Regularity of solutions for the critical N-dimensional Burgers equation

Transcription

1 Ann I H Poincaré AN 7 ) 47 5 wwwelseviercom/locate/anihpc Regularity of solutions for the critical N-dimensional Burgers equation Chi Hin Chan a,, Magdalena Czuba b, a Department of Mathematics, The University of Texas at Austin, University Station C Austin, TX , USA b Department of Mathematics, University of Toronto, 4 St George St Toronto, Ontario, M5S E4, Canada Received 4 March 9; received in revised form August 9; accepted August 9 Available online November 9 Abstract We consider the fractional Burgers equation on with the critical dissipation term We follow the parabolic De-Giorgi s method of Caffarelli and Vasseur and show existence of smooth solutions given any initial datum in L ) 9 Elsevier Masson SAS All rights reserved Résumé Nous considérons l équation de Burgers avec diffusion fractionnelle dans Nous montrons l existence de solutions globales regulières pour toute donnée initiale dans L ), en utilisant une version parabolique de la méthode de De Giorgi introduite par Caffarelli et Vasseur 9 Elsevier Masson SAS All rights reserved Introduction In this paper we investigate the regularity of the solutions to the critical N-dimensional Burgers equation The equation is given by N t θ + θ j θ = ) θ, j= ) where θ :[, ) R ) is called critical, because of the invariance with respect to the scaling transformation given by θ λ t, x) = θλt,λx), * Corresponding author Fax: addresses: cchan@mathutexasedu CH Chan), czuba@mathtorontoedu M Czuba) Fax: /$ see front matter 9 Elsevier Masson SAS All rights reserved doi:6/janihpc98

2 47 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 and is a special case of t θ + N θ j θ = ) α θ, j= where <α< In a recent paper Kiselev, Nazarov and Shterenberg [6] have done an extensive study for the - dimensional Burgers equation in the periodic setting, which covers the subcritical case <α<, the critical case α =, and also the supercritical case <α< Among the results obtained in [6], the authors prove the global in time existence of locally Hölder continuous solutions for the critical case α = with respect to periodic initial datum θ L p R) with <p< On the other hand, Dong, Du and Li [] consider wellposedness questions also for all the values of α, but both with and without the periodic setting, and with the emphasis on the finite time blow up in the supercritical case Finite time blow up for the supercritical case is also established by Alibaud, Droniou and Vovelle [] Asymptotics for the subcritcal case are investigated by Karch, Miao and Xu [4], and for the critical and supercritcal by Alibaud, Imbert and Karch [3] Uniqueness of wea solutions for the subcritical equation is addressed by Droniou, Gallouët and Vovelle [3] and of nonuniqueness in the supercritical case by Alibaud and Andreainov [] In another recent wor, Miao and Wu [7] show global wellposedness of the critical Burgers equation in critical Besov spaces Ḃ /p p, R) For further bacground and motivation for the fractional Burgers equation we refer our readers to [6,,,4,3,3,,7], and references therein The main goal of this paper is to establish the following theorem Theorem Given any initial datum u L ) there exists a global wea solution L, ; L )) L, ; H )) of the critical Burgers equation ) such that θ, ) = θ in the L )-sense For every t>, we have θt, ) L ) θ is locally Hölder continuous Corollary The solution θ obtained in Theorem is smooth As far as we now, this is the first result for the regularity of the N-dimensional critical Burgers equation as well as the first regularity result in non-periodic setting for initial data in L Although we are aware of the -dimensional regularity results in the periodic setting in [6], we do not now whether their method of modulus of continuity can be readily generalized to the N-dimensional and non-periodic setting We would lie to emphasize that the method of our proof relies on the methods of Caffarelli and Vasseur [5] In [5] authors develop a very delicate parabolic De-Giorgi s method, which leads them to the global smooth solutions for the critical quasi-geostrophic equation in the N-dimensional setting We add here that Kiselev, Nazarov, and Volberg [5] also obtain the same existence result as [5] in -dimensional setting by using the method of modulus of continuity Moreover the method of modulus of continuity is also employed in [6] and [7] Before we explain the way in which our paper originates from [5], we briefly remar on the regularity problem of solutions for the quasi-geostrophic equation, which is a question parallel to the regularity problem of solutions for the Burgers equation Since the finding of the global wea solutions by Resnic in his thesis [8], there has been a significant amount of wor devoted to addressing the existence and uniqueness of smooth solutions See for example [6,7,9,,] Of course, we have to mention that the existence of global smooth solutions for the critical quasi-geostrophic equation with respect to initial datum L ) has recently been established independently by [5] and [5] We are now ready to clarify the relationship between our wor and the wor of Caffarelli and Vasseur [5] As we have mentioned, the purpose of this paper is to perform suitable modifications on the parabolic De-Giorgi s method developed in [5], so that, after our modifications, the method will give the existence of locally Hölder continuous solutions to the critical N-dimensional Burgers equation with respect to the initial datum θ L ) We would lie to bring to the readers attention the following main issue In [5] the authors study the following critical N-dimensional quasi-geostrophic equation t θ + v θ = ) θ,

3 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) div v =, v j = R j [θ], where θ :, ) R is a scalar-valued solution and v is the velocity field related to θ by some selected singular integral operators R j Besides characterizing the fractional Laplacian ) via harmonic extension of functions to the upper half plane see [4] for more on the harmonic extension), one of the ey stepping stones in [5] is the following local energy inequality appearing in Section 3 of [5] Proposition See Caffarelli and Vasseur [5]) Let θ :,T) R be a wea solution for the critical N- dimensional quasi-geostrophic equation for which the respective velocity field v :,T) verifies v L,T ;BMO )) + sup vt,x)dx C v t [,T ] B Then it follows that θ satisfies the following local energy inequality for some universal constant Φ v depending only on C v t B ) ηθ + dxdzds + ηθ + ) t, x) dx B ηθ + ), x) dx + Φ v B t B η θ + dxds + t B η θ +) dxdzds, ) where, t),t), B ={x : x < }, B = B [, ], θ is the harmonic extension of θ to [, ), θ + = θ χ {θ >}, θ + = θχ {θ>}, and η is some cut off function supported in B In order to use the above local energy inequality ) freely Caffarelli and Vasseur mae the ey observation that: if θ is a solution of the critical quasi-geostrophic equation, then any other function u = β{θ L}, with arbitrary constants β> and L R, gives another solution of the same quasi-geostrophic equation Such an observation is of crucial importance since this allows the authors to use the above local energy inequality with the same universal constant Φ v for any functions in the form of u = β{θ L} that is, not just for the solution θ itself) This provides a lot of advantage whenever it is necessary to shift the focus from the solution θ to some appropriate u = β{θ L} Unfortunately, in the case of the critical Burgers equation such a ey observation is no longer valid This is the main obstacle we are facing in borrowing the parabolic De-Giorgi s method from [5] However, we can overcome this difficulty by maing the following important observation: after the local energy inequality ) was established in [5], the authors actually relied only on the local energy inequality ), rather then the critical quasi-geostrophic equation itself Because of this observation, when we are dealing with a solution θ of the critical Burgers equation, we are motivated to focus on the more general function u = β{θ L}, with constants β > and L R, and we try to obtain the corresponding local energy inequality satisfied by u = β{θ L} Indeed, we will find that: if θ solves the N-dimensional critical Burgers equation, then, u = β{θ L} will satisfy the following local energy inequality t B ηu ) + dxdzds + ηu + ) t, x) dx B t ηu + ) [ ], x) dx + NC N L + θ L [ 4,] R 3 ) η u + dxds B B t + η u +) dxdzds, B 3)

