{k,p} ({{\mathbb {R}}^n}))\), for any \(k \in {\mathbb {N}}\) and \(p \in [1,\infty ]\). Furthermore, \(\rho \) is a classical solution defined for all \(t > 0\). In fact, if in addition \(\rho _0 \in {\mathcal {W}}^{s,p} ({{\mathbb {R}}^n})\), then \(\rho \in C([0,T]; {\mathcal {W}}^{s,p} ({{\mathbb {R}}^n}))\) for any \(s \geqq 0\) and \(p\in [1,\infty ]\).
Before presenting the proof, let us first introduce some preliminaries. The proof of the regularity result is based on an iteration argument in fractional Sobolev spaces \({\mathcal {W}}^{s,p}\), whose definition and basic properties can be found in “Appendix A”. The reason to use fractional spaces is that our iterative scheme does not seem to be able to jump between \( \rho \in C( (0,T^*) , L^\infty )\) and \( u( \cdot ; \rho (\nabla W*\rho ) ) \in C((0,T^*) , {\mathcal {W}}^{1,1})\), but we can gain fractional regularities to bridge the integer gap.
In each step of the iteration, assuming that \(\rho \in C((0,T]; {\mathcal {W}}^{s,p})\) for certain \(s\geqq 0\), \(p\in [1,\infty )\), we aim to use the formula (2.2) to upgrade the regularity to a higher order. This will be done by controlling the fractional Sobolev norm of \(\nabla G_{t-s} * F(s)\), where \(F(s)=\rho (s)(\nabla W*\rho (s))\). The following two key ingredients will be used in this estimate:
-
1.
To obtain estimates on fractional Sobolev norms of a convolution, we need a Young’s inequality between fractional Sobolev spaces. We could not locate such a result in the literature, so we provide a proof in Theorem A.1, which might be of independent interest.
-
2.
In order to control the fractional Sobolev norms of \(F(s)=\rho (s)(\nabla W*\rho (s))\) itself, we need a product estimate in fractional Sobolev spaces. An estimate of this kind was obtained by Brezis and Mironescu [13]:
$$\begin{aligned} \Vert f g \Vert _{{\mathcal {W}}^{\theta s, p}} \leqq C ( \Vert f \Vert _{L^\infty } \Vert g \Vert _{{\mathcal {W}}^{\theta s,p}} + \Vert g \Vert _{L^r} \Vert f \Vert _{\mathcal W^{s,\sigma }} ^\theta \Vert f \Vert _{L^\infty }^{1-\theta }), \end{aligned}$$(2.11)where \(p,r,\sigma \in (1,\infty )\), \(s \in (0,\infty )\), \(\theta \in (0,1) \) are such that \( \frac{1}{r} + \frac{\theta }{\sigma }= \frac{1}{p}.\) However, a delicate issue is that we only assume \(\nabla W \in L^\infty \) in this section, thus \(\nabla W*\rho (s)\) can only belong to \(L^\infty \)-based spaces (such as \(C^s={\mathcal {W}}^{s,\infty }\)). In particular, it is impossible to show it belongs to \(W^{s,p}\) for any \(p<\infty \). For this reason, we could not apply (2.11) since it requires \(p,\sigma < \infty \). In the following lemma, we derive a product estimate for the fractional Sobolev norm of fg where \(f\in {\mathcal {W}}^{s,p}\) and \(g\in C^s\). It can be seen as a minor generalisation of (2.11) with \(t=\infty \), and we give a short direct proof.
Lemma 2.3
Let \(p \in [1,\infty )\), \(s, \theta \in [0,1)\). If \(f \in C^s\) and \(g \in W^{\theta s, p}\), then \(f g \in W^{\theta s, p}\), and we have the estimate
where \(\Vert f \Vert _{C^{s}} = [f]_{C^s} + \Vert f \Vert _{L^\infty }\).
Proof
If \(s=0\) or \(\theta =0\), clearly \(\Vert f g \Vert _{L^p} \leqq \Vert f \Vert _{L^\infty } \Vert g \Vert _{L^p}\). For \(s,\theta \in (0,1)\), we write
Since \(\int _{|y|< 1} |y|^{-n+(1-\theta )sp} \mathop {}\!\textrm{d}y , \int _{|y| \geqq 1} |y|^{-n-\theta sp} \mathop {}\!\textrm{d}y < C(p,s,\theta )\) we conclude the result. \(\square \)
We now have all the machinery needed for the proof of the main result of this section.
Proof of Theorem 2.2
Iterating in (2.6) and using Young’s inequality, we get \(\rho \in C( (0,T^*) ; L^p ({{\mathbb {R}}^n}))\) for any \(p \in [1,\infty )\). To recover higher regularity we pass through fractional Sobolev spaces. We begin by proving some further regularity estimates for u(t; F). Applying Theorem A.1, we have that
where \(\gamma = \alpha + \beta \). The time term is integrable if \( 1 \leqq p < \frac{n}{n+\alpha -1} . \) Hence, necessarily \(\alpha < 1\), and we deduce that
Applying (2.13) with \(\beta = 0, p = 1, \alpha \in (0,1), q = r \in [1,\infty )\) we recover that \(\rho \in C((0,T^*); W^{\alpha ,q} ({{\mathbb {R}}^n}))\).
Let us reinterpret (2.12) for \(f = \frac{\partial }{\partial x_i} (W*\rho )\) and \(g = \rho \). For \(s, \theta \in (0,1)\), applying Lemma 2.3, we have that
Using the standard Young inequality
Then, we have that
This allows us to show, applying again (2.13), that \(\rho \in C((0,T^*); W^{2s,p} ({{\mathbb {R}}^n}))\) for \(s \in (0,1)\) and \(p \in [1,\infty )\). We can repeat the argument for \(s \in (1,2)\) by noticing that \(\frac{\partial }{\partial x_j} (\rho \tfrac{\partial W}{\partial x_i} * \rho ) = \frac{\partial \rho }{\partial x_j} \tfrac{\partial W}{\partial x_i} * \rho + \rho \tfrac{\partial W}{\partial x_i} * \frac{ \partial \rho }{\partial x_j}\), and the reasoning above works in each element. Similar formulas hold for higher derivatives of F, and hence the argument can be extended to any \(s > 0\). Once we have space regularity, through (P) time regularity follows.
It remains to show that the solution is global in time. Towards this end, we will show that \(\rho _-(\cdot ,t) \equiv 0\) for all \(t \in [0,T^*)\). For a smooth and convex function \(j:{\mathbb {R}}\rightarrow {\mathbb {R}}\), we can write
Let us approximate the convex (but non-smooth) function \(j(s)=\max \{-s,0\}\) by a sequence of smooth convex functions \(\{j_\varepsilon \}_{\varepsilon >0}\), where \(j_\varepsilon '=j'\) in \([-\varepsilon ,0]^c\) (so \(j_\varepsilon ''\equiv 0\) in \([-\varepsilon ,0]^c\)) and satisfies \(0\leqq j_\varepsilon ''\leqq 2\varepsilon ^{-1}\) in \([-\varepsilon ,0]\). Hence \(J_\varepsilon (s) := \int _0^s j_\varepsilon ''(\sigma ) \sigma \mathop {}\!\textrm{d}\sigma \) satisfies \(|J_\varepsilon (s)|\leqq |s|\) for all \(0<\varepsilon <1\), and \(\lim _{\varepsilon \rightarrow 0^+}J_\varepsilon (s)=0\) for all s. Since \(\Delta (W * \rho ) \in L^\infty ({{\mathbb {R}}^n})\) for \(t > 0\), sending \(\varepsilon \rightarrow 0^+\) and applying the dominated convergence theorem to the right hand side of (2.16) gives \(\rho _-(\cdot ,t)\equiv 0\) during its existence. Hence, \(\Vert \rho (t)\Vert _{L^1} = \Vert \rho _0 \Vert _{L^1}\) for all \(t \in [0,T^*)\), and due to the blow-up criteria (2.7) we know there is no blow-up in finite time, that is, \(T^* = +\infty \).
When \(\rho _0 \in L^1 ({{\mathbb {R}}^n}) \cap {\mathcal {W}}^{s,p} ({{\mathbb {R}}^n})\), we want to extend the regularity to \( \rho \in C([0,T]; L^1 ({{\mathbb {R}}^n}) \cap \mathcal W^{s,p} ({{\mathbb {R}}^n}))\). The first step is to notice that (2.13) works also for \(\delta = 0\). Since (2.14) and (2.15) are point-wise in t, they hold up to \(t = 0\). And thus the result is proven. \(\square \)
3 Sharp Decay of the Free Energy and the Entropy
First, we give the sharp decay rate of the free energy functional in original variables E(t) given by (1.5) for a bounded interaction potential W. From now on, we will always assume that the interaction potential W is even without specifying it.
