Eventual regularization of the slightly supercritical fractional Burgers equation

We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Holder continuous for large time.

There are parallel results for the quasi-geostrophic equation. In the subcritical case, the solutions are smooth [5]. In the critical case the solutions are also smooth, which was proved independently by Kiselev, Nazarov and Volberg [8] and Caffarelli and Vasseur [3] using different methods. The proof by Kiselev, Nazarov and Volberg is based on their previous work on the Burgers equation and consists of showing that certain modulus of continuity (that is essentially Lipschitz for nearby points) is preserved by the flow. The proof by Caffarelli and Vasseur is more involved and consists in proving a Hölder continuity result using classical ideas of De Giorgi.
The two different methods were also used in the context of the critical Burgers equation. The method of modulus of continuity was used in [9] to show smoothness of solutions in the periodic setting. On the other hand, the parabolic De Giorgi method developed in [3] was used in [4] to show smoothness of solutions in the non-periodic setting.
For the case of the supercritical quasi-geostrophic equation, it was shown that the solutions are smooth for large time if s = 1/2 − ε for a small ε [13] extending the methods of Caffarelli and Vasseur. More precisely the idea is to use the extra room in the improvement of oscillation lemma to compensate for the bad scaling.
In this article, we prove that the solutions of a slightly supercritical fractional Burger's equation become regular for large time. It is a similar result to the one shown in [13] for the quasi-geostrophic equation.
It is important to point out that in [9], [1], [7] it was shown that singularities indeed occur for any s < 1/2. What we show here is that they disappear after a certain amount of time. Even though singularities may (and sometimes do) appear during an interval of time [0, T ], for t > T they do not occur any more. The amount of time T that we need to wait depends on the initial data and the value of s. For any given initial data, T → 0 as s → 1/2. The essential idea of the proof is to combine the ideas from [4] and [13]. On the other hand, we can present a completely self contained proof which has been simplified considerably.
The idea in the proofs in this paper is still to make the improvement of oscillation in parabolic cylinders compete with the deterioration of the equation due to scaling. The improvement of oscillation lemma is the lemma which allows us to show Hölder continuity when we iterate it at different scales (as in the classical methods of De Giorgi). We present a simple and completely self contained proof of this crucial lemma in this paper (section 4). An alternative approach could be to redo the proof in [4] adapted to general powers of the Laplacian using the extension in [2].
We find a few advantages in the choice of presenting this new proof of the oscillation lemma in this article. One is that it makes the paper self contained. It also provides a proof that does not use the extension argument and thus it could be generalized to other integral operators instead of the fractional Laplacian. The new proof is essentially a parabolic adaptation of the ideas in [12]. This proof uses strongly that the equation is non-local. This idea is also used in [11] to obtain a Hölder estimate for critical advection diffusion equations for bounded flows that are not necessarily divergence free.
We now state the main result.
Remark 1.2. We note that we believe this could be extended to data in any L p , 1 ≤ p < ∞, but for simplicity we do not pursue this here. Notation:

The notion of a solution and vanishing viscosity approximation
By a solution of (1.1) we mean a weak solution (a solution in the sense of distributions) that can be obtained through the vanishing viscosity method. In other words it is a limit as ε 1 → 0 of solutions satisfying where θ 0 is an initial data for (1.1). For every ε 1 > 0 and θ 0 ∈ L 2 , the equation (2.1) has a solution θ which is C ∞ for all t > 0. We list the properties of such solution in the next elementary lemma. Lemma 2.1. For every ε 1 > 0 and θ 0 ∈ L 2 , the equation (2.1) is well posed and its solution θ satisfies 1. θ(·, t) ∈ C ∞ for every t > 0.