4 474 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 where u is the harmonic extension of u, u + = u χ {u >}, u + = uχ {u>}, and η is some cut off function supported in B Now, let us compare inequalities ) and 3) In the case of the critical Burgers equation, the constant NC N [ L + θ L [ 4,] R 3 ) ] plays the same role as the universal constant Φ v appearing in ) However, the universal constant Φ v in ) remains unchanged while we replace the solution θ by β{θ L} In contrast, inequality 3) does not enjoy this stability property, since the quantity NC N [ L + θ L [ 4,] R 3 ) ] might become large compared with θ L [ 4,] R 3 ) when the shifting-level L is changing Because of this, we have to mae sure that the constant NC N [ L + θ L [ 4,] R 3 ) ] is under control by a certain integer multiple of θ L [ 4,] R 3 ) at any time we need to employ 3) in our paper In fact, once we succeed in applying inequality 3) to β{θ L} the main obstacle we are facing disappears and the parabolic De-Giorgi s method as developed in [5] leads to the proof of Theorem The set up of the paper is as follows In Section we show existence of the L bounded wea solution Section 3 is devoted to the proof of the local energy inequality In Section 4 we have some fundamental lemmas, which when combined together with Theorem see below) result in the proof of Theorem In Section 5 we discuss how to extend the Hölder continuity to higher regularity Finally, in Appendix A we provide proofs of the two technical lemmas from Section 4 Existence of L -bounded wea solutions To prove the existence of Hölder continuous solutions for the N-dimensional critical Burgers equation ) it is necessary for us to establish the existence of L -bounded solutions first To that end, we provide a proof for the following theorem Theorem For any given initial datum θ L ), there exists a wea solution θ L, ; L )) L, ; H )) of the critical Burgers equation ) which satisfies the following two properties θ, ) = θ in the L )-sense For every t>, we have θt, ) L ) C N θ t N L ), where C N is some universal constant depending only on N Proof We start by considering the following modified critical Burgers equation, t θ + N ψ R θ) j θ = ) θ + ɛ θ ) j= In the above, an artificial diffusion term ɛ θ is included, and the nonlinear term θ j θ is now replaced by ψ R θ) j θ, where R> is an arbitrarily chosen quantity, and ψ R : R R is the continuous piecewise linear function given by ψ R λ) = λ χ { R<λ<R} + R χ { λ R} Due to the addition of the artificial diffusion term ɛ θ, it is not hard to convince ourselves that the existence of Leray Hopf) wea solutions for the above modified Burgers equation ) can easily be established through an application of the standard Galerin approximation Because of this, for the rest of this proof we freely employ the wea solutions of the modified Burgers equation ) Now, given an initial datum θ L ), we consider a wea solution θ of ) in the Leray Hopf class L, ; L )) L, ; H )) satisfying θ, ) = θ in the L )-sense We will employ the standard De-Giorgi s method to prove that θ is L -bounded over [t, ), for every t > Before this can be done, it is necessary to show that our solution θ of ) verifies the following vanishing property for almost every t, ), and at every truncation level L> ψ R θ) j θ {θ L} + dx =, )

5 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) where {θ L} + ={θ L}χ {θ>l} For the sae of convenience, we write θ L ={θ L} + We then observe ψ R θ) j θ θ L = [ j ψr θ)θl { ] [ θ L j ψr θ) ]} = { [ j ψr θ)θ ] L θ L χ { R<θ<R} j θ } By taing the integral over of the above identity, we yield { } ψ R θ) j θ θ L dx = θl χ { R<θ<} j θdx+ θl χ {<θ<r} j θdx, so we will succeed in justifying ), if we can show θl χ { R<θ<} j θdx= θl χ {<θ<r} j θdx= Start with the second term Without the loss of generality we assume that L R We note θl χ {<θ<r} j θ = θl χ {θ L <R L} j θ L = θl j [R L θ L ] + Then a computation shows θl χ {<θ<r} j θ = R L) [ j [R L θ L ] + + R L) j R L θl ) ] + 3 [ j R L θl ) 3 +] 3) Next [R L θ L ] + = R L) { θ L χ {θl <R L} + R L)χ {θl R L}} Observe θ L χ {θl <R L} L ) and R L)χ {θl R L} L ), and since these functions are also in L ),theyareinl p ), <p<, so it follows from 3) that we must have θl χ {<θ<r} j θdx= In exactly same way, we can also show that θl χ { R<θ<} j θdx= Hence the validity of property ) is established We are now ready to apply the De-Giorgi s method to the solution θ :, ) R of the modified critical Burgers equation ) To begin, let M>beanarbitrary large positive number to be chosen later) We consider the following sequence of truncations θ = [ θ M )], + By multiplying ) by θ, and then taing integral over, we obtain ) t θ dx + ɛ θ dx = ) θ θ dx, 4) in which we no longer see the term N j= ψ R θ) j θ)θ dx, thans to the vanishing property ) This is because θ L χ {<θ<r} j θ automatically vanishes if L>R