Lemma 3.1
Assume \(W \in {\mathcal {W}} ^{1,\infty } ({{\mathbb {R}}^n})\), and \(\rho _0\in L^1_+({\mathbb {R}}^n)\) satisfy \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\) and \(E[\rho _0]<\infty \), as introduced in (1.5). Then there exists a constant \(c > 0\) depending on \(\Vert W\Vert _{L^\infty }\) and n, such that
Proof
For the length of the proof, let us denote \(E(t) := E[\rho (t)]\). Taking the time derivative of E(t), we have
If we define the auxiliary function u(x, t) as \( u := \rho e ^{ W * \rho }, \) the above becomes
where the last identity follows from the fact that \(u |\nabla \log u|^2 = 4 |\nabla \sqrt{u}|^2\). For bounded W, we have \(\Vert W * \rho (t) \Vert _{L^\infty } \leqq \Vert W \Vert _{L^\infty } \Vert \rho (t) \Vert _{L^1} \leqq \Vert W\Vert _{L^\infty }\), where we used that \(\Vert \rho (t)\Vert _{L^1}=\Vert \rho _0\Vert _{L^1}=1\). Applying this to (3.2) yields
In the rest of the proof, we aim to obtain a lower bound on the integral \(\int _{{\mathbb {R}}^n} |\nabla \sqrt{u} |^2 \mathop {}\!\textrm{d}x\) in terms of E itself. Recall that E can be written as
where \(p>1\) will be determined momentarily. Applying Jensen’s inequality gives
From now on, let us fix \(p:=\frac{2}{n}\). For such p, the Gagliardo-Nirenberg inequality gives that
Combining this with (3.3) and (3.4), we have
This means
where \(c(n,\Vert W\Vert _{L^\infty })=4 C(n)^{-1} e^{-(2+\frac{3}{n})\Vert W\Vert _{L^\infty }}\). Solving this differential inequality yields the inequality (3.1), finishing the proof. \(\square \)
We now focus on using these estimates to obtain uniformspsinspstimeboundsfortherescaledequation (1.3). Following [21], we perform a time-dependent rescaling with the new time and spatial variables being
where \(\lambda (t) = \sqrt{2t+1}\). Let the rescaled density \({{\widetilde{\rho }}}(\tau ,y)\) be related to \(\rho (t,x)\) by
or, equivalently,
Note that \(\rho (0,\cdot )={{\widetilde{\rho }}}(0,\cdot )\) and the \(L^1\) norm of \({{\widetilde{\rho }}}(\tau ,\cdot )\) is preserved under the rescaling. In addition, if \(\rho (t,x)\) satisfies the heat equation \(\partial _t \rho = \Delta _x \rho \), it is well-known (see [21] for example) that \({\tilde{\rho }}(\tau ,y)\) satisfies the Fokker–Planck equation \(\partial _\tau {{\widetilde{\rho }}}= \Delta _y{{\widetilde{\rho }}}+ \nabla _y\cdot ({{\widetilde{\rho }}}y).\)
Next let us derive the equation satisfied by \({{\widetilde{\rho }}}\) when \(\rho \) solves (P). Compared to the heat equation, \(\partial _\tau {{\widetilde{\rho }}}\) has an additional term \(e^{(n+2)\tau } \nabla _x \cdot (\rho \nabla _x(W*\rho ))(t,x)\), and it suffices to express it in terms of the new variables \(\tau ,y\) as well as \({{\widetilde{\rho }}}\). Using the definition of \(\tau ,y\) and \({{\widetilde{\rho }}}\), the convolution \((W*\rho )(t,x)\) can be expressed as
using the change of variables \(y':=\lambda ^{-1}(t)x'\) and (3.6), where \({{{\widetilde{W}}}}(\tau ,y) := W(e^\tau y)\). As a result, the additional term in \(\partial _\tau {{\widetilde{\rho }}}\) can be written as
where we used that \(\nabla _y=e^\tau \nabla _x\), as well as (3.6) and (3.7). Finally this leads to the equation for \({{\widetilde{\rho }}}\) in rescaled variables:
Remark 3.2
In the rescaled variables, even though \({{{\widetilde{W}}}}(\tau ,\cdot )=W(e^\tau \cdot )\) is \(\tau \)-dependent, its \(L^\infty \) norm remains uniformly bounded as long as \(W\in L^\infty \), and one can easily check that
However, the \({\mathcal {W}}^{m,q}\) norm of \({{{\widetilde{W}}}}(\tau ,\cdot )\) can be exponentially growing/decaying in \(\tau \), depending on the values of m and q. More precisely, for any multi-index \(\alpha \) and \(q\geqq 1\), we have
which leads to
As a result, if \(W \in {\mathcal {W}}^{m,q}\) with \(m \leqq n / q\), then \(\Vert {{{\widetilde{W}}}}(\tau ,\cdot )\Vert _{{\mathcal {W}}^{m,q}}\) is uniformly bounded above for all \(\tau \geqq 0\). Furthermore, if \(m < n / q\) then \(\Vert {{{\widetilde{W}}}}(\tau ,\cdot )\Vert _{{\mathcal {W}}^{m,q}}\) decays to zero as \(\tau \rightarrow \infty \). The same kind of estimates holds for fractional Sobolev norms (see “Appendix A.1”).
The next step is to establish uniform-in-time bounds for the free energy, the second moment and, as a consequence, the entropy in rescaled variables. Throughout the rest of this paper, we will focus on the analysis of the rescaled equation (1.3). For notational simplicity, we will suppress the y subscript from \(\nabla _y\) and \(\Delta _y\). Also, all the time and spatial variables below related to \({{\widetilde{\rho }}}\) will be the rescaled variables, unless specified otherwise. For example, “taking the time derivative” stands for taking the \(\tau \)-derivative; and when y appears below in \({{\widetilde{\rho }}}(\tau ,y)\), it will stand for the rescaled spatial variable rather than the original one.
Let us point out that one of the main difficulties to study the rescaled equation (1.3) is the lack of a monotone-decreasing free energy functional. If \({{{\widetilde{W}}}}\) were known to be independent of \(\tau \), it is well-known that there would be a natural free energy functional \({\widetilde{F}}(\tau )\) associated to (1.3), given by
But, since \({{{\widetilde{W}}}}(\tau ,\cdot )=W(e^\tau \cdot )\) is \(\tau \)-dependent, \({\widetilde{F}}(\tau )\) is not necessarily decreasing in time. In fact, taking the time derivative of \({\widetilde{F}}(\tau )\) yields
where \(\frac{ \partial {{{\widetilde{W}}}}}{\partial \tau } (\tau , y) = \frac{ \partial }{\partial \tau } [ W(e^\tau y) ] = e^\tau y \cdot \nabla W(e^\tau y) = y\cdot \nabla {{{\widetilde{W}}}}(\tau , y)\). Plugging this into the above yields
where the right hand side is not necessarily negative due to the additional double integral.
Instead of looking for a monotone free energy for the rescaled equation, let us consider a new free energy functional
Even though this functional is not monotone in \(\tau \), as we will show below, it has a natural relation with the free energy \(E(t) = E[\rho (t)]\) defined in (1.5) in the original variable, and the sharp rate of decay of E(t) that we established in Lemma 3.1 implies a uniform-in-\(\tau \) bound of \({{{\widetilde{E}}}}(\tau )\).
Lemma 3.3
Assume \(W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\), and \(\rho _0\in L^1_+({\mathbb {R}}^n)\) satisfy \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\) and \(E[\rho _0]<\infty \). The energy functionals E(t) in (1.5) and \({{{\widetilde{E}}}}(\tau )\) in (3.11) satisfy that
where t and \(\tau \) are related by (3.5). As a consequence, Lemma 3.1 implies that
Proof
Let us write the original energy \(E(t)= \int _{{\mathbb {R}}^n} \left( \rho \log \rho + \frac{1}{2} \rho (W * \rho ) \right) \mathop {}\!\textrm{d}x\) in terms of \({{\widetilde{\rho }}}\). For the entropy term, using (3.5) and (3.6) we have
where in the last step we used that \(\tau =\log \lambda (t) \) as well as \(\int _{{\mathbb {R}}^n} {{\widetilde{\rho }}}(\tau ,y)\mathop {}\!\textrm{d}y=1\). As for the interaction energy, using (3.5) and (3.6) together with (3.7), we have
Combining the above two identities together yields (3.12). Using (3.12) and the inequality (3.1) for E(t) we have, recalling \(\tau =\log \sqrt{2t+1}\),
where the last inequality follows from the fact that for all \(t\geqq 0\), the fraction in the second line is uniformly bounded above by some constant only depending on \(\Vert W\Vert _{L^\infty },n\) and \(E[\rho _0]\). This finishes the proof of (3.13).\(\square \)
Remark 3.4
For \(W\in L^\infty \), since the interaction energy satisfies
the bound (3.13) on \({{{\widetilde{E}}}}\) immediately implies that the entropy for the rescaled equation is uniformly bounded above:
Let us now prove a uniform-in-time bound of the second moment in rescaled variables. In (3.14), we have obtained a uniform-in-time bound of the entropy \(\int {{\widetilde{\rho }}}(\tau ,y)\log {{\widetilde{\rho }}}(\tau ,y) dy\). In order to upgrade it into a uniform-in-time \(L \log L\) norm of \({{\widetilde{\rho }}}\), we need some uniform-in-time tightness of \({{\widetilde{\rho }}}(\tau ,\cdot )\). Our next goal is to obtain a uniform-in-time bound of the second moment of \({{\widetilde{\rho }}}(\tau ,\cdot )\), given by
A natural starting point is to track the evolution of \(\mathcal N_2(\tau )\) in time. Taking its time derivative and integrating by parts in space, we deduce that
where the last identity is obtained by exchanging y and z in the integrand and taking average with the original integral.
Note that if W is attractive (i.e. W is radially increasing), we have that \(x\cdot \nabla W(x)\geqq 0\) for all x, and the same is true for the rescaled potential \({{{\widetilde{W}}}}(\tau ,\cdot )\). This leads to the differential inequality \(\frac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}\tau }{\mathcal {N}}_2(\tau ) \leqq 2n-2{\mathcal {N}}_2(\tau )\), which yields a uniform-in-time upper bound of \({\mathcal {N}}_2(\tau )\). However, this argument fails for a general bounded potential W that is not necessarily attractive.
To overcome this difficulty, instead of tracking the time derivative of \({\mathcal {N}}_2(\tau )\) itself, the idea is to take a linear combination with the functional \({\widetilde{F}}(\tau )\) in (3.9), so that the double integral involving \(\nabla {{{\widetilde{W}}}}\) will be cancelled in their time derivatives. The result is as follows:
Theorem 3.5
Let \(W \in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^n)\), and assume \( \rho _0\in L^1_+({\mathbb {R}}^n)\) with \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\), \(E[\rho _0]<\infty \), and \({\mathcal {N}}_2[\rho _0]<\infty \). Then we have
Proof
Let \({\widetilde{F}}(\tau )\) be defined as in (3.9), and recall that its time derivative is given by (3.10). Comparing (3.10) with (3.15), we observe that for the linear combination \({\widetilde{F}}(\tau )+\frac{1}{2}{\mathcal {N}}_2(\tau )\), the double-integrals in their time derivative exactly cancel each other. More precisely, we have
Recall that \({\widetilde{F}}(\tau )={{{\widetilde{E}}}}(\tau )+\frac{1}{2}{\mathcal {N}}_2(\tau )\), and \({{{\widetilde{E}}}}(\tau )\) has a uniform-in-time upper bound due to (3.13). Therefore
Multiplying by \(e^{\tau }\) and integrating, we have that
for all \(\tau \geqq 0\), where in the second inequality we used (3.13) and the fact that \({{{\widetilde{E}}}}(0)=E[\rho _0]\).
