# Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces

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## Abstract

We prove the sharp local \(L^1-L^\infty \) smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known \(L^p-L^\infty \) estimate for \(p>1\). It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.

## Keywords

Logarithmic fast diffusion equation Ricci flow Smoothing estimate## Introduction

### Theorem 1.1

The theorem gives an interior *sup* bound for the logarithmic fast diffusion equation, depending only on the initial data, and *not* on the boundary behaviour of *u* at later times. This is in stark contrast to the situation for the normal linear heat equation on the ball, whose solutions can be made as large at the origin as desired in as short a time as desired, whatever the initial data \(u_0\). We are crucially using the nonlinearity of the equation.

The theorem effectively provides an \(L^1-L^\infty \) smoothing estimate. It has been noted [13] that no \(L^1-L^\infty \) smoothing estimate should exist for this equation, because such terminology would normally refer to an estimate that gave an explicit *sup* bound in terms of \(t>0\) and \(\Vert u_0\Vert _{L^1}\), and this is impossible as we explain in Remark 1.7. Our theorem circumvents this issue, and almost immediately implies the following improvement of the well-known \(L^p-L^\infty \) smoothing estimates for \(p>1\) (see in particular Davis-DiBenedetto-Diller [3] and Vázquez [13]) in which the constant *C* is universal, and in particular does not blow up as \(p\downarrow 1\).

### Theorem 1.2

*universal*\(C<\infty \) such that for any \(t\in (0,T)\) we have

In fact, in Section 4 we will state and prove a slightly stronger result.

*B*evolves under the Ricci flow equation [8, 9] \(\partial _tg = -2Kg\), where \(K=-\frac{1}{2u}\Delta \log u\) is the Gauss curvature of

*g*, is equivalent to the conformal factor

*u*solving (1.1). Meanwhile, the unique complete hyperbolic metric (the Poincaré metric) has a conformal factor

^{1}

*h*is given by

### Theorem 1.3

In other words, if we define \(\hat{\alpha }:= C\left( \alpha +\Vert (u_0-\alpha h)_+\Vert _{L^1(B)}\right) \), then *u*(*t*) will be overtaken by the self-similar solution \((2t+\hat{\alpha })h\) after a definite amount of time.

### Remark 1.4

To fully understand Theorem 1.3, it is important to note the geometric invariance of all quantities. In general, given a Ricci flow defined on a neighbourhood of some point in some surface, one can choose local isothermal coordinates near to the point in many different ways, and this will induce different conformal factors. However, the ratio of two conformal factors, for example \(\frac{u(t)}{h}\), *is* invariantly defined. In particular, if the flow is pulled back by a Möbius diffeomorphism \(B\mapsto B\) (i.e. an isometry of *B* with respect to the hyperbolic metric, which thus leaves *h* invariant but changes *u* in general) then the supremum of this ratio is unchanged. Similarly, the quantity \(\Vert (u_0-\alpha h)_+\Vert _{L^1(B)}\) is invariant under pulling back by Möbius maps, which is instantly apparent by viewing it as the \(L^1\) norm of the invariant quantity \((\frac{u_0}{h}-\alpha )_+\) with respect to the (invariant) hyperbolic metric rather than the Euclidean metric.

### Remark 1.5

An immediate consequence of Remark 1.4 (and the fact that one can pick a Möbius diffeomorphism mapping an arbitrary point to the origin) is that to control the supremum of \(\frac{u(t)}{h}\) over the whole ball *B*, we only have to control it at the origin.

### Remark 1.6

*B*, but wish to apply the theorem on some smaller sub-ball, for example on \(B_\rho \) where the hyperbolic metric has larger conformal factor

This viewpoint is also helpful in order to appreciate that our *local* results apply to arbitrary Ricci flows on arbitrary surfaces, even noncompact ones: One can always take local isothermal coordinates \((x,y)\in B\) and apply the result.

*k*we set \(\alpha =k/4\). Then \(\alpha h\ge k\) on

*B*, so \((u_0-k)_+\ge (u_0-\alpha h)_+\), and any time valid in Theorem 1.1 will also be valid in Theorem 1.3, i.e.

*B*, and restricting to \(B_{1/2}\), where \(h\le 64/9\), completes the proof.

### Remark 1.7

*t*(here via the diffeomorphism \(\mathbf{{x}}\mapsto e^{2t}\mathbf{{x}}\)). Geometrically, the metric looks somewhat like an infinite half cylinder with the end capped off. It is more convenient to consider the scaled version of this solution given by

*u*(0,

*t*) (for example) that only depends on \(\delta \), and not \(\mu \). However, we see that

*before*the special time \(t=1\).

## Proof of the Main Theorem

In this section, we prove Theorem 1.3, which implies Theorem 1.1 as we have seen. The proof will involve considering a potential that is an inverse Laplacian of the solution *u*. Note that this is different from the potential considered by Hamilton and others in this context, which is an inverse Laplacian of the curvature. Indeed, the curvature arises from the potential we consider by application of a *fourth* order operator. Nevertheless, our potential can be related to the potential considered in Kähler geometry. Our approach is particularly close to that of [7], from which the main principles of this proof are derived. In contrast to that work, however, our result is purely local, and will equally well apply to noncompact Ricci flows.