6 476 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 To manage the term ) θ θ dx, it is necessary to use a recent result of Córdoba and Córdoba [8], which states that for any convex function φ : R R,wehave φ θ) ) θ ) φθ) ) To employ such a result, we consider the convex function [ φ λ) = λ M )] + with φ λ) = χ {λ>m Then it follows from 5) 5) [ ) θ ] θ = φ θ) ) θ θ [ ) θ ] θ We use this in 4) to get ) t θ dx + ɛ θ dx θ ) θ dx 6) Recall we wish to prove θ is L -bounded over [t, ] for every t > Let t > be fixed, and consider the increasing sequence T = t ),, which approaches the limiting value t as + Alsofix and t verifying T T t< We then integrate 6) over [,t] to obtain θ t t, )dx+ θ ) θ dxds θ, )dx, 7) θ dxds, since we should not use it in estimating in which we purposely drop the artificial energy term ɛ t θ L [t, ) ) Next, by taing the average over [T,T ] among the terms in the above inequality and then taing the sup over t [T, ),wehave sup t [T, ) θ t, )dx+ T θ ) θ dxds t We now consider the following sequence of quantities U = sup t [T, ) θ t, )dx+ Then, our last inequality tells us that U t T θ, )dxd T θ ) θ dxds T T θ, )dxd Our goal is to build up a nonlinear recurrence relation for U, by relying on the above inequality By employing Sobolev embedding, interpolation and Hölder s inequality we now that our solution θ of ) satisfies the following inequality for all θ L + N ) [T, ) ) CU, 8)

7 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) for some constant C depending only on N Because of this, we can raise up the index for follows T θ dxd T ) θ χ {θ > M } N M Hence 8) together with our last inequality gives T θ θ N M ) N CU + N T θ, )dxd as U + N ) t M N CU + N 9) We can now choose M = t ) N so that t M N = Hence U + N ) CU + N, ) From the nonlinear recurrence relation ), we now that there exists some constant δ N, ), depending only on N, such that U as, provided we have U <δ N Due to this observation, if the initial datum θ = θ, ) verifies ) N θ L ) < δn N+, + N C then, we must have that U + N CU + N + N C θ + N L ) <δ N Note we use U θ, which holds because of the energy inequality that can be obtained in a standard way for L ) the Leray Hopf solutions of ) For such a θ, we have lim U =, and hence θ M = t ) N is valid almost everywhere on [t, ) By applying the same De-Giorgi s method to θ, we should also get θ M = t ) N almost everywhere on [t, ) At this point, let us summarize what we have done so far: If θ :, ) R is a wea solution of the modified critical Burgers equation with initial datum θ, ) = θ L ) verifying θ L ) < δ N ) + N N+ N, then it follows that θ L [t, ) ) t C ) N for every t > Next, we need to remove the smallness condition imposed on θ L ) in the above statement To this end, let θ :, ) R be a given wea solution of the modified critical Burgers equation ), and let λ>bethe unique positive number such that λ N θ L ) = δn + N C ) N N+ ) For such a λ>, we consider the rescaled function θ λ t, x) = θλt,λx), which solves the following rescaled modified Burgers equation in the wea sense N t θ λ + ψ R θ λ ) j θ λ = ) θλ + ɛ λ θ λ j= At first glance, it seems to be troublesome that θ λ no longer solves the original equation ) However, this is not problematic at all since the energy term ɛ θ dx is purposely dropped from inequality 6) before we apply the De-Giorgi s method to θ This means that all the estimates starting from 7) in the above process are

8 478 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 independent of the artificial diffusion term ɛ θ This tells us, in particular that if θ is replaced by {θ λ M )} + in inequality 7), all the estimates thereafter remain unchanged This observation, together with the fact that by ) θ λ, ) L ) < ) N δn N+, + N C give us θ λ L [ t λ, ) ), t t λ ) N > Since θ λ L [ t λ, ) ) = θ L [t, ) ), it follows from ) that the following inequality is valid for every t > θ L [t, ) ) + N C δ N In summary, we have established: ) N N+ θ L ) t N There exists some universal constant C N, ), depending only on N, such that for every wea solution θ ɛ,r) :, ) R of the modified critical Burgers equation ) with initial datum θ ɛ,r), ) = θ L ),wehave θ ɛ,r) L [t, ) ) C N θ L ), for every t > t N Now, the solution θ ɛ,r) θ of the modified critical Burgers equation ) satisfies the uniform bound C L ) N By passing to the limit, as ɛ + and R +, it follows that θ ɛ,r) converges to some wea solution θ :, ) θ R of the critical Burgers equation ), which must also satisfy the same uniform bound C L ) N So, we are finished with the proof of Theorem 3 Harmonic extension to [, ) and the local energy inequality We begin by introducing the harmonic extension see [5] and [4] for more details) Operator ) θ is not a local operator However it can be localized Indeed, define the harmonic extension operator H : C RN ) C RN R + ) by Hθ)= in, ), H θ)x, ) = θx), x Then it can be shown we can view ) θ as the normal derivative of Hθ) on the boundary {x, ): x } ie, ) θx)= ν H θ)x) From now on we use θ to denote the harmonic extension of θ or more precisely θ t,x,z)= H θt, ) ) x, z) Now we are ready to proceed to the local energy inequality and its proof, which closely follows [5] Proposition 3 Local energy inequality) Let θ :[ 4, ] R be a wea solution of the Burgers equation ) Then, for any function u in the form of u = β{θ L}, with β>, and L R we have t B 4 ηu ) + dxdzds + ηu + ) t, x) dx t N t N

9 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) t ηu + ) ), x) dx + NC N L + θ L [ 4,] R 3 ) t + B 4 η u +) dxdzds, η u + dxds where η can be any cut off function supported in B4 = [, 4], =[ 4, 4] N, and C N = 8 C N where C N is the constant appearing in the Sobolev inequality f C L N N N f ) H ) Proof Start with = η u + u dxdz B 4 = B 4 = B 4 η u +) u dxdz+ η u +) u dxdz+ η u + zu dx 4 η x, )u + ) udx, 3) where we use ) u = z u, ) A calculation shows that 3) is equivalent to ) = ηu + dxdz+ η u ) + dxdz + η x, )u + ) udx 3) B 4 Now if θ solves ), u solves B 4 t u + N j= β u + Lβ) j u = ) u 33) Also observe 3 η u + β u + Lβ) j udx = Hence for the third term on the RHS in 3) we have η u + ) u = t + L 3 j ) η u + dx 3 N j= η ) u + u + Lβ) dx + L β N j= j η u + j udx η ) u + u + Lβ) dx β η u + j udx 34) Substitute 34) into 3), integrate between and t, and tae the absolute value of the RHS to obtain