Note that this inequality does not yield an upper bound for \({\mathcal {N}}_2(\tau )\) yet, since we do not know whether \({{{\widetilde{E}}}}\) is bounded below. Now we write \( {{{\widetilde{E}}}}(\tau ) + {\mathcal {N}}_2(\tau )\) back into \({\widetilde{F}}(\tau ) + \frac{1}{2}{\mathcal {N}}_2(\tau )\), and use the crucial fact that \({\widetilde{F}}\) is bounded below by a constant only depending on n and \(\Vert W\Vert _{L^\infty }\), since
where the integral in the second inequality is the free energy of the Fokker–Planck equation, which is minimised at the Gaussian profile. Combining the lower bound of \({\widetilde{F}} (\tau )\) with the upper bound of \({\widetilde{F}} (\tau ) + \frac{1}{2}{\mathcal {N}}_2(\tau )\) in (3.17), we finish the proof of (3.16). \(\square \)
Finally, we obtain a uniform bound in \(L \log L\) in rescaled variables. Joining (3.14) and (3.16) we recover a uniform-in-time bound of \(\int _{{\mathbb {R}}^n}{{\widetilde{\rho }}}(\tau ) |\log {{\widetilde{\rho }}}(\tau ) |\) by classical techniques [7, 10] (we give a general result in Lemma B.1 which may be of independent interest).
Corollary 3.6
If \(W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\), then the solution of (1.3) satisfies
4 Propagation of Regularity for the Rescaled Density \({{\widetilde{\rho }}}\)
4.1 Uniform-in-Time Bounds of \(L^2\) and \(H^1\) Norms
Theorem 4.1
Let \(n \geqq 1\), \( W \in {\mathcal {W}}^{1,\infty }({\mathbb {R}}^n)\) and \(\nabla W \in L^n ({{\mathbb {R}}^n})\). Assume \( \rho _0\in L^1_+({\mathbb {R}}^n)\) with \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\). Let \(\rho (x,t)\) be the solution constructed in Theorem 2.1 with initial data \(\rho _0\). Then the rescaled density \({{\widetilde{\rho }}}(\tau ,y)\) defined in (3.5)–(3.6) satisfies the following:
-
1.
If \(\rho _0 \in L^2 ({\mathbb {R}}^n)\) with \({\mathcal {N}}_2[\rho _0]<\infty \), then \({{\widetilde{\rho }}}\in L^\infty (0,\infty ; L^2 ({\mathbb {R}}^n))\) with the estimate
$$\begin{aligned} \Vert {{\widetilde{\rho }}}(\tau ) \Vert _{L^2} \leqq C(n, \Vert W\Vert _{L^\infty },\Vert \nabla W\Vert _{L^n}, \Vert \rho _0 \Vert _{L^2}, {\mathcal {N}}_2 [\rho _0] ) \quad \text { for all }\tau \geqq 0.\nonumber \\ \end{aligned}$$(4.1) -
2.
If \(\rho _0 \in H^1 ({\mathbb {R}}^n)\) with \({\mathcal {N}}_2[\rho _0]<\infty \), then \({{\widetilde{\rho }}}\in L^\infty (0,\infty ; H^1 ({\mathbb {R}}^n))\) with the estimate
$$\begin{aligned} \Vert {{\widetilde{\rho }}}(\tau ) \Vert _{H^1} \leqq C(n,\Vert W\Vert _{L^\infty }, \Vert \nabla W\Vert _{L^n},\Vert \rho _0 \Vert _{H^1}, {\mathcal {N}}_2 [\rho _0]) \quad \text { for all }\tau \geqq 0.\nonumber \\ \end{aligned}$$(4.2)
With this result, by compactness we can easily prove that
Corollary 4.2
If \(W \in L^\infty ({{\mathbb {R}}^n})\) and \(\nabla W \in L^n\) for \(\rho _0 \in L^2 ({\mathbb {R}}^n)\) (resp. \(H^1 ({\mathbb {R}}^n)\)) with \({\mathcal {N}}_2[\rho _0]<\infty \), there exists a mild solution of (P) that satisfies the estimates above.
Proof of Theorem 4.1
By the instant regularisation result in Theorem 2.2 and the relation between \(\rho \) and \({{\widetilde{\rho }}}\), we know that \({{\widetilde{\rho }}}\in C^1 ((0,T]; H^2 ({\mathbb {R}}^n))\), even if we do not have an estimate of \(\Vert \rho (t)\Vert _{H^2}\). To obtain uniform-in-time estimates on the \(L^2\) norm of \({{\widetilde{\rho }}}\), we will track the \(L^2\) norm evolution of \({{\widetilde{\rho }}}_k := ({{\widetilde{\rho }}}- k)_+\), where \(k>1\) is a constant to be determined later. To begin with, we list some properties of \({{\widetilde{\rho }}}_k\).
Step 1. Relation between \({{\widetilde{\rho }}}_k = ({{\widetilde{\rho }}}- k)_+\) and \({{\widetilde{\rho }}}\). Due to (3.18) we have, for any \(k > 1\) and \(\tau \geqq 0\), that
where \(C_0 = C(n, \Vert W\Vert _{L^\infty },E [\rho _0], {\mathcal {N}}_2 [\rho _0])\). Note that since \(\Vert \rho _0\Vert _{L^1}=1\), \(E[\rho _0]\) can be bounded above using \(\Vert \rho _0\Vert _{L^2}\) and \(\Vert W\Vert _{L^\infty }\).
Next we state an inequality relating the \(L^p\) norm of \({{\widetilde{\rho }}}_k(\tau ,\cdot )\) and \({{\widetilde{\rho }}}(\tau ,\cdot )\) (below the \(\tau \) dependence is compressed for notational simplicity). Since \({{\widetilde{\rho }}}= {{\widetilde{\rho }}}_k + \min \{{{\widetilde{\rho }}},k\}\), combining the triangle inequality on the \(L^p\) norm with Hölder’s inequality gives
where the second inequality follows from \(\Vert \min \{{{\widetilde{\rho }}},k\}\Vert _{L^\infty }\leqq k\) and \(\Vert \min \{{{\widetilde{\rho }}},k\}\Vert _{L^1}\leqq \Vert \rho _0\Vert _{L^1} = 1\). For \(p = \infty \) we simply use the fact that
Hence, if \({{\widetilde{\rho }}}_k \in L^\infty (0,T; L^2 ({{\mathbb {R}}^n}))\), then so is \({{\widetilde{\rho }}}\).
Step 2. Evolution of \(L^2\) norm of \({{\widetilde{\rho }}}_k\). We compute
We first deal with the more complicated term \(J_1\). We have
where in the second step we used that \(\Vert \nabla {{{\widetilde{W}}}}\Vert _{L^{n}} = \Vert \nabla W \Vert _{L^{n}}\), which is due to (3.8). Note that the above computation holds for all \(n\geqq 1\), where for \(n = 1\) we use the notation \(\frac{n}{n-1} = \infty \). Applying (4.4) (or (4.5) for if \(n = 1\)) and using \(k\geqq 1\), we recover
The Gagliardo-Nirenberg inequality yields
and plugging these two inequalities into (4.7) gives
By applying Young’s inequality for products to each element in the second term we recover that for any \( 0< \delta < 1 \),
We now deal with the term \(J_2\). This can be computed explicitly as
Using (4.8) as well as the fact that \(\Vert {{\widetilde{\rho }}}_k\Vert _{L^1}\leqq 1\), we have
Plugging the \(J_1\) and \(J_2\) estimates into (4.6), we have that for any \( 0< \delta < 1\) and \(k\geqq 1\),
Due to (4.3), \(\Vert {{\widetilde{\rho }}}_k\Vert _{L^1}\) can be made arbitrarily small for large k. Thus we can find a sufficiently large \(k = k(n, E[\rho _0], \mathcal N_2[\rho _0], \Vert W \Vert _{L^\infty })\) and a sufficiently small \(\delta =\delta (n,\Vert \nabla W\Vert _{L^n})\), such that for such \(\delta \) and k,
where the second inequality follows from (4.8) and the fact that \(\Vert {{\widetilde{\rho }}}_k\Vert _{L^1}\leqq 1\). Therefore, \(X(\tau ):= \Vert {{\widetilde{\rho }}}_k(\tau )\Vert _{L^2}^2\) satisfies the differential inequality
with \(c_1=c(n)\) and \(C_2 = C(n, k, \Vert \nabla W\Vert _{L^n})\), thus \(X(\tau )\) is decreasing whenever \(X \geqq (C_2/c_1)^{\frac{n}{n+2}}\). In other words, \(X(\tau )\) has the upper bound \(X(\tau ) \leqq \max \{ X(0), (C_2/c_1)^{\frac{n}{n+2}} \}\). This means that
where we used that \(E[\rho _0]\) can be bounded above using \(\Vert \rho _0\Vert _{L^2}\) and \(\Vert W\Vert _{L^\infty }\). Through (4.4), we obtain a uniform-in-time bound of \(\Vert {{\widetilde{\rho }}}(\tau )\Vert _{L^2}\), finishing the proof of (4.1).