Energy equality:
whereḢ s stands for the homogeneous Sobolev space.
Proof. We consider the operator that maps θ to the solution of Then we see that the map A : θ →θ is a contraction in the norm To see that we note (This is an elementary computation using Fourier transform). Given θ 1 and θ 2 such that |||θ i ||| ≤ R for i = 1, 2, we estimate |||Aθ 1 − Aθ 2 ||| using Duhamel formula. On one hand we have Using the interpolation inequality: ||f || L ∞ ≤ ||f || On the other hand, we also estimate Thus, if we choose T small enough (depending on R), A will be a contraction in the ball of radius R with respect to the norm ||| · |||.
Therefore, the equation (2.1) has a unique solution locally in time for which the norm ||| · ||| is bounded. A standard bootstrap argument proves that moreover |||∂ k x θ||| L 2 ≤ Ct −k/2 for all k ≥ 0. This proves 1. and 3. for short time.
The energy equality 2. follows immediately by multiplying equation (2.1) by θ and integrating by parts. Since the L 2 norm of the solution is non increasing, the solution can be continued forever, thus 1. and 3. hold for all time.
If we let ε 1 → 0, the energy estimate allows us to obtain a subsequence of solutions of the approximated problem that converges weakly in L ∞ (L 2 ) ∩ L 2 (Ḣ s ) to a weak solution for which the energy inequality holds. In a later section, we will also prove a bound of the L ∞ norm of θ(·, t) for t > 0, that is also independent of ε 1 , thus we can also find a subsequence that converges weak- * in L ∞ ((t, +∞) × R) for every t > 0.

A word about scaling
There is a one-parameter group of scalings that keeps the equation invariant. It is given by θ r = r 2s−1 θ(rx, r 2s t). If θ solves (1.1), then so does θ r . In the critical case s = 1/2, the scaling of the equation keeps the L ∞ norm fixed. This case is critical because the scaling coincides with the a priori estimate given by the maximum principle.
We can consider a one parameter scaling that preserves Hölder spaces. The function θ r = r −α θ(rx, r 2s t) has the same C α semi-norm as θ. If we want to prove that θ ∈ C α , we will have to deal with this type of scaling, but in this case the equation is not conserved. Instead, if θ satisfies We have an extra factor in front of the nonlinear term. Note that if α > 1 − 2s (only slightly supercritical) and r < 1 (zoom in), this factor is smaller than one.
In the case of the equation with the extra term ε 1 ∆θ, the viscosity will have a larger effect in smaller scales. Indeed, if θ satisfies (2.1), θ r satisfies

L ∞ Decay
First, as an immediate consequence of the energy equality in Lemma 2.1 we have the following lemma.
where C(s) = 2s C 1/4s s 2 1+4s , and C s is the constant appearing the integral formulation of the fractional Laplacian below.
Proof. Let T > 0 and suppose θ is a solution of (2.1). Define for some p to be chosen later. By Lemma 2.1 there must exist a point (x 0 , t 0 ) such that Observe that F satisfies the following equation Next by Cauchy Schwarz where the last inequality follows from Lemma 3.1 andC s = ( 2 1+4s ) Let p = 4s, and choose R so that Cs with C(s) as in the statement of the theorem. Finally, from the definition of F and since the estimate is independent of ǫ 1 and T is arbitrary, the theorem follows (note this gives an upper bound for θ. To obtain a lower bound we can redo the proof with F defined by −t 1 p θ(x, t).).

Remark 3.3.
Note that an estimate like (3.5) could be obtained using any L p norm instead of L 2 . We chose to use L 2 because it is the norm that is easiest to show that it stays bounded (using the energy inequality).