10 48 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 t B 4 ηu ) + dxdzds + t B 4 + N 3 η u ) + dxdzds + t j= j η u + t) dx η u + ) dx η ) u + u + Lβ) dx ds β + L N 3 t j= We examine the last two terms Both can be written as a constant multiple of N t η j ηu + vdxds, j= η u + j udxds 35) where v = β u + Lβ) = θ and the constant equal to 3 for the first term, and v = L and the constant equal to 3 for the second Following [5] by Hölder s inequality in space and Cauchy s inequality with ɛ in time we obtain N t t η j ηu + vdxds ɛ ηu + L N N ds + N t ηvu + ɛ L N+ N ds j= By Sobolev embedding and the energy minimization property of Hχ B4 ηθ + ), it is not difficult to see 3 t ɛ ηu + L N N ds ɛ C N ) t B 4 ηu + ) dxdzds, which means it can be combined with the LHS of 35) if ɛ is small enough, say ɛ = Next, since N C N N+ < and η has compact support within B4 we have N ɛ t as needed ηvu + L N+ N ds NC N 4 Proof of Theorem t NC N v L ηvu + dxds t t ) NC N L + θ L ηu + dxds ηu + dxds Theorem proven in Section gives us the first part of Theorem What remains is to establish the Hölder continuity for solutions of Eq ) For this purpose, we need the following three lemmas In what follows, we use the abbreviations that Q r =[ r, ] B r and Q r =[ r, ] B r 3 See Section 3 of [5] for more details

11 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) Lemma 4 Given any C θ, ), there exists some ɛ > depending only on N and C θ ), and some λ, ) depending only on N), such that for every solution θ :[ 4, ] RN R of Eq ) which verifies θ L [ 4,] RN ) C θ, we have the following implication for every function u in the form of u = β{θ L}, with β min{, C θ }, and L 6C θ : If u = β{θ L} verifies u on [ 4, ] B 4, and B 4 4 ) u u + ɛ, then it follows that u λ on [ 6, ] B 6 Lemma 4 Given any C θ, ), and any sufficiently small ɛ >, there exists some δ >, and also some constant D θ, ) depending only on C θ and N), such that for every solution θ :[, ] R of Eq ) which verifies θ L [,] ) C θ, we have the following implication for all function u in the form of u = β{θ L}, with β C θ, and L 6C θ : If u = β{θ L} verifies the following three conditions: i) u on Q =[, ] B, ii) {t,x,z) Q : u t,x,z) } Q, iii) {t,x,z) Q :<u t,x,z)<} δ, then it follows that u ) + dxdt + Q 4 Q 4 u ) + dxdzdt D θ ɛ 4) Lemma 43 Oscillation Lemma) Given any C θ, ), there exists some λ > depending only on N and C θ ), such that for every solution θ :[, ] R of Eq ), which verifies θ L [,] ) C θ, we have the following implication for any function u in the form of u = β{θ L}, with β C θ, and L 3C θ : If it happens that u on Q, and {t,x,z) Q : u } Q, then it follows that u λ on Q 3 Remar The above lemmas correspond to Lemma 6, Lemma 8 and Proposition 9 in [5] respectively However, here they are not stated for the solution θ of the equation, but for the function u = β{θ L} since this is the function that we actually apply the lemmas to Most of all, the above lemmas require restrictions for the constants β and L, which were not needed in [5] This is a result of the main difficulties of dealing with the Burgers equation explained in the introduction Remar The proof of Lemma 4 and Lemma 4 relies on the local energy inequality as established in Proposition 3 The two lemmas are technical tools needed to establish the Oscillation Lemma and are proved in Appendix A Remar 3 The Oscillation Lemma gives us the Hölder continuity We describe this next, and then give the proof of the Oscillation Lemma It is very important to observe that the universal constant λ in the Oscillation Lemma is invariant under the natural scaling θ λ t, x) = θλt,λx) for solutions of the N-dimensional critical Burgers equation This observation is of crucial importance since it allows us to employ the Oscillation Lemma at different scales in the proof of Hölder continuity see below) The scale-invariant property of the Oscillation Lemma is due to the invariance of solutions for the N-dimensional critical Burgers equation under above scaling This in particular explains why our method wors in the critical case

12 48 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 Proof of Hölder continuity Set r = 3 Letθ :[, ] RN R be a solution of ) with θ L [,] ) C θ for some C θ, ) In order to use Lemma 43, it is necessary to consider the function u = β {θ L }, where the constants β and L are given by β = sup Q θ inf Q θ and L = sup θ + inf Q Q θ Note β C θ, and L C θ Now, we are going to construct a sequence of functions u = β {θ L } inductively in a way that is dependent on u To begin the inductive process, we observe that u = β {θ L } verifies the condition that u onq To construct a suitable u = β {θ L } from u, we split our discussion into two cases: Case : {t,x,z) Q : u } Q We apply Lemma 43 to u over Q and deduce that u λ on Q r, where r = 3 and λ is the constant in Lemma 43 Hence, we have } λ {u + onq λ r Let a =, and define u λ to be } } u = a {u + λ = aβ {θ L + λ β Case : {t,x,z) Q : u } Q In this case, we apply Lemma 43 to u over Q and deduce that u λ on Q r Hence, we have } λ {u onq λ r As before, we write a =, and define in this case that λ } } u = a {u λ = aβ {θ L λ β We observe that in either case { u = aβ θ L + ) λ } β, {, } u onq r aβ C θ, and L ) λ β Cθ + β λ 3 C θ This means that we can apply Lemma 43 to u over Q r in order to construct u = a{u + ) λ } in exactly the same way For the reasons of transparency and completeness we describe now the inductive step Suppose that at step N +, we have a function u given by { ) } u = a β θ L + λ ) s s, β a s= which verifies the required condition that u onq r Here, let us mae the crucial observation that a β β C θ