Step 3. Uniform-in-time \(H^1\) bound. In the rest of the proof we aim to check (4.2), where it suffices to control the time-evolution of \(\Vert \nabla {{\widetilde{\rho }}}(\tau )\Vert _{L^2}^2\). Taking its time derivative gives
For \(J_1 := \int _{{\mathbb {R}}^n} \Delta {{\widetilde{\rho }}}\nabla {{\widetilde{\rho }}}\cdot \nabla ({{{\widetilde{W}}}}* {{\widetilde{\rho }}})\), using the fact that \(\Vert \nabla W\Vert _{L^n} = \Vert \nabla {{{\widetilde{W}}}}\Vert _{L^n}\), we have
Applying the Gagliardo-Nirenberg inequalities (see, e.g., [33])
where we recall the well-known fact that \(\Vert D^2 {{\widetilde{\rho }}}\Vert _{L^2} \leqq C(n) \Vert \Delta {{\widetilde{\rho }}}\Vert _{L^2}\), the inequality for \(J_1\) becomes
where we also use that \(\Vert {{\widetilde{\rho }}}\Vert _{L^1}=1\). Note that the power of \(\Vert \Delta {{\widetilde{\rho }}}\Vert _{L^2}\) on the right hand side is strictly less than 2. Likewise, \(J_2 := \int _{{\mathbb {R}}^n} (\Delta {{\widetilde{\rho }}}) {{\widetilde{\rho }}}\Delta ({{{\widetilde{W}}}}* {{\widetilde{\rho }}})\) satisfies
The last term on the right hand side can be controlled as
where the second inequality follows from the Gagliardo-Nirenberg inequality
Thus
and again the power of \(\Vert \Delta {{\widetilde{\rho }}}\Vert _{L^2}\) is less than 2. Finally, the term \(J_3 := \int _{{\mathbb {R}}^n}\Delta {{\widetilde{\rho }}}\nabla \cdot ({{\widetilde{\rho }}}y) \mathop {}\!\textrm{d}y\) can be explicitly computed as
thus
Since in the estimates for \(J_1, \ldots , J_3\), the powers of \(\Vert \Delta {{\widetilde{\rho }}}\Vert _{L^2}\) are all strictly lower than 2, plugging the estimates into (4.9) and applying Young’s inequality for products gives
Since we already have the uniform-in-time bound of \(\Vert {{\widetilde{\rho }}}(\tau ) \Vert _{L^2}\) in Step 2, the above differential inequality yields the uniform-in-time \(H^1\) bound (4.2). \(\square \)
Remark 4.3
We expect that the propagation of \(H^k\) regularity for any integer \(k> 1\) follows from a similar procedure as Step 3, although the computation becomes more involved. We leave the computation to interested readers.
4.2 Uniform-in-Time Bounds of \(C^\alpha \) Norm
In this subsection, we aim to derive the propagation of regularity via an alternative approach. Instead of tracking the evolution of some integral-based quantities such as the \(L^2\) or \(H^1\) norm, which has been done in a vast amount of literature, we will track the evolution of point-wise quantities such as the modulus of continuity. In the context of nonlocal PDEs, such idea has been successfully used by Kiselev–Nazarov–Volberg [31] to establish the global-wellposedness for the SQG equation with critical dissipation.
Our approach is similar to [31]: in order to show that \({{\widetilde{\rho }}}\) has a certain modulus of continuity for all times, we will carefully look at the first “breakthrough” time \(\tau _0\) where the modulus of continuity is about to be violated, and aim to derive a contradiction. While [31] constructed a piecewise modulus of continuity to treat the criticality of SQG equation, for our application to (1.3) it turns out the simple Hölder continuity would work.
Throughout this paper, for any \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\), we denote its Hölder seminorm \([f]_{C^\alpha }\) and Hölder norm \(\Vert f\Vert _{C^\alpha }\) as follows:
Theorem 4.4
Let \( W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\), \(\alpha \in (0,1)\), \(\rho \in C_+ ([0, T); C^\alpha ({\mathbb {R}}^n))\) be a classical solution of (P) and assume
-
(a)
\(n\geqq 2\), and W satisfies \(\Vert W\Vert _{L^\infty } \leqq C_W\), \(\Vert \nabla W\Vert _{L^n} \leqq C_W\) and \(\Vert \Delta W \Vert _{ L^{\frac{n}{2}} } \leqq C_W \).
-
(b)
\(\rho _0 \in L^1_+({\mathbb {R}}^n)\) satisfies that \(\int \rho _0 = 1\), \({\mathcal {N}}[\rho _0]<\infty \), and \(\Vert \rho _0\Vert _{C^\alpha }<\infty .\)
Then, the rescaled density \({{\widetilde{\rho }}}(\tau ,y)\) defined in (3.5)–(3.6) is \(C^\alpha \) Hölder continuous uniformly in time, in the sense that
Before presenting the proof, let us first state and prove a simple lemma that will be useful in the proof. It shows that if a function has a bounded \(C^\alpha \) seminorm, as well as an \(L \log L\) bound, it must have an \(L^\infty \) bound.
Lemma 4.5
For any function \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) and \(\alpha \in (0,1)\), if \([f]_{C^\alpha } \leqq K\) for some \(K> 1\) and \(\int _{{\mathbb {R}}^n}f|\log f| \leqq C_0\), then we have
Proof
Let \(A := \Vert f\Vert _{L^\infty }\), and it suffices to obtain an upper bound of A when \(A>2\). Take any \(x_0\in {\mathbb {R}}^n\) such that \(f(x_0)\geqq \frac{3}{4}A\). Using the modulus of continuity \([f]_{C^\alpha }\leqq K\), we have that \( f(x)\geqq \frac{3A}{4} - K |x-x_0|^\alpha \) for all \(x\in {\mathbb {R}}^n\), thus
Combining this with the bound \(\int f |\log f| \leqq C_0\) and the fact that \(\frac{A}{2}>1\), we have
where \(\omega _n\) is the volume of a unit ball in \({{\mathbb {R}}^n}\). This inequality can be rewritten as
Setting \(a:= (A/2)^{\frac{(n+\alpha )}{\alpha }}\) and \(b:= C_1 K ^{\frac{n}{\alpha }}\), where \(C_1 :=\max \{ C (n , \alpha ) C_0 ,1\}\), the above inequality implies that \(a\log a \leqq b\). To bound a, it suffices to estimate the solution \({{\bar{a}}}\) to the equation \({{\bar{a}}} \log {{\bar{a}}} = b\) for \(b>0\). (Note that the function \(a\log a\) is increasing for \(a>1\), thus \(1\leqq a<{{\bar{a}}}\).) Since \(\log {{\bar{a}}} \leqq {{\bar{a}}}\) for \({{\bar{a}}}>1\), we have that \(b = {{\bar{a}}} \log {{\bar{a}}} \leqq {{\bar{a}}}^2 \), hence \(\log {{\bar{a}}} \geqq \frac{1}{2} \log b \). This leads to
Plugging the definition of a and b into above, we have
where in the second inequality we use the fact that \(C_1=\max \{ C (n , \alpha ) C_0 ,1\}\geqq 1\). Solving this inequality yields (4.11) and finishes the proof. \(\square \)
Remark 4.6
Note that if we replace the \(L\log L\) bound \(\int f|\log f| \leqq C_0\) in Lemma 4.5 by an \(L^1\) bound \(\Vert f\Vert _{L^1}\leqq C_0\) instead, then an estimate very similar to (4.11) would still hold, except that we would lose the \((\log K ) ^{- \frac{ \alpha }{n + \alpha }}\) factor. As we will see soon, this negative power of \(\log K\) plays an essential role in the proof of Theorem 4.4.
Proof of Theorem 4.4
By continuous dependence in \(L^1\) of the initial data, without loss of generality we can assume that \(\rho _0 \in {\mathcal {W}}^{2,\infty } ({{\mathbb {R}}^n})\), and hence \(\rho \in C([0,T]; {\mathcal {W}}^{2,\infty } ({{\mathbb {R}}^n}))\) with \(\partial \rho / \partial t \in C([0,T] \times {\mathbb {R}}^n)\) for any \(T>0\). Note that these regularity properties are also inherited by \({{\widetilde{\rho }}}\), since it is a (smooth) rescaling of \(\rho \) given by (3.5)–(3.6). Once (4.10) is proved for \({\mathcal {W}}^{2,\infty }\) initial data (note that the bound K is independent of \(\Vert \rho _0\Vert _{{\mathcal {W}}^{2,\infty }}\)), the \(L^1\) continuous dependence on initial data in Theorem 2.1 allow us to approximate a \(C^\alpha \) initial data and pass to the limit.
Recall that \({{\widetilde{\rho }}}\) solves the rescaled equation (1.3), and Corollary 3.6 give a uniform-in-time \(L\log L\) bound of \({{\widetilde{\rho }}}\), namely
Using (1.3) and the bound (4.13), our goal is to show that \([{{\widetilde{\rho }}}(\tau )]_{C^\alpha }\leqq K\) for all \(\tau \geqq 0\), where \(K>\Vert \rho _0\Vert _{C^\alpha }\) is a sufficiently large constant to be determined later, which depends on \(n, \alpha , C_W, C_0\) and \(\Vert \rho _0\Vert _{C^\alpha }\). Once this is shown, combining it with (4.13) and applying Lemma 4.5 yields the \(L^\infty \) bound of \(\rho \), finishing the proof.
Towards a contradiction, assume that \([{{\widetilde{\rho }}}(\tau )]_{C^\alpha }\leqq K\) is not satisfied for all \(\tau \geqq 0\). Let us set
and define \(\tau _0\) as the first time such that the modulus of continuity \(\omega \) is about to be violated, i.e.,
Note that \(\tau _0>0\) since \(\Vert \rho _0\Vert _{C^\alpha }<K\) and \({\tilde{\rho }} \in C([0,T];C^\alpha )\) for any \(T>0\). Assuming \(\tau _0<\infty \), let us take a closer look at \({{\widetilde{\rho }}}\) at the “breakthrough” time \(\tau _0\) and derive various estimates on \({{\widetilde{\rho }}}(\tau _0)\) in the next 4 steps, and we will finally obtain a contradiction at the end of Step 4.