The main inspiration leading to the statement of Theorem 1.1 was provided by the examples constructed by the first author and Giesen [5, 6].

### Reduction of the Problem

In this section, we successively reduce Theorem 1.3 to the simpler Proposition 2.2. Consider first, for \(m\ge 0\), \(\alpha \ge 0\), the following assertion.

**Assertion**\({\mathcal {P}}_{m,\alpha }\): For each \(\delta \in (0,1]\), there exists \(C<\infty \) with the following property. For each smooth solution \(u:B\times [0,T)\rightarrow (0,\infty )\) to the equation \(\partial _tu=\Delta \log u\) with initial data \(u_0:=u(0)\) on the unit ball \(B\subset {\mathbb {R}}^2\), if

### Claim 1

Theorem 1.3 follows if we establish \(\{{\mathcal {P}}_{m,\alpha }\}\) for every \(m\ge 0\), \(\alpha \ge 0\).

### Proof of Claim 1

*B*at time \(t_0= \frac{m}{4\pi }(1+\delta )\) (unless \(t_0\ge T\), in which case there is nothing to prove). But

It remains to prove the assertions \({\mathcal {P}}_{m,\alpha }\), but first we make some further reductions. To begin with, we note that the assertions \({\mathcal {P}}_{0,\alpha }\) are trivial, because \(u_0\le \alpha h\) and \(t_0=0\) in that case, so we may assume that \(m>0\).

By parabolic rescaling by a factor \(\lambda >0\), more precisely by replacing *u* with \(\lambda u\), \(u_0\) with \(\lambda u_0\), and *t* with \(\lambda t\), we see that in fact \({\mathcal {P}}_{m,\alpha }\) is equivalent to \({\mathcal {P}}_{\lambda m,\lambda \alpha }\). Thus, we may assume, without loss of generality, that \(m=1\) and prove only the assertions \({\mathcal {P}}_{1,\alpha }\) for each \(\alpha \ge 0\).

Next, it is clear that assertion \({\mathcal {P}}_{1,1}\) implies \({\mathcal {P}}_{1,\alpha }\) for every \(\alpha \in [0,1]\). Indeed, in the setting of \({\mathcal {P}}_{1,\alpha }\) (\(\alpha \le 1\)), we can apply assertion \({\mathcal {P}}_{1,1}\) to deduce that \(u(t_0)\le C(1+1)h\le (2C)(1+\alpha )h\).

What is a little less clear is:

### Claim 2

Assertion \({\mathcal {P}}_{1,1}\) implies \({\mathcal {P}}_{1,\alpha }\) for every \(\alpha \ge 1\).

### Proof of Claim 2

By the invariance of the assertion \({\mathcal {P}}_{1,\alpha }\) under pull-backs by Möbius maps (see Remark 1.5), it suffices to prove that \(u(t_0)\le C(1+\alpha )h\) at the origin in *B*, where \(t_0=\frac{1}{4\pi }(1+\delta )\). This in turn would be implied by the assertion \(u(t_0)\le C h_{\alpha ^{-1/2}}\) at the origin in *B*.

Thus our task is reduced to proving that Assertion \({\mathcal {P}}_{1,1}\) holds.

Keeping in mind Remark 1.5 again, we are reduced to proving:

### Proposition 2.1

*C*depends only on \(\delta \).

In this proposition, we make no assumptions on the growth of *u* near \(\partial B\) other than what is implied by (2.2). However, we may assume without loss of generality that *u*(*t*) is smooth up to the boundary \(\partial B\). To get the full assertions, we can apply this apparently weaker case on the restrictions of the flow to \(B_\rho \), for \(\rho \in (0,1)\), and then let \(\rho \uparrow 1\). (Recall Remark 1.6.)

*u*(

*t*), we will estimate a larger solution

*v*(

*t*) of the flow arising as follows. We would like to define new initial data \(v_0\) on

*B*by

*v*(

*t*). This is certainly possible, but it will be technically simpler to consider a smoothed out (and even larger) version of this. Indeed, for each \(\mu >0\) (however small) consider a smooth function \(\gamma :{\mathbb {R}}\rightarrow [0,\infty )\) such that \(\gamma (x)=0\) for \(x\le -\mu \), \(\gamma (x)=x\) for \(x\ge \mu \), and \(\gamma ''\ge 0\), as in Fig. 2.

*v*(

*t*) to the logarithmic fast diffusion equation – starting with \(v_0\), and existing for all time \(t\ge 0\). This flow will have bounded curvature, not just initially, but for all time, because \(v_0\le Ch\) for some large

*C*(see [4], in contrast to [6]). Moreover, that solution will be maximally stretched [4] and in particular, we will have

### Proposition 2.2

*B*. Suppose further that for some \(\delta \in (0,1]\) we have

*C*depends only on \(\delta \).