13 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) L λ β s= ) s a )s C θ + λ β a ) = C θ + λ β 4 λ 3C θ, where in the second term, we have implicitly used the fact that = 4 a ) λ The above two inequalities simply tell us that we can apply Lemma 43 to u over Q in either one of the following two cases: r Case : {t,x,z) Q : u r } Q r We apply Lemma 43 to u over Q and deduce that u r λ on Q Hence we have {u r + λ + λ } onq We define u r + + as u + = a{u + λ } So, we have u + = a + β { θ L + λ β ) s s= ) s + λ a β ) } a Case : {t,x,z) Q : u r } Q r We can apply Lemma 43 to u over Q, and deduce that u r λ over Q Hence, we have {u r + λ λ } onq Because of this, we define u r + + = a{u λ } So, we have u + = a + β { θ L + λ β ) s s= ) s λ a β ) } a From the above inductive process, we have a sequence of functions { ) } u = a β θ L + λ ) s s, β a s= which verify the following conditions u onq, for any r a β C θ, for any L β λ s= ) s a )s 3Cθ, for any Therefore we can deduce for all a β sup θ inf θ ) = sup u Q Q r r Q inf u Q 4 r r Thus sup θ inf θ 4 ) ) 4C θ Q Q β r r a a At this point, we note that the above inequality and the shift-invariant property of solutions of ) give us the conclusion that θ is C α at any t,x,z), and hence θ itself must be C α This completes the proof of Theorem 4 Proof of the Oscillation Lemma The proof closely follows [5] Assume Lemmas 4 and 4 hold, and let θ :[, ] R be a solution to Eq ) with θ L [,] ) C θ for some C θ, ), as well as let ɛ depending only on N and C θ ), and λ, ) depending only on N) be the two constants appearing in Lemma 4 Also, consider the constant D θ depending only on C θ ), which appears in Lemma 4 We choose ɛ ={ 4D ɛ θ }, so that we have 4D θ ɛ ) = ɛ <ɛ With such an ɛ, we have a small number δ depending only on ɛ ) as it appears in the statement of Lemma 4 With these preparations, let u = β{θ L}, with β C θ, and L 3C θ, and suppose that u verifies

14 484 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 u onq {t,x,z) Q : u } Q Now, let us define K + N + to be the largest nonnegative integer for which K + + δ We then define a list of functions w,for K + by w = w ), K +, with w = u Then for every K + we have { w = {u }+ = β θ L β + } β Now, it is easy to see that for each K +, u onq implies w = {u } + onq Moreover, since {t,x,z) Q : u } is always a subset of {t,x,z) Q : w }, we always have {t,x,z) Q : w } Q Besides these, we also have to mae the crucial observation that, for every K +,wehave β β C θ, and L + β β 6C θ This means that we can apply Lemma 4 and Lemma 4 to w if we find that such an application is needed At this point, we need to separate our discussion into two cases in the following way First, if it happens that, for every K +,wehave {t,x,z) Q :<w < } δ, we then observe that we must have { t,x,z) Q : w } = { t,x,z) Q : w } δ + { t,x,z) Q : w }, for every K + Because of the above estimate, we can deduce inductively that {t,x,z) Q : w K + } K + δ Q, which in turn tells us that w K + = K +{u }+ almost everywhere on Q Hence we have u K +, almost everywhere on Q So, we are done in the first case Second, let us suppose the case in which there exists some N with K +, such that {t,x,z) Q : <w < } <δ We can then apply Lemma 4 to w and deduce w ) + + w ) + D θ ɛ ), Q 4 Q 4 which simply means ) w w +) + 4D θɛ ) <ɛ Q 4 Q 4 Now, the above inequality tells us that we can apply Lemma 4 directly to w + over Q, and deduce that w + 4 λ on Q, which implies 6 u λ + on Q 6 To finish the argument, we consider the barrier function b 3 : B 6 R characterized by the following conditions

15 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) b 3 =, on B 6 b 3 = on all the sides of B except the one for z = 6 b 3 = λ, on the side for z = + Then, by a simple application of the maximum principle, we now that there exists some constant λ, with <λ < K + min{,λ}, such that b 3 λ on B 3 Since u=β{θ L} is harmonic and that u is bounded above by b 3 along the sides of the cube B 6, it must follow that u b 3 λ on B 3 So we are done in the second case 5 Higher regularity: Proof of Corollary Extending Hölder continuity to higher regularity is not very difficult Indeed what is done in [5, Appendix B] can be applied here as well Therefore we only show how to set up the proof However, since some technical details are omitted in [5] for showing the solution is C α for all α<, we illuminate them here Let θ :, ) R be a solution of the N-dimensional critical Burgers equation which is essentially bounded and locally Holder s continuous on [τ, ) That is, θ L [τ, ) ) Cloc α [τ, ) R N ) for some <α< Fix y = t,x ) τ, ) Without the loss of generality, we may assume 4 θy ) = Consider now the Poisson ernel t Pt,x)= C N, t + x ) N+ which is the fundamental solution for the operator t + ) Ifθ = θ, ) is the initial datum for our solution θ, using Duhamel s principle we have t N θt,x)= Pt, ) θ x) Pt t, ) θ j θ)t, )x) dt = Pt, ) θ x) N t j Pt t,x x )θ t,x )dx dt j= Since Pt, ) θ is nown to be C, we just need to examine gt,x) = N t j= j= j Pt t,x x )θ t,x )dx dt For convenience, we now extend Pt,x) to the whole space-time R by requiring that Pt, ) =, whenever t< With such an extension gt,x) = N χ { t t t} j Pt t,x x )θ t,x )dx dt j= = N t,x )dx dt j= + χ {t } j Pt t,x x )θ 4 Otherwise replace the solution by wt,x) = θt,x + ut t )) θy ),whereu is a vector in with each entry equal to θy )

16 486 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 Our first tas is to estimate the difference gy + he) gy ), where e + with e =, and h is some sufficiently small positive number We observe that gy + he) gy ) = N χ {t } j= + { j Py + he y ) j Py y ) } θ y )dy Therefore, we need to estimate the following two terms for each j N A = χ {t } j Py + he y )θ y )dy χ {t } j Py y )θ y )dy, A = By,h) + By,h) By,h) χ {t }{ j Py + he y ) j Py y ) } θ y )dy, where By, h) ={y + : y y < h} Start with A Since θy ) = θy ) θy ) C y y α, and x Pt,x) C N N + ) t + x ) N+ = C N N + ) y N+, we recognize that the second integral in the expression for A can be controlled by χ {t } j Py y )θ y )dy C y y N+ α dy By,h) = C By,h) h r N+) r N+ α dr = CN,α)hα 5) Next, to control the first term in the expression for A, we need the following observation For every t>, j Pt, ) is an odd function in the x-variable Hence, the average value of j Pt, ) over any disc {t} {x : x <R} centered at the t-axis must be zero By the virtue of the above observation, it is easy to see χ {t } j Py + he y )dy = By +he,h) Then observe we can write χ {t } j Py + he y ) θy ) ) dy = A + A A 3, where By,h) A = By,h) By +he,h) χ {t } j Py + he y )θy )) dy A = By,h) By +he,h) χ {t } j Py + he y ){θy )) θy + he)) } dy A 3 = By +he,h) By,h) χ {t } j Py + he y )θy + he)) dy We first loo at A By the Holder s continuity of θ,wehave