Step 1. We claim that
and there exist \(z_1, z_2 \in {{\mathbb {R}}^n}\) such that \(z_1\ne z_2\), and
Indeed, by definition of \(\tau _0\), (4.14) holds when \(\tau _0\) is replaced by any \(\tau <\tau _0\), thus (4.14) also holds at \(\tau _0\) due to the continuity of \({{\widetilde{\rho }}}\) in time. Also by definition of \(\tau _0\), there exists a sequence \((\tau _k, y_1^{(k)} , y_2^{(k)})\) of points such that \(\tau _k\in (\tau _0,\tau _0+1)\), \(\tau _k\searrow \tau _0\), and
where \(\tilde{C_1}:= \sup _k \Vert \nabla {{\widetilde{\rho }}}(\tau _k, \cdot ) \Vert _{L^\infty } < \infty \) since \({\tilde{\rho }} \in C([0,\tau _0+1];{\mathcal {W}}^{2,\infty })\). Using that \(\omega (r)=Kr^\alpha \), the above inequality becomes
so \(y_1^{(k)} - y_2^{(k)} \not \rightarrow 0\). On the other hand, using \({{\widetilde{\rho }}}\geqq 0\) we have that
where \(\tilde{C_2}:= \sup _k \Vert {{\widetilde{\rho }}}(\tau _k, \cdot ) \Vert _{L^\infty } < \infty \) again due to \({\tilde{\rho }} \in C([0,\tau _0+1];{\mathcal {W}}^{2,\infty })\). This leads to the estimate
meaning that \(y_1^{(k)}\) and \( y_2^{(k)}\) cannot be too far apart either.
To obtain a convergent subsequence of \((y_1^{(k)}, y_2^{(k)})\), we need to show that the sequence is uniformly bounded in k. Towards this end, recall that the second moment \({\mathcal {N}}_2[{{\widetilde{\rho }}}]\) is known to be bounded by Theorem 3.5, and we will use this to show that \(\{y_1^{(k)}\}\) are uniformly bounded. First note that \({{\widetilde{\rho }}}(\tau _k,y_1^{(k)})\) is uniformly positive since
where the second inequality follows from (4.18). As a result, by definition of \({{\tilde{C}}}_1= \sup _k \Vert \nabla {{\widetilde{\rho }}}(\tau _k, \cdot ) \Vert _{L^\infty }\), we have, using the mean value theorem, that
Combining this with the uniform bound of the second moment in Theorem 3.5 provides an upper bound of \(|y_1^{(k)}|\) independent of k. Notice that
Thus \(|y_2^{(k)}|\) are also uniformly bounded due to (4.19). Hence, there exists a convergent subsequence of \((y_1^{(k)}, y_2^{(k)})\), and let its limit be \(z_1\) and \(z_2\). Note that \(z_1\ne z_2\) due to (4.18). Using the first inequality in (4.17) and passing to the limit, we have that \({{\widetilde{\rho }}}(\tau _0,z_1)-{{\widetilde{\rho }}}(\tau _0,z_2)\geqq \omega (|z_1-z_2|)\), and combining it with (4.14) yields (4.15).
Finally, to show (4.16), recall that for any \(h\in (0,\tau _0)\) we know that \({{\widetilde{\rho }}}(\tau _0-h, \cdot )\) has modulus of continuity \(\omega \). Combining this with (4.15) gives that
Passing to the limit as \(h \rightarrow 0^+\) finishes the proof of (4.16).
Step 2. Set \(r_0 := |z_1 - z_2|\) and assume WLOG that \( z_1 - z_2 = r_0 \textbf{e}_1. \) In this step we aim to prove the following:
To show (4.20) and (4.21), define
Since in step 1 we showed that \({{\widetilde{\rho }}}(\tau _0,\cdot )\) has modulus of continuity \(\omega \) achieved at \(z_1\) and \(z_2\), it implies that \(g(y)\geqq {{\widetilde{\rho }}}(\tau _0,y)\) for all \(y\in {{\mathbb {R}}^n}\), with equality achieved at \(y=z_1\). This yields \(\nabla {{\widetilde{\rho }}}(\tau _0, z_1) = \nabla g(z_1) \) and \(\partial _{11}{{\widetilde{\rho }}}(\tau _0, z_1) \leqq \partial _{11} g(z_1)\). A parallel argument can be applied similarly to \((\tau _0, z_2)\), which finishes the proof of (4.20) and (4.21). Finally, to show (4.22), define
Again, the fact that \({{\widetilde{\rho }}}(\tau _0,\cdot )\) has modulus of continuity \(\omega \) achieved at \(z_1\) and \(z_2\) gives that \(h(v)\leqq \omega (|z_1-z_2|)\) for all \(v\in {{\mathbb {R}}^n}\), and it achieves its maximum at \(v = 0\). Thus we recover the estimate (4.22) for \(i =2,\dots ,n\). Notice that this is valid also for \(i = 1\), but in (4.21) we have better quantitative information.
Step 3. Let us estimate \(A := \Vert \rho (\tau _0, \cdot ) \Vert _{L^\infty }\) and \(r_0 := |z_1-z_2|\) in terms of K, which will be helpful for us to obtain a contradiction later. Namely, we will prove that
where \(C_1, C_2 > 0\) depend only on \(C_0, n , \alpha \).
Estimate (4.23) directly follows from Lemma 4.5, where we also used (4.13). Since \({{\widetilde{\rho }}}(\tau _0,z_1) = \omega (r_0) + {{\widetilde{\rho }}}(\tau _0, z_2) \geqq \omega (r_0) + 0 = K r_0^\alpha \) we have that \( r_0 \leqq A^{\frac{1}{\alpha }} K^{-\frac{1}{\alpha }}. \) Combining this with (4.23) yields (4.24).
Step 4. In this step, we will show that
if K is sufficiently large (depending on \(C_0,n,\alpha , C_W\)), which would lead to a direct contradiction with (4.16). Since \({{\widetilde{\rho }}}\) satisfies the rescaled equation (1.3), \(\frac{ \partial {{\widetilde{\rho }}}}{\partial \tau } (\tau _0, z_1) - \frac{ \partial {{\widetilde{\rho }}}}{\partial \tau } (\tau _0, z_2)\) can be written as \(T_1+T_2+T_3+T_4\), where the four terms are defined below. Our claim is that they satisfy the inequalities
where \(C_3, C_4 > 0\) depend only on \(C_0, n , \alpha \) and \(C_5, C_6 > 0\) depend only on \(C_0, n , \alpha , C_W\).
To recover (4.25), note that (4.21)-(4.22) yields that \(T_1 \leqq 2\omega ''(r_0)=2\alpha (\alpha -1) Kr_0^{\alpha -2}\), which is negative since \(\alpha \in (0,1)\). Combining this with (4.24) gives
For (4.26), we apply (4.20) and (4.24) to get
To compute (4.27) we use (4.20) to deduce
We can then apply Young’s convolution inequality to obtain
This lead to the bound
and we conclude (4.27) by using (4.23) and (4.24).
To compute (4.28) we proceed similarly
and plugging (4.23) into this inequality yields (4.28).
Finally, comparing the powers in \(T_1,\ldots ,T_4\), note that \(|T_1|\) (coming from \(\Delta {{\widetilde{\rho }}}\)) has the fastest growth as \(K\rightarrow \infty \), since it has a larger power of \(\log K\) compared to the powers of \(T_3, T_4\). By choosing K large enough we have that \(T_1 + T_2 + T_3 + T_4 < 0\). This is a contradiction with (4.16). \(\square \)
Remark 4.7
Note that in step 4, the “good contribution from diffusion” \(T_1\) (4.25) and the “bad contribution from aggregation” \(T_3\) and \(T_4\) (4.27)–(4.28) carry exactly the same power of K, although they have different powers of \(\log K\). This subtle difference in the logarithm powers is the key for us to show that \(T_1\) dominates \(T_3\) and \(T_4\) for \(K\gg 1\). In this sense, the a priori \(L \log L\) bound in Corollary 3.6 is playing a crucial role since it contributes the logarithm term in Lemma 4.5. Also, the assumptions on \(\Vert \nabla W\Vert _{L^n}\) and \(\Vert \Delta W\Vert _{L^{n/2}}\) are sharp in the sense that if the assumptions were to be made in \(L^p\) spaces with any lower p, it would result in a higher power of K in (4.27) and (4.28), and the proof would not go through since \(T_1\) would not dominate \(T_3\) and \(T_4\) for \(K\gg 1\).
5 Convergence to the Gaussian
In this section we focus on obtaining the asymptotic behaviour based on the uniform estimates in the previous two sections. We first concentrate on the \(L^1\) relative entropy approach as introduced in the linear Fokker–Planck equation in [2, 39] based on the crucial use of the logarithmic Sobolev inequality. As usual the \(L^2\) relative entropy strategy can also be applied similarly, replacing the log-Sobolev by Poincaré’s inequality with respect to the Gaussian measure (see [2]). For an elementary presentation in this direction we send the reader to [41].
5.1 \(L^1\) Relative Entropy
Going back to the notion of \(L^1\) relative entropy given by (1.7), we can can compute the time derivative as
where \(I_1\) is the relative Fisher information
and \(J_1\) and \(J_2\) are given by
Remark 5.1
For the heat equation i.e. \(W = {{{\widetilde{W}}}}= 0\), the above becomes \( \frac{\mathop {}\!\textrm{d}}{\mathop {}\!\textrm{d}\tau } E_1 ({{\widetilde{\rho }}}\Vert G) = - I_1 [{{\widetilde{\rho }}}\Vert G] . \) From the logarithmic Sobolev inequality (see [26]), it is classical that, when \(W = 0\) we have
From which \(\dot{E}_1 \leqq - 2 E_1\) and we recover the exponential decay \(E_1({{\widetilde{\rho }}}\Vert G) \leqq E_1 ({{\widetilde{\rho }}}_0 \Vert G ) e^{-2\tau }\).