### The Potential Function and the Differential Harnack Estimate

To prove Proposition 2.2, we consider a potential function that will be constructed using the following lemma, which will be proved in Section 3 along with other technical aspects of the proof.

### Lemma 2.3

*B*, and let \(v:B\times [0,\infty )\rightarrow (0,\infty )\) be the unique complete solution to the equation \(\partial _tv=\Delta \log v\) with \(v(0)=v_0\). Then there exists \(\psi \in C^\infty (B\times [0,\infty ))\cap C^0(\overline{B}\times [0,\infty ))\) such that for all \(t\ge 0\), we have

### Lemma 2.4

*v*, we have

### Remark 2.5

*v*rather than \(\varphi \), using (2.10) and (2.9) gives

We end this section by noting the consequences of the Harnack Lemma 2.4 for *v*. Only the first, simpler, estimate will be required, but it will be required twice.

### Corollary 2.6

The stronger statement (2.18) is nothing more than a rearrangement of (2.16) and the inequality \({\mathcal {H}}\le 0\). The weaker statement (2.17) then follows by recalling that \((1+1/x)^x\le e\) for all \(x>0\), and noting that the equation (2.7) for \(\psi \) implies \(\Delta \psi \ge 0\) on *B*, with \(\psi =0\) on \(\partial B\), so the maximum principle implies that \(\psi (t)\le 0\).

### Exponential Integrability

### Theorem 2.7

*v*(

*t*) at time \(t_0=\frac{1}{4\pi }(1+\delta )\), but to do that, we first apply what we have just learned from Theorem 2.7 at time \(\tilde{t}:=\frac{1}{4\pi }(1+\delta /2)\). This then gives us \(L^p\) control in (2.17) for \(1<p<\frac{4\pi \tilde{t}}{1+\delta /100}=\frac{1+\delta /2}{1+\delta /100}\), and in particular we can set \(p=1+\delta /3\), and conclude that

*h*near the boundary \(\partial B\) would cause a problem; we avoid this by working only on the interior ball \(B_{1/2}\) and comparing the flow with the hyperbolic metric \(h_{1/2}\). Or equivalently, we make a rescaling of the domain coordinates so that the ball \(B_{1/2}\) becomes a unit ball.

*B*for \(t\ge 0\). Our objective of proving the bound (2.6) would then be implied by a bound

## Proofs of Lemmas

We first want to prove Lemma 2.3, giving the existence and properties of \(\psi \), but that proof will in turn use the following lemma, which describes the asymptotic behaviour of *v* near \(\partial B\).

### Lemma 3.1

*v*as in Lemma 2.3, we have

Assuming for the moment that Lemma 3.1 is true, we give a proof of Lemma 2.3.

### Proof

*v*, and (1.2).

### Proof of Lemma 3.1

*B*, and that there exists \(a\in (0,1)\) such that on the annulus \(B\backslash \overline{B_a}\) we have

*v*(

*t*) starting at \(v_0\) is maximally stretched on

*B*, and so \((2t+1)h\le v(t)\), while on \(B\backslash \overline{B_a}\), the solution \((2t+1)h_a\), being complete, must also be maximally stretched and so \(v(t)\le (2t+1)h_a\) on this annulus (see [11]). Therefore, we conclude that for all \(t\ge 0\), we have

We now prove the Harnack Lemma 2.4, but first we need to carefully state an appropriate maximum principle, which follows (for example) from the much more general [2, Theorem 12.22] using the Bishop-Gromov volume comparison theorem.

### Theorem 3.2

Suppose *g*(*t*) is a smooth family of complete metrics defined on a smooth manifold *M* of any dimension, for \(0\le t\le T\), with Ricci curvature bounded from below and \(\left| \partial _t g\right| \le C\) on \(M\times [0,T]\).

*f*(

*x*,

*t*) is a smooth function defined on \(M\times [0,T]\) that is bounded above and satisfies

### Proof of Lemma 2.4

Let *g*(*t*) be the family of smooth metrics corresponding to *v*(*t*). Since we know the curvature of *g*(*t*) is bounded, it satisfies the assumptions in Theorem 3.2. Moreover, as discussed in Remark 2.5, \({\mathcal {H}}\) is bounded and equals zero initially, so Theorem 3.2 then implies that \({\mathcal {H}}\le 0\) for all \(t\ge 0\). \(\square \)

## \(L^p{-}L^\infty \) Smoothing Results

In this section, we prove the following result that implies Theorem 1.2 by setting \(1+\delta =4\pi \).

### Theorem 4.1

### Proof

*k*in terms of

*t*, we compute

## Footnotes

- 1.
We abuse terminology occasionally by referring to the function

*h*itself as the hyperbolic metric.

## Notes

### Acknowledgements

The first author was supported by EPSRC Grant No. EP/K00865X/1 and the second author was supported by NSFC 11471300.

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