17 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) θy + he) ) θy ) ) θy + he) { θy + he) θy ) } + θy ) { θy + he) θy ) } C h α + y y α) y + he y α If we further use that y By, h) By + he, h), the above inequality tells us θy + he) ) θy ) ) { C + α } h α y + he y α Thus A C By +he,h) h α y + he y N+ α dy = Ch α 5) On the other hand, the terms A and A 3 can be handled in the following way A C j Py + he y ) y y α dy and A 3 By,h) By +he,h) h) α C Ch α By,h) By +he,h) {h y +he y h} ) = Ch α log, By +he,h) By,h) Ch α {9h y +he y h} y + he y N+ dy y + he y N+ dy j Py + he y ) θy + he) ) dy y + he y N+ dy ) = Ch α log 9 By combining 5), 53), 54), we can conclude χ {t } j Py + he y ) θy ) ) dy Ch α, By,h) which with 5) implies A Ch α To complete the estimate for gy + he) gy ), we also need to control A For this purpose, we first recall the derivatives of j P i j Pt,x)= C N N + )t { δ ij N+3)x ix j } t + x ) N+3 { t j Pt,x)= C N N + )x j t + x ) N+3 t + x ) N+3 + N+3)t t + x ) N+3 + Then, it follows directly from the above two identities that j Py) C y N+, y = t, x) RN+, ) } 53) 54)

18 488 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 Therefore for all y + By, h) j Py + he y ) j Py y ) h j Py + λe y ) dλ h C dλ y + λe y N+ h C 9 y y dλ = C h N+ y y N+, where use the fact that for any y + By, h), wehave y + λe y y y h 9 y y Asa result, we can control A as follows if we have α< A { y y h} { y y h} = Ch h j Py + he y ) j Py y ) θy ) ) dy Ch y y N+ α dy r +α dr = h α So we conclude that if α satisfies α<, then, we must have gy + he) gy ) Ch α, for all sufficiently small h> This means that θ must be of class C α also, provided θ is of class C α By bootstrapping the above argument, we may now conclude that our locally Holder s continuous function θ is of class C γ,for any <γ < To go beyond Lipschitz and obtain the C,β regularity for θ, we just need to follow the argument in the second part of [5, Appendix B] Acnowledgements Both authors are extremely grateful to Professor Alexis Vasseur for suggesting the problem and for his guidance and support throughout the process Without him this wor would not have come into existence Appendix A A Proof of Lemma 4 The proof closely follows [5] except when the local energy inequality is employed Also we provide more details in Step Two below step 7 in [5]) For convenience, the following proof is given in the setting in which the L solution θ of the N-dimensional critical Burgers equation is defined on [ 4, ] The desired conclusion of Lemma 4 can be obtained by rescaling Step One: Determination of the constant λ and of the sequence of truncated energy terms A We begin by constructing the universal constant λ For this purpose, we consider the barrier function b : B4 R which verifies the following conditions b = onb 4

19 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) b =, on all the sides of the cube B4, except for the one with z = b =, on the side of B4 specified by z = Since b is harmonic on B4, we use the maximum principle to deduce that there exists some sufficiently small λ with <λ<, such that b 4λ is valid over B We note that λ depends only on N Next, let θ :[ 4, ] R be a solution of ), which verifies θ L [ 4,] ) C θ Wesetu = β{θ L}, with β C θ, and L 6C θ, and define for each u ={u C } +, and u = { u C }+, where C = λ + ) We now consider the following quantity for each A = δ T η u ) dxdzdt + η u L T,;L )), A) where T =, and {η } = is a fixed) sequence of functions in C c RN ) such that χ B+ + ) η χ B+ ), and η C, The integral along the z-direction in A) is taen over [,δ ], for some sufficiently small δ We will select such δ,in a way depending only on λ We choose δ in Step Four Now, we observe that the conclusion of Lemma 4 follows at once, provided we succeed in building up a nonlinear recurrence relation on A by using the De-Giorgi s technique We are now going to build up such a nonlinear recurrence relation for A under the assumption that the following two conditions are valid η u =, z [ δ, ], η + u + [ η u ) Pz) ] η +, A) x, z) B + ) [,δ ] A3) The symbol Pz)appearing in condition A3) stands for the Poisson ernel P,z) Step Two: Establishing the nonlinear recurrence relation for A by assuming the validity of conditions A) and A3) To begin, we observe that for each, we may express the function u C as { u C = β θ L C }, β with β C θ, and L + C β {6 + C }C θ 8C θ This means that we can apply Proposition 3 directly to u C = β{θ L C β }, and deduce that the following inequality is valid for every t B ηu ) dxdzds + ηu ) t, x) dx B B ηu ), x) dx + Φ t B η u dxds + t B η u ) dxdzds, A4) where, in the above inequality, we have Φ = NC N {8C θ + C θ } and η is some smooth cut off function compactly supported inside B By assuming the validity of condition A) at step, that is η u =,

20 49 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 for all z [δ, ], we now that the function η u χ { z δ } = η u χ { z } has no jump-discontinuity at z = δ We now choose some smooth function ψ :[, ) R which verifies the following conditions ψz), for all z [, ) ψz)=, for all z [, ] ψz)=, for all z [, ) dψ dz, for all z [, ) We then apply inequality A4) with the cut off function η ψ and deduce that the following inequality is valid for all, t with T T t where T = ) t B η ψu ) + η u ) t, x) dx B B η u ), x) dx + Φ t B η u dxds + t B η ψ) u ) dxdzds A5) We next notice that η ψu = η u χ { z δ } = η u χ { z }, and this implies that t t η ψu ) = t η u χ ) { z } = δ η u ) B Next, let us recall that according to the definition of η we have η χ B+ ) χ B+ η C χ B+ ) So, it follows that ) η χ B+ ) η ψ) η ψ + η dψ dz C + )χ B+ ) C + )η η C χ B+ ) C η Combining all these, it follows from A5) that the following inequality is valid for all, t with T T t t δ η u ) + η u ) t, x) dx t η u ), x) dx + Φ t C η u ) dxds + C η u ) dxdzds By taing the average among all the terms appearing in the above inequality over the variable [T,T ], and then taing the sup over t [T, ], we yield the following