Remark 5.2
Applying the Csiszar–Kullback inequality [2, 22, 32], since \(1 + 2t = e^{2\tau }\), we have
where \( U(t,x) = K\left( t + \frac{1}{2} , x \right) \). Using the standard decay of the heat equation for even initial data with bounded second moment, we know that \(\Vert U(t) - K(t) \Vert _{L^1} \leqq Ct^{- 1}\) (see [24]). This means that, as long as \(\alpha < 1\), we can always replace U by K as an intermediate asymptotics profile preserving the rate. Notice also that one can get the convergence in 2-Wasserstein distance by using Talagrand inequality (see, e.g., [20, 37]). It is a challenging problem to decide whether the Fisher information \(I_1({{\widetilde{\rho }}}\Vert G)\) also decays to 0 as \(\tau \rightarrow \infty \) with an explicit rate.
To prove our convergence results, we will show that \(|J_i| \leqq C_i e^{-\alpha _i \tau }\) for some \(\alpha _i>0\) under certain assumption on W. Once this is shown, using the logarithmic Sobolev inequality, we recover
Solving the differential inequality, we conclude that
With this approach, it remains to obtain the best possible rate of decay in \(J_1\) and \(J_2\). In (3.8), one can easily check that \(\Vert D^\alpha {{{\widetilde{W}}}}(\tau ) \Vert _{L^q} = e^{(|\alpha | - \frac{n}{q}) \tau }\Vert D^\alpha W \Vert _{L^q}\) has the fastest decay when \(\alpha \) is the smallest (i.e. 0) and q is the lowest (i.e. 1). Therefore, the best possible decay of \(J_1\) and \(J_2\) is obtained by moving all derivatives away from \({{{\widetilde{W}}}}\), and only let \(\Vert {{{\widetilde{W}}}}(\tau )\Vert _{L^1}\) appear in the estimate (note that it requires \(\Vert W\Vert _{L^1}\) be finite). Since \(\Vert {{{\widetilde{W}}}}(\tau ) \Vert _{L^1} = e^{-n\tau } \Vert W \Vert _{L^1}\), \(J_1\) and \(J_2\) also decay with this rate (the detailed proof will be done in Theorem 5.3). Plugging this into the inequality (5.4) for \(E_1\), we get the decays: \(E_1\leqq e^{-\tau }\) if \(n=1\), \(\tau e^{-2\tau }\) if \(n = 2\), and \(e^{-2\tau }\) if \(n \geqq 3\). It is a challenging open problem to prove or disprove if these decay rates are sharp in dimensions \(n=1,2\) under the assumptions of Theorem 5.3.
In the next two theorems, we will use two different ways to prove the decay of \(E_1\) under different assumptions of W. We first prove Theorem 5.3 assuming \(W\in L^1\), which leads to the best possible rate of decay using the argument in the previous paragraph. However, the assumption \(W\in L^1\) is a bit too restrictive, since it requires W to have fast decay at infinity. We then prove Theorem 5.4 with weaker assumptions on W, where W is allowed to have arbitrarily slow power-law decay such as \(W(|x|)\sim |x|^{-\varepsilon }\) for \(|x|\gg 1\) for any \(\varepsilon >0\). This is done at the expense of a slower convergence rate; in Remark 5.5 we will explain why it is natural to expect slower convergence when W has slower decay at infinity.
Theorem 5.3
Let \(n\geqq 1\). Assume \(W\in {\mathcal {W}}^{1,\infty }({{\mathbb {R}}^n})\cap L^1({{\mathbb {R}}^n})\) with \(\nabla W \in L^n({{\mathbb {R}}^n})\). If \(n\geqq 2\), further assume that \(\Delta W \in L^\frac{n}{2}({{\mathbb {R}}^n})\). Suppose \( \rho _0\in L^1_+({\mathbb {R}}^n)\) with \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\), \(E[\rho _0]<\infty \), and \({\mathcal {N}}_2[\rho _0]<\infty \). Let \(\rho (x,t)\) be the solution constructed in Theorem 2.1 with initial data \(\rho _0\). Then the rescaled density \({{\widetilde{\rho }}}(\tau ,y)\) defined in (3.5)–(3.6) satisfies
where \(C<\infty \) depends on \(\rho _0\) and W. In addition, \(|{\mathcal {N}}_2 [{{\widetilde{\rho }}}(\tau )] - n|\) also has exponential decay in \(\tau \), with the same upper bound as in \(E_1\).
Proof
From the instantaneous regularisation result in Theorem 2.2, the solution \({{\widetilde{\rho }}}\) of (1.3) is in \(C((0, \infty ),{\mathcal {W}}^{k,p}({{\mathbb {R}}^n}))\) for any \(k\geqq 0\) and \(p\in [1, \infty ]\). In particular, we have \({{\widetilde{\rho }}}(1,\cdot )\in H^1({{\mathbb {R}}^n})\cap C^\alpha ({{\mathbb {R}}^n})\) for any \(\alpha \in (0,1)\). Applying the uniform-in-time propagation of \(H^1\) regularity proved in Theorem 4.1 with \(\tau =1\) being the initial time yields that
In other words, we can also say that C depends on \(\rho _0\) and W in a quite non-explicit manner.
For \(n=1\), combining the Gagliardo-Nirenberg inequality \( \Vert {{\widetilde{\rho }}}\Vert _{L^\infty } \leqq \Vert \nabla {{\widetilde{\rho }}}\Vert _{L^2}^{\frac{2}{3}} \Vert {{\widetilde{\rho }}}\Vert _{L^1}^{\frac{1}{3}}\) with (5.5) directly yields that \(\sup _{\tau \geqq 1}\Vert {{\widetilde{\rho }}}(\tau )\Vert _{L^\infty }< C\).
For \(n\geqq 2\), note that \(H^1({{\mathbb {R}}^n})\) is not embedded in \(L^\infty ({{\mathbb {R}}^n})\). To show that \(\sup _{\tau \geqq 1}\Vert {{\widetilde{\rho }}}(\tau )\Vert _{L^\infty }< C\) for \(n\geqq 2\), one way is to obtain uniform-in-time propagation of the \(H^k\) regularity, which we will not prove here (see Remark 4.3). Instead, let us apply the uniform-in-time propagation of \(C^\alpha \) regularity in Theorem 4.4 with \(\tau =1\) being the initial time. It yields that for any \(\alpha \in (0,1),\)
which directly yields that \(\sup _{\tau \geqq 1}\Vert {{\widetilde{\rho }}}(\tau )\Vert _{L^\infty }< C\) for \(n\geqq 2\). We point out that so far we have not used \(W\in L^1({{\mathbb {R}}^n})\).
Now that we have obtained the uniform-in-time bounds (for all \(\tau >1\)) of the \(L^\infty \) and \(H^1\) norms of \({{\widetilde{\rho }}}\) for all \(n\geqq 1\), we will use these to prove the decay with \(J_1\) and \(J_2\) with the optimal rate when \(W\in L^1({{\mathbb {R}}^n})\). For \(J_1\), using Hölder’s and Young’s inequalities we have
where in the last inequality we use the uniform-in-time bound on \({\mathcal {N}}_2[{{\widetilde{\rho }}}(\tau )]\) in Theorem 3.5, as well as the fact that \(\Vert {{{\widetilde{W}}}}\Vert _{L^1}=\Vert W\Vert _{L^1}e^{-n\tau }\) from (3.8). Likewise we can estimate \(J_2\) as
Plugging these estimates on \(J_1\) and \(J_2\) into (5.3), we obtain (5.4) with \(\alpha _1=\alpha _2=n\), finishing the proof for \(E_1 ({{\widetilde{\rho }}}\Vert G)\).
Finally, note that the convergence of \(|{\mathcal {N}}_2 (\tau )-n|\) immediately follows from the above estimates for \(J_1\). In fact, from (3.15) we have
and solving this differential equation gives
Using (5.6) into the right hand side gives the exponential decaying bound of \(|{\mathcal {N}}_2 [{{\widetilde{\rho }}}]-n|\), finishing the proof. \(\square \)
Theorem 5.4
Let \(n\geqq 1\). Assume \( W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\) satisfies \(\nabla W\in L^n({{\mathbb {R}}^n})\). If \(n\geqq 2\), further assume that \(\Delta W \in L^\frac{n}{2}({{\mathbb {R}}^n})\). Suppose \( \rho _0\in L^1_+({\mathbb {R}}^n)\) with \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\), \(E[\rho _0]<\infty \), and \({\mathcal {N}}_2[\rho _0]<\infty \). Let \(\rho (x,t)\) be the solution constructed in Theorem 2.1 with initial data \(\rho _0\). Then the rescaled density \({{\widetilde{\rho }}}(\tau ,y)\) satisfies the following:
-
(a)
For \(n=1\), if in addition we assume that \((-\Delta )^{\frac{1}{2}-\varepsilon }W \in L^1({\mathbb {R}})\) for some \(\varepsilon \in (0,\frac{1}{2})\), then for all \(\tau \geqq 1\) we have that
$$\begin{aligned} E_1 ({{\widetilde{\rho }}}\Vert G) \leqq C \Big ( e^{-2\tau } + \Vert (-\Delta )^{\frac{1}{2}-\varepsilon } W \Vert _{L^{1}} F_{ 2\varepsilon } (\tau ) \Big ) \leqq Ce^{-2\varepsilon \tau }\nonumber \\ \end{aligned}$$(5.8)$$\begin{aligned} |{\mathcal {N}}_2 [{{\widetilde{\rho }}}] - n| \leqq Ce^{-2\varepsilon \tau }, \end{aligned}$$where \(C<\infty \) depends on \(\rho _0\) and W.
-
(b)
For \(n= 2\), if in addition we assume that \(\nabla W \in L^{p_1}({\mathbb {R}}^2)\) for some \(p_1\in [1,2)\), then for all \(\tau \geqq 1\) we have that
$$\begin{aligned} E_1 ({{\widetilde{\rho }}}\Vert G) \leqq C \Big ( e^{-2\tau } + \Vert \nabla W \Vert _{L^{p_1}} F_{ \frac{2}{p_1} - 1} (\tau ) \Big ) \leqq Ce^{(\frac{2}{p_1}-1)\tau } \end{aligned}$$(5.9)$$\begin{aligned} |{\mathcal {N}}_2 [{{\widetilde{\rho }}}] - n| \leqq Ce^{(\frac{2}{p_1}-1)\tau }, \end{aligned}$$where \(C<\infty \) depends on \(\rho _0\) and W.