The oscillation lemma
Then, if ε 0 is small enough (depending only on µ and M 0 ) there is a λ > 0 (depending only on µ , 0]. We will apply the lemma above only to the case when M is constant in Q 1 . This is not necessary to prove the lemma as it will be apparent in the proof. We are not aware of any possible application of the lemma with variable M (even discontinuous).
Proof. Let m : [− 2 M0 , 0] → R be the solution of the following ODE: (4.1) The above ODE can be solved explicitly and m(t) has the formula We will show that if c 0 is small and C 1 is large, then θ M 0 µ/2 for ε 0 small. Let β : R → R be a fixed smooth nonincreasing function such that β(x) = 1 if x ≤ 1 and β(x) = 0 if x ≥ 2. Moreover, we can take β with only one inflection point between 0 and 2, so that if β ≤ β 0 then β ′′ ≥ 0.
, 0]. We will arrive to a contradiction by looking at the maximum of the function We are assuming that there is one point in Let (x 0 , t 0 ) be the point that realizes the maximum of w: w(x, t).
where the last inequality is valid since 5 1+2s ≤ 25 for 1 4 ≤ s ≤ 1 2 . We choose the constant c 0 in order to make sure that m(t) stays below 1/4 (simply by choosing c 0 < 1/8), and we choose Note that the constant C s in the integral form of the fractional Laplacian stays bounded and away from zero as long as s stays away from 0 and 1. We can consider C s bounded above and below independently of s as long as s stays in a range away from 0 and 1, like for example s ∈ [1/4, 1/2]. Now we recall that w = θ + mb − ε 0 (1 + t) and we rewrite the inequalities in terms of θ.
We consider two cases and obtain a contradiction in both.
Let us start with the latter. If b(x 0 , t 0 ) ≤ β 1 , then ∆b(x 0 , t 0 ) ≥ 0 and (−∆) s b(x 0 , t 0 ) ≤ 0, then where in the last inequality, we have implicitly use the fact that but this is a contradiction with (4.1) for any C 1 ≥ 0. Let us now analyze the case b(x 0 , t 0 ) > β 1 . Since b is a smooth, compactly supported function, there is some constant C (depending on M 0 ), such that |∆b| ≤ C and |(−∆) s b| ≤ C. Then we have the bounds We replace the value of m ′ (t 0 ) in the above inequality using (4.1) and obtain Recalling that b(x 0 , t 0 ) ≥ β 1 , we arrive at a contradiction if C 1 is chosen large enough.
There is no deep reason for the choice of the number 500 in the above lemma. But the smaller the cube is, say Q 1 400 , on which the improved oscillation occurs, we need a number, say 500, which is greater than 400 in order to make inequality (5.2) hold. In principle, 500 can be replaced by any number greater than 400.
Proof. We want to apply Lemma 4.1 to θ. We check if we have the required hypothesis. We set M 0 = 2 · 10 1/2 . (The reason for this choice will become clear shortly.) Next, θ will be either nonnegative or nonpositive in half of the points in [− 10,10] . (Otherwise, we would continue the proof with −θ instead of θ and −M .) Next, the hypothesis that we are missing is that θ may be larger than 1 outside Q 1 . Thus we define θ = min(θ, 1).
However, in order to apply Lemma 4.1, we need to rescale so that we can have that the inequality holds on [−5, 5] × [− 2 M 0 , 0]. Since we also need to preserve the condition θ ≤ 1 after rescaling, we choose to work with the function θ * (x, t) = θ( 1 10 x, 1 10 2s t). Observe that θ * satisfies the following differential inequality over Q 5 .

Proof of the main result
To simplify the exposition of the proof of theorem 5.2, we first state and establish the following technical but elementary lemma.
for some constant C (independent of ε 1 ) and for all points such that |x − y| > cε 2−2s 1 .
We have to construct the sequence θ k . We start with θ 0 = θ and M 0 = 1 which clearly satisfy the assumptions. Now we define the following ones recursively. Let us assume that we have constructed up to θ k and let us construct θ k+1 .
Now the proof of the main result follows immediately.
Proof of Theorem 1.1. For any initial data θ 0 ∈ L 2 , by Theorem 3.2 θ(−, t) L ∞ (R) decays. So all we have to do is wait until it is less than one, and we can apply Corollary 5.3.
Remark 5.4. The only part of the paper where we use that the solution is in L 2 is in the proof of the decay of the L ∞ norm (Theorem 3.2). For the rest of the paper, all we use is that the L ∞ norm of θ will eventually become smaller than one so that we can apply Corollary 5.3. Of course there is nothing special about the number one, and a similar estimate can be obtained just by assuming that θ L ∞ ≤ C. However, the value of α would depend on this C.