21 A T T η u ), x) dx d { + C + Φ) CH Chan, M Czuba / Ann I H Poincaré AN 7 ) T η u ) dxds + T } η u ) dxdzds Our goal is to raise up the index for the three terms appearing in the right-hand side of the above inequality We just focus on T η u ) which is the most difficult among the three), and we remember our lucy number ) from the process of applying the De-Giorgi s method in Section N+ N Now, by using the facts χ {u >} χ {u > λ } λ u, and η χ B+ η u ) ) χb+ u ) χ{u >} χ B+ ) u ) χ B+ ) λ λ λ u ) N ) N ) u N+ N ) ) N η u ) N+ N ) Hence, it follows at once from the above inequality that T η u ) dxdzds T ) N λ T ) N = λ T λ ) N η u ) N+ N ) η u δ η u N+ N ) N+ L N ) ) N+ N ) N+ L N ) ) ) η, we can deduce that dzdt dzdt In the last line of the above estimate, we implicitly employ A) at step Now, by assuming the validity of A) at step 3, that is, η 3 u 3 =, for all z [δ 3, ], we now that the function η 3 u 3 χ { z δ 3 } = η 3 u 3 χ { z } has no jump-discontinuity at z = δ 3 and has the same trace as η 3 u 3 ) at z = Therefore, we can use the energy minimization property of harmonic extension to deduce that the following estimate is valid at step 3 δ 3 η 3 u ) 3 dxdz = η 3 u 3 χ { z δ 3 }) dxdz A6) = { η 3 u 3 ) } dxdz η 3 u 3 ) η 3 u 3 )dx

22 49 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 Because of this last inequality, we can use the Sobolev embedding and Hölder s inequality to obtain η 3 u 3 N+ A L N ) [T 3,] ) 3 By comparing the above inequalities, we see that we need a passage from the term η u N+ to the L N ) ) term η 3 u 3 N+, and such a passage is provided to us by condition A3) at step 3) Indeed, by L N ) ) assuming the validity of condition A3) at step 3, Young s inequality tells us that, for every t [, ] and every z,δ ),wehave η u N+ Pz) N ) ) L ) η 3u 3 N+ L L N ) ) = P) L ) η 3u 3 N+, L N ) ) where the last equality is valid just because we always have Pz) L ) = P) L ) So, it follows from A6) that T η u ) dxdzds λ ) N T ) N λ T δ δ δ P) N+ P) N+ N ) L ) N ) L ) T η u N+ N ) N+ L N ) ) dzdt P) N+ N ) L ) η 3u 3 λ λ N+ N ) N+ L N ) ) dzdt ) N N+ η 3u N ) 3 N+ L N ) [T 3,] ) ) N A + N 3 So, we have raised up the index for η u ) dxdzds The other two terms, namely T T η u ), x) dx d and T η u ) dxds can be treated in a similar way As a result, with the assistance of condition A) and condition A3), we are able to obtain the following nonlinear recurrence relation at step A C A+ N 3, A7) where, in the above nonlinear recurrence relation, C stands for some constant depending only on C θ and N More precisely, we can summarize what we have done in the following way For every 3, if condition A) is valid at steps 3,,, and condition A3) is valid at step 3, then it follows that the nonlinear recurrence relation A7) is valid at step also Step Three: Establishing condition A3) at step by assuming the validity of condition A) at step We need to introduce another barrier function b :[, ) [, ] R, which verifies b = on, ), ) b,z)=, for z, ) b x, ) = b x, ) =, for x, ) Now, by assuming the validity of A) at step, we are ready to establish condition A3) at step by controlling the behavior of u over B + ) [,δ ], where the suitable δ will be chosen once and for all, and in a way + depending only on N) during this procedure

23 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) Indeed, a direct application of the maximum principle together with A) at step ) yields the following expression on B + ) [,δ ] + N x u + η x i u ) Pz)+ b δ, z ) xi x δ + b δ, z ) δ, i= where x + = + ), x = x +, and that b + verifies b x, z) ) e x Hence, when the scaling factor gets δ involved in the variables of b, we yield the following inequality which is valid over the same set B + ) [,δ ] + x + x i b δ, z ) xi x δ + b δ, z ) δ ) [ x + x i ) e δ + e x i x ) ] δ Now, if we restrict x to be in B + ),wehavemin{ x + x + i, x i x }, for all x B + ) +) + So, the above inequality implies that the following holds over the smaller set B + ) [,δ ] + x + x i b δ, z ) xi x δ + b δ, z ) δ )) e 4 +) δ Hence, the following inequality is also valid over B + ) [,δ ] + u η u ) Pz)+ N )) e 4 Since u + [u +) δ λ ] + +, the above inequality implies that we have over B + ) [,δ ] + u + [η u ) Pz)+ N )) e 4 +) δ λ + ] A8) + Here, let us discuss what we have done The above way of arriving at inequality A8) is just a simple application of the maximum principle with the participation of the barrier function b with its width in the z-direction being compressed by the scaling factor However, A8) eventually forces us to compare N δ ))e 4 +) δ with λ This motivates us to choose δ to be sufficiently small so that the following holds for all + N ) ) e 4{) +}δ) λ + A9) Observe that δ, which maes A9) valid for all, depends only on N Once δ is chosen and fixed, A8) at step ) together with the assistance of A9) give u + [η u ) Pz) λ ] + + over B + ) [,δ ], and this in turn gives us the validity of condition A3) at step Now, let us summarize what we have achieved + in this step We can always select a sufficiently small δ>forwhichcondition A9 is valid for all For such δ>, the validity of condition A) at step directly implies the validity of condition A3) at step Step Four : Propagation of condition A) Now, let δ> be the fixed, sufficiently small constant which maes condition A9) valid for all Now, we attempt to derive condition A) at step + byassuming the validity of A) at step To do this, let us recall that inequality A8) at step and condition A9 together give u + [η u ) Pz) λ ] + + over B + ) [,δ ] In order to obtain A) at step +, we may just tae advantage of the inequality we just mentioned and + deduce that η u ) Pz) L ) η u L ) Pz) L ) A ) z N P) L ),