-
(c)
For \(n\geqq 3\), if in addition we assume that \(\nabla W \in L^{p_1}({{\mathbb {R}}^n})\) and \(\Delta W \in L^{p_2}({{\mathbb {R}}^n})\) for some \(p_1\in [1,n)\) and \(p_2 \in [1,\frac{n}{2})\), then for all \(\tau \geqq 1\) we have that
$$\begin{aligned} E_1 ({{\widetilde{\rho }}}\Vert G) \leqq C \Big ( e^{-2\tau } + \Vert \nabla W \Vert _{L^{p_1}} F_{ \frac{n}{p_1} - 1} (\tau ) + \Vert \Delta W\Vert _{L^{p_2}} F_{\frac{n}{p_2}-2}(\tau ) \Big ),\nonumber \\ \end{aligned}$$(5.10)$$\begin{aligned} |{\mathcal {N}}_2 [{{\widetilde{\rho }}}] - n| \leqq C \big ( e^{-2\tau } + \Vert \nabla W \Vert _{L^{p_1}} F_{ \frac{n}{p_1} - 1} (\tau ) \big ), \end{aligned}$$where \(C<\infty \) depends on \(\rho _0\) and W.
In particular, for \(n\geqq 2\) if \( W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\), \(\nabla W \in L^{n-\varepsilon } ({{\mathbb {R}}^n})\), \(\Delta W \in L^{\frac{n}{2}} ({{\mathbb {R}}^n})\) (and also \(\Delta W \in L^{\frac{n}{2} - \varepsilon } ({{\mathbb {R}}^n})\) if \(n\geqq 3\)) for some \(\varepsilon > 0\), and \(\rho _0\) satisfies the above assumptions, then the rescaled solution \({{\widetilde{\rho }}}\) satisfies \(E_1 ({{\widetilde{\rho }}}\Vert G) \rightarrow 0\) and \({\mathcal {N}}_2[{{\widetilde{\rho }}}]\rightarrow n\) as \(\tau \rightarrow \infty \).
Proof of Theorem 5.4
To begin with, note that the same argument as in the first half of the proof of Theorem 5.3 gives
where the constant C again depends on \(\rho _0\) and W in a quite non-explicit manner. Note that we only need \( W \in \mathcal W^{1,\infty } ({{\mathbb {R}}^n})\), \(\nabla W\in L^n({{\mathbb {R}}^n})\), and \(\Delta W \in L^\frac{n}{2}({{\mathbb {R}}^n})\) (for \(n\geqq 2\)) to get these bounds; in particular they do not rely on the extra assumptions in parts (a,b,c).
Next we will prove part (a) by obtaining decay estimates for \(J_1\) and \(J_2\) for \(\tau >1\), under the additional assumption that \((-\Delta )^{\frac{1}{2}-\varepsilon }W \in L^1({\mathbb {R}})\) for some \(\varepsilon \in (0,\frac{1}{2})\). We start with controlling \(J_1\) as in the first line of (5.11), which yields \(|J_1|\leqq C\Vert \nabla {{{\widetilde{W}}}}*{{\widetilde{\rho }}}\Vert _{L^2}\), thus
where in the last inequality we used (3.8), and that \(\sup _{\tau >1}\Vert {{\widetilde{\rho }}}(\tau )\Vert _{H^1}\leqq C\). For \(J_2\), we have
where we used the \(J_1\) estimate to control \(\Vert \nabla {{{\widetilde{W}}}}*{{\widetilde{\rho }}}\Vert _{L^2}\), and we also used that \(\sup _{\tau >1}\Vert {{\widetilde{\rho }}}(\tau )\Vert _{H^1}\leqq C\). Plugging these into (5.3) gives (5.8). The decay estimate for \(|{\mathcal {N}}_2[{{\widetilde{\rho }}}]-n|\) follows in the same way as the last paragraph of the proof of Theorem 5.3.
We now move on to part (b), under the assumption that \(\nabla W\in L^{p_1}({{\mathbb {R}}^n})\) for some \(p_1\in [1,2)\). Again, using that \(|J_1|\leqq C\Vert \nabla {{{\widetilde{W}}}}*{{\widetilde{\rho }}}\Vert _{L^2}\), for \(p_1 \in [1,2)\) we have
For \(J_2\), taking \(q_1 = \frac{2p_1}{3p_1 - 2} \in (1,2]\) we have
which has the same decay rate as the \(J_1\) estimate. Plugging these into (5.3) gives (5.9). Again, the decay estimate for \(|{\mathcal {N}}_2[{{\widetilde{\rho }}}]-n|\) follows in the same way as the last paragraph of the proof of Theorem 5.3.
To prove part (c), we start with the \(J_1\) estimate. If \(p_1\in [1,2)\), the estimate (5.12) still holds. And if \(p_1\geqq 2\), we control \(J_1\) as
which gives the same decay rate as (5.12). For \(J_2\) we apply the usual Young inequality
Plugging these into (5.3) gives (5.10). Again, the decay estimate for \(|{\mathcal {N}}_2[{{\widetilde{\rho }}}]-n|\) follows in the same way as above.
Once we finish part (b,c), the last statement in the theorem follows as a direct consequence, since these assumptions of W are covered by part (b) for \(n=2\), and part (c) for \(n\geqq 3\). This finishes the proof. \(\square \)
The proof of Theorem 1.2 follows directly from from (5.4) and (5.2) (using the Csiszar–Kullback inequality and the change of variables in (1.2).
Remark 5.5
Note that the assumptions in Theorem 5.4 allows W to have arbitrarily slow power-law decay at infinity, which is much less restrictive than the \(W\in L^1({{\mathbb {R}}^n})\) assumption in Theorem 5.3. To see this, let \(W \in C^\infty ({{\mathbb {R}}^n})\) be a smooth potential with \(W = -|x|^{-\varepsilon }\) in \(B(0,1)^c\) for some \(0<\varepsilon \ll 1\). For \(n=1\), the definition of fractional Laplacian gives \((-\Delta )^{\frac{1}{2}-\delta }W\sim -|x|^{-\varepsilon +2\delta -1}\) for \(|x|>1\), thus \((-\Delta )^{\frac{1}{2}-\delta }W \in L^1({\mathbb {R}})\) for \(\delta \in (0,\frac{\varepsilon }{2})\), which satisfies the assumptions in Theorem 5.4(a). For \(n\geqq 2\), one can easily check that \(\nabla W \in L^{p_1}({{\mathbb {R}}^n})\) for all \(p_1 > \frac{n}{1+\varepsilon }\) and \(\Delta W\in L^{p_2}({{\mathbb {R}}^n})\) for all \(p_2>\frac{n}{2+\varepsilon }\), thus there exists \(p_1\in (\frac{n}{1+\varepsilon },n)\) (and \(p_2\in (\frac{n}{2+\varepsilon },\frac{n}{2})\) if \(n\geqq 3\)) that satisfy the assumptions in Theorem 5.4(b,c). Applying Theorem 5.4 gives \(E_1 ( {{\widetilde{\rho }}}\Vert G )\rightarrow 0\) for any \(\varepsilon >0\), although the decay rate goes to 0 as \(\varepsilon \rightarrow 0\).
From the above example, the assumptions on W in Theorem 5.4 is sharp in the sense that \(W=\log |x|\) is the \(\varepsilon \rightarrow 0\) limit of \(W_\varepsilon =\frac{-|x|^{-\varepsilon }+1}{\varepsilon }\), but for such W (even if we modify it to be smooth near the origin) it is well-known that the steady state for the rescaled equation (1.3) is different from the Gaussian, thus \(E_1 ( {{\widetilde{\rho }}}\Vert G )\) has no decay as \(\tau \rightarrow \infty \). For this reason, as \(p_1\) and \(p_2\) approach n and \(\frac{n}{2}\) respectively in Theorem 5.4, it is natural to expect that the convergence becomes arbitrarily slow.
5.2 \(L^2\) Relative Entropy
We also look at the convergence of the \(L^2\) relative entropy under different assumptions on the interaction potential. In order to study the \(L^2\) convergence, we define the \(L^2\) relative entropy as
Recall that G solves the stationary Fokker–Planck equation. In fact, notice that we can rewrite (1.3) as
It is natural that \({{\widetilde{\rho }}}/ G\) will provide good estimates. In fact, it well-known that the space \(L^2 (G^{-1} \mathop {}\!\textrm{d}y)\) is natural because it makes the Fokker–Planck operator self-adjoint. Notice that that \(G^{-1} \geqq (2\pi )^{\frac{n}{2}} > 0\) so the \(L^2(G^{-1}\mathop {}\!\textrm{d}y)\) convergence is stronger than the usual \(L^2\).
Theorem 5.6
Let \(n\geqq 2\). Let \({{\widetilde{\rho }}}\) be a classical solution of (1.3) for \(\tau \geqq 1\) such that
and \(\nabla W \in L^1 ({\mathbb {R}}^n) \). Then,
as \(\tau \rightarrow +\infty \). In particular, if W satisfies \( W \in {\mathcal {W}}^{1,\infty } ({{\mathbb {R}}^n})\) with \(\nabla W \in L^1({{\mathbb {R}}^n})\) and \( \rho _0\in L^1_+({\mathbb {R}}^n)\) satisfies \(\int _{{\mathbb {R}}^n}\rho _0 \mathop {}\!\textrm{d}x = 1\), \(E[\rho _0]<\infty \), and \({\mathcal {N}}_2[\rho _0]<\infty \), then the solution constructed in Theorem 2.1 is such that \(E_2({{\widetilde{\rho }}}\Vert G) \rightarrow 0\) with the above rates.
Remark 5.7
Notice that in this setting we do not use the uniform-in-time bound of \({\mathcal {N}}_2 [{{\widetilde{\rho }}}]\), nor integrability of \(D^2 W\).