24 494 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 where the second inequality comes from the definition of A and the fact that Pz) L ) = z N P) L ) But this eventually tells us that over B + ) [δ +,δ ] we have + u + [A ) P) z N L ) λ ] [ + A ) P) + δ N +) L ) λ ] + A) + We note that the second inequality is valid because δ + z δ Here, we have to eep in mind that condition A) at step + is what we want So, by taing a closer loo at A), it is natural that we want to have the following inequality because we want u + = onb + ) [δ +,δ ]) + A ) P) L δ N +) ) λ A) + But this simply forces us to admit that the following two conditions should be true for any sufficiently large M M should be greater than { } ) δ N A M, A) P) L M δ N +) ) λ, + A3) The reason is that we have already seen a sequence { δ N ) } = appearing in inequality A), which is a sequence decaying to as increases If we would lie to construct another sequence decaying in a rate faster than { δ N ) } =, the best thing to do is to choose some { } M =, with M to be large when compared with Hence { } M = will decay faster than { δ N ) } = This observation more or less explains the origins of conditions A) and A3) However, it is important to observe that condition A3) automatically becomes valid for any sufficiently large M>, while condition A) and condition A) mutually depend on each other in a delicate way, just as we will see in our next step But now, let us summarize the result we have obtained in this step For the fixed choice of sufficiently small δ as selected in Step Three, and for any sufficiently large M>as selected in Step Four, condition A) at step, together with condition A) at step, will imply the validity of condition A) at step + Step Five : Propagation of condition A) and its relation to the nonlinear recurrence relation A7) for the truncated energy terms A In this step, by assuming condition A) at step 3 and also condition A) at steps 3,,, we attempt to deduce the validity of condition A) at step In our present circumstance, by applying the conclusion of Step Four to conditions A) and A) at steps 3,, and successively, we can deduce that our assumptions will imply the validity of condition A) at steps,, and also Hence, we can invoe the conclusion obtained in Step Two to deduce that A C A+ N 3 is valid at step This, together with the validity of condition A) at step 3, will in turns imply that we have the following inequality to be valid { } A C A+ N + 3 N C M 3 Because of the above inequality, we will have the validity of condition A) at step, provided if M is chosen to be sufficiently large so that the following condition becomes valid for all N for more details about this, see Lemma 7 of [5]) { } + M N C M 3 More precisely, we obtain the following conclusion in this step δ N A4)

25 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) If M is chosen to be large enough, condition A4) will become valid for all N for a formal proof of this fact, see Lemma 7 of [5]) For any such sufficiently large M being selected, condition A) at step 3, together with condition A) at steps 3,,, will give the validity of condition A) at step Step Six : Completing the argument by taing the initial steps Before we complete the proof of Lemma 4), let us summarize what we have achieved from Step One to Step Five We recall that after the universal constant λ, ) is chosen in Step One, we have determined δ>which depends only on N), and some sufficiently large M> which depends only on N and C θ ) such that the following three conditions which are conditions A9), A3), and A4) respectively) are valid at the same time for all 4{) +}δ) λ + N) )e P) M δ N +) L ) M C { M 3 } + N λ + With the technical support of the three conditions listed as above, we have also demonstrated that the propagation of condition A) and the propagation of condition A) mutually rely on each other in the following way Condition A) at step, together with condition A) at step, will give condition A) at step + Condition A) at step 3, together with condition A) at steps 3,,, will give condition A) at step Because of this, we can conclude our proof for Lemma 4 by selecting some sufficiently small ɛ in a way depending only on N and the given constant C θ ) such that the following two statements are true This is sufficient because the validity of condition A) for all immediately gives the desired conclusion of Lemma 4) A, for every N M η u =, for all z [, ] To see the way in which the ɛ is selected, we just recall that the function u = β{θ L} with β C θ, and L 6C θ ) under consideration is required to satisfy the hypothesis that u = β{θ L} verifies u on[ 4, ] B4, and that 4 B4 u + ) dxdzds + 4 u + dxds ɛ So, we may invoe inequality A5), which we obtained in Step Two, to deduce that u = β{θ L} must satisfy the following inequality for every N A T B η ψu ) + sup t [T,] B η u ) t, x) dx C 4N + Φ)ɛ, where, in the above inequality, C stands for some constant depending only on N, and we have implicitly use the fact that η C, and N Because of the above inequality, we now that if ɛ satisfies <ɛ< { M N 4N C + Φ) }, then it follows at once that condition A) is valid for N On the other hand, we also need to control the behavior of u over B [, ] by the upper bound λ because this will give u ={u λ)} + on

26 496 CH Chan, M Czuba / Ann I H Poincaré AN 7 ) 47 5 B [, ], and hence the validity of condition A) at step ) To achieve this, we use the local energy inequality to observe that the following estimate is valid for all z [, ] u + χ B Pz) L ) C θ Pz) L ) ɛ ), where, in the above estimate, C θ stands for some constant depending on N and C θ We now recall that the barrier function b as constructed in Step verifies b 4λ on B So, a simple application of the maximum principle to u will give the following bound for u over B [, ] u 4λ + C θ P) L ) ɛ ) on B [, ] λ This means that if we select any ɛ with <ɛ < { } C, it will follow at once that u θ P) ={u λ)} + L ) onb [, ], and hence the validity of condition A) at step So finally, we conclude that the ɛ required in Lemma 4 can be any positive number less than M min N 4N C + Φ) ) ) λ ), C θ P) L ) and we have completed the proof for Lemma 4 A Proof of Lemma 4 The proof uses the following lemma, the proof of which can be found in Appendix A of [5] Lemma A See De-Giorgi Isoperimetric Lemma [5]) Let ω H [, ] N+ ), and then A ={x: ωx) }, B ={x: ωx) }, C ={x: <ωx)<}, A B C N ω H C We now present the proof of Lemma 4 The proof is the same as the proof of Lemma 8 in [5] However, we mae changes to its presentation, which we believe mae it easier to follow For convenience, the following proof is also given in the setting in which the L solution θ of the N-dimensional critical Burgers equation is defined on [ 4, ] The desired conclusion of Lemma 4 can be obtained by rescaling Start with choosing ɛ Next, since u onq 4, by the local energy inequality there exists some constant C such that 4 B u + dxdzdt C Now we mae two observations First, since u onq 4, then u ) + Second, if we let U = { t,x,z) Q : u t,x,z) }, then Q u ) + dt dxdz = U A5) u ) dt dxdz A6) +