Proof of Theorem 5.6
We have
We point out that
We now write
Hence
The last term converges to zero for all \(n>1\), because of the scaling \( \Vert \nabla {{{\widetilde{W}}}}\Vert _{L^1} \leqq e^{ (1 - n) \tau } \Vert \nabla W \Vert _{L^1}. \) Let us define \(w = \frac{{{\widetilde{\rho }}}}{G}\). In the rescaled heat equation, this converts the Fokker–Planck into the Ornstein-Uhlenbeck semigroup. We recall the Gaussian Poincaré inequality
noticing that G and \({{\widetilde{\rho }}}= w G\) have mass equal to 1. To simplify the notations, let
and under these notations (5.15) becomes \( 0\leqq u \leqq v\). We can also rewrite (5.14) in these new notations as
Let \(\tau _0>1\) be such that \(Ce^{(1-n)\tau _0} < 1\), and we claim that \(\sup _{\tau \geqq \tau _0} u(\tau ) \leqq \max \{1 , u(\tau _0) \}\). In fact, if \(u(\tau ) \geqq 1\) for some \(\tau \geqq \tau _0\), using the facts that \(Ce^{(1-n)\tau }<1\) and \(0\leqq u\leqq v\), we have \(\frac{\mathop {}\!\textrm{d}u}{\mathop {}\!\textrm{d}\tau } \leqq -2v + (u+1)^{\frac{1}{2}} v^{\frac{1}{2}} \leqq -2v + \sqrt{2}v< 0\), proving the claim. Using this estimate, we have that
If \(n= 2\), we isolate the \(v^{\frac{1}{2}}\) in the above last term and use Young’s inequality to obtain the following for \(\tau >\tau _0\):
Applying the inequality \(0\leqq u\leqq v\) to the right hand side, and multiplying both sides of the inequality by the obvious integrating factor \(A(\tau ) = \frac{e^{2\tau }}{1+\tau } \), we have
If \(n\geqq 3\), we proceed slightly differently in isolating v using Young’s inequality again
Using the corresponding integrating factor \(A(\tau ) = e^{2\tau + e^{-\tau }}\) we obtain
This completes the proof. \(\square \)
Remark 5.8
We point out that the \(L^2\) norm of \(\rho - U\) decays faster than that of \(\rho \) itself. Using the change of variables (1.2), we can translate the result above to
On the other hand, the \(L^2\) decay of \(\rho \) itself is only
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Acknowledgements
The research of JAC and DGC was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 883363). JAC was also partially supported by the EPSRC Grant Numbers EP/T022132/1 and EP/V051121/1. The research of DGC was partially supported by Grant PGC2018-098440-B-I00 from the Ministerio de Ciencia, Innovación y Universidades of the Spanish Government. The research of YY was partially supported by Grants NSF DMS 1715418, 1846745 from the National Science Foundation of the US, the Sloan Research Fellowship, the NUS startup Grant A-0008382-00-00 and MOE Tier 1 Grant A-0008491-00-00. The research of CZ was partially supported by Grant NSF DMS 19000083 from the National Science Foundation of the US. The problem in this paper was discussed at the AIM workshop “Nonlocal differential equations in collective behavior” in Jun 2018. JAC and YY thank AIM for support and collaborative opportunity, and thank the workshop participants for stimulating discussions.
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Appendices
Some Comments on Fractional Sobolev Spaces
We recall the definition of the Sobolev-Slobodecki semi-norm for \(s \in (0,1)\) below. For \(p\in [1,\infty )\), let us define
and for \(p=\infty \) let
For \(s \in (k,k+1)\) we define
The complete norm is constructed via
A discussion on these norms can be found in [11, 23]. When \(s \notin {\mathbb {N}}\) the Sobolev-Slobodecki spaces also coincide with the Besov spaces \(B_{p,p}^s ({{\mathbb {R}}^n}) = {\mathcal {W}}^{s,p} ({{\mathbb {R}}^n})\).
1.1 Scaling of Fractional Sobolev Norm
It is a direct computation that, if \(s \in (0,1)\)
Hence, for \(s \in (0,1)\), we have that
Therefore, still for \(s \in (0,1)\), we conclude that
1.2 A Result for Fractional Laplacians
The fractional Laplacian is defined through the Fourier transform as the operator of symbol \(|\xi |^{2s}\)
Due to the properties of the convolution
Through the standard Young inequality we have that
Furthermore, it is known (see [38, Proposition 2.1.7 and Proposition 2.1.8]) that
whenever \(\varepsilon , 2s+\varepsilon \notin {\mathbb {N}}\).
1.3 A Result in Norms
Theorem A.1
Let \(s_0, s_1 \geqq 0\), and \(p_0, p_1 \in [1,\infty ]\). Then, there exists C such that
We will use K-interpolation. We introduce some definitions and results that can be found in [5, Chapter 5].
K-interpolation Given \(X_0, X_1\) we say they are compatible spaces if they can be embedded into a common Hausdorff topological space Z. We define, for \(p \in [1,\infty )\)
and
where
We define
Operators in interpolation spaces The key result we will use is that if for compatible pairs \((X_0,X_1)\) and \((Y_0, Y_1)\) and an operator \(T: X_i \rightarrow Y_i\) for both \(i = 0,1\) we have that \(T : (X_0, X_1)_{K;p,\theta } \rightarrow (Y_0, Y_1)_{K;p,\theta }\) and we have
This is proved in [5, Theorem 1.12, Chap 5].
Embedding We will also use the embedding formula [5, Proposition 1.10, Chap 5], which says that
Hence, there exists \(C_r(\theta , q, r, X_0, X_1)\) such that
Interpolation reiteration An important result [5, Theorem 2.4, Chap 5] states that taking interpolation of interpolations is, in itself, and interpolation of the original spaces. Let \(0 \leqq \theta _0 <\theta \leqq 1 \) and \(q, q_0, q_1 \in [1,\infty ]\)
Interpolation of Sobolev spaces According to [5, Theorem 4.17, Chap 5], if \(s_0 \ne s_1\), \(\theta \in (0,1)\) and \(p,q \in [1,\infty ]\) we have
Here and below \( s_\theta = (1-\theta ) s_0 + \theta s_1. \) In particular, we obtain that
This can be computed from the interpolation between \((L^p, \mathcal W^{k,p})\) and the reiteration formula (A.6). To be precise, this means that
Proof of Theorem A.1. Step 1. \(s_0=k_0,s_1=k_1 \in {\mathbb {N}}\). Due to the standard Young inequality, we know that
Applying the result for derivatives, let us write \(s = k = k_0 + k_1\) and \(\alpha = \alpha _0 + \alpha _1 \)
Thus, we have as expected that
Step 2. \(s_0=0, 0 < s_1 \notin {\mathbb {N}}\). We define the map \( T_f : g \mapsto f*g. \) If \(s_1 < k_1 \in {\mathbb {N}}\) and \(s_1 = (1-\theta )0 + \theta k_1\), then
By interpolation, we get
Notice that the interpolation for g is with p, and not \(p_1\) as we would like. Using the embedding formula (A.5) since \(p \geqq p_1\) we have
Using (A.7) we deduce the result, multiplying the constant \(C_e\) to our right-hand side.
Step 3. \(s_0 \in {\mathbb {N}}, 0 < s_1 \notin \mathbb N\). We proceed as in Step 1, followed by Step 2.
Step 4. \(s_0, s_1 \notin {\mathbb {N}}\). Now we must interpolate in f. For g fixed we define \(T_g : f \mapsto f * g\). Let \(s_0 < k_0 \in {\mathbb {N}}\) and \(\theta \) such that \(s_0 = (1-\theta )0 + \theta k_0\). By Steps 2 and 3 we have that
Hence, by interpolation we obtain that
We apply again the embedding (A.5) and (A.7) to conclude the result. \(\square \)
Relating \(\rho \log \rho \) and the Second Moment
We provide a general result relating moment-type bounds and the integrability of \(F(\rho )\) with the integrability of \(|F(\rho )|\). This is very useful to obtain equi-integrability results. In the case of \(F(s) = s\log s \) and the second moment, this is very classical (see, e.g. [7, 10]). For the proof we will use symmetrisation techniques (see [29, 35].
Lemma B.1
Let \(F : {\mathbb {R}} \rightarrow {\mathbb {R}}\) continuous be such that:
-
1.
For some \(\rho _1, \rho _2 > 0\) \( |F| \text { is non-decreasing in } [0, \rho _1] \), \(F\geqq 0 \text { in } [\rho _2 , +\infty ) \)
-
2.
There exists g non-decreasing and locally integrable such that \( \rho \in L^1 ({{\mathbb {R}}^n})\),
$$\begin{aligned} {\mathcal {N}}_g [\rho ] = \int _{{\mathbb {R}}^n}g(|x|) \rho (x) < \infty \qquad \text { and define } G(s) = \int _0^s g(r) r^{n-1} \mathop {}\!\textrm{d}r. \end{aligned}$$
Then, there exists \(C_0, r_0\) depending only on \(\rho _1, \rho _2\) and \(\Vert \rho \Vert _{L^1}\) such that
Proof
We decompose the F into its positive and negative parts \( F = F_+ - F_- . \) We consider \(\rho ^*\) the decreasing rearrangement of \(\rho \). We have that
First, let us estimate \(\int F_- (\rho )\). Since \(\rho ^*\) is decreasing, we can apply Lieb’s trick (see [34]) to show that
where \(|S^{n-1}|\) is the measure of the \((n-1)\)-dim sphere. Thus, we have that
Since \(\rho ^*\) is decreasing and tends to 0 there exists \(r_1, r_2 > 0\) such that \( \overline{\{ \rho ^* > \rho _i \}} = \overline{ B_{r_i}}. \) Notice that
Thus, we can estimate
We take
We have that
Since \(F_+ = F + F_-\) we have that \(|F| = F + 2 F_-\) and this proves the result. \(\square \)
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Carrillo, J.A., Gómez-Castro, D., Yao, Y. et al. Asymptotic Simplification of Aggregation-Diffusion Equations Towards the Heat kernel. Arch Rational Mech Anal 247, 11 (2023). https://doi.org/10.1007/s00205-022-01838-5
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DOI: https://doi.org/10.1007/s00205-022-01838-5