GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WITH CRITICAL AND SUPERCRITICAL DISSIPATION

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1 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WITH CRITICAL AND SUPERCRITICAL DISSIPATION KAZUO YAMAZAKI 2 Abstract. We study the transort euation with nonlocal velocity introduced in [4]. We rove its global well-osedness under critical and suercritical dissiation, the last case under the smallness condition, in Besov saces with critical and subcritical regularity indexes, using the Fourier localization method and modulus of continuity. Keywords: Besov Sace, Modulus of Continuity, Hilbert Transform. Introduction We study a system of euations concerning the transort euation with nonlocal velocity in R defined as follows: () t θ + uθ x + νλ 2α θ =, θ(, x) = θ (x) where ν > is the dissiative coefficient which hereafter we shall assume to be one, with H a Hilbert transform: u = Hθ π P.V. θ(y) x y dy, and the oerator Λ has its Fourier symbol Λf = ξ ˆf with α (, 2 ]. Originally, the divergence-form of () was studied in [6] as a D model of Quasi-geostrohic euation (QG). The authors in [4] studied () as a simle model related to the vorticity form of 3D Euler euation, Constantin, Lax and Majda model and Birkhoff-Rott euations. Hereafter let us denote () by CCF model for Córdoba-Córdoba-Fontelos model as known in the literature. We note that the original model consisted of a minus sign for the nonlinear term; however, to simlify our comutation and due to the roerty of Hilbert transform, we decided to consider (). We shall also refer to the case α > 2 the subcritical, α = 2 the critical and α < 2 Date: August 4, 24. 2MSC : 35B65, 35Q35, 35Q86 2 Deartment of Mathematics, Oklahoma State University, 4 Mathematical Sciences, Stillwater, OK 7478, USA, (45) , kyamazaki@math.okstate.edu

2 2 KAZUO YAMAZAKI the suercritical CCF model due to its rescaling θ λ (x, t) = λ 2α θ(λx, λ 2α t) and the fact that the su norm of the solution to () is conserved (cf. []). In [4], the authors showed that with symmetric ositive initial data θ (x) such that max x θ = θ () = and Su(θ (x)) [ L, L], the solutions to () with ν = exerience θ x L = su x θ x (, t) blow u in finite time. On the other hand, with α > /2, and θ H 2 (R), the global regularity of the solution to () was shown as well as for α = /2 with initial data sufficiently small. Subseuently, the regularity results were imroved in [2] and []. While the author in the former roved the global well-osedness of the critical CCF model with eriodic smooth initial data, the author in the latter roved the global well-osedness of the critical and subcritical CCF model with an arbitrary initial data in H max{ 3 2 2α,} while the suercritical CCF model locally well-osed in H 3 2 2α and globally well-osed if the initial data is sufficiently small. Some olynomial-in-time decay estimates are also discussed in []. The blowu result was also imroved. In [6], the authors showed that for α [, 4 ), < δ < 4α there exists a C = C(α, δ, s) such that if θ (x) dx > C x+δ where M = θ L, the solution to () blows u in finite time. In comarison, it is well-known that in the case of the Burgers euation which is essentially () with u = θ, the existence of initial data such that the solution blows u in finite time in the suercritical case and global well-osedness in the critical and subcritical cases are well-known (cf. [], [4], []). We also note a recently introduced Burgers-Hilbert euation: θ t + θθ x = Λ 2α Hθ of which its blowu in the suercritical case was shown in [5]. The issue concerning the regularity and blowu of suercritical QG remains an imortant oen roblem (cf. [4]). In this aer we obtain the local well-osedness of the CCF model in generalized Besov sace, namely Ḃs, for all s + 2α with deending on α and extend to globally in time for the case α 2 as well as α (, 2 ) with small initial data, namely theorem.. Let θ Ḃs, (R), s + 2α { [, ) α = 2 [ 2α, ] α (, 2 ). Then there exists T > such that the CCF model () has a uniue solution θ L T Ḃs, L T Ḃs+2α, Moreover, for the case α = 2, for all β R+, t β θ L T Ḃs+β,.

3 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI theorem.2. For α 2, the CCF model () has a uniue global solution θ C(R + ; Ḃs,) L loc (R+ ; Ḃs+, ), where s + 2α theorem.3. For α (, 2 ), [ 2α, ], there exists a constant η > such that if θ Ḃs, < η, s + 2α then the CCF model () admits a uniue global solution θ C(R + ; Ḃs,) L T (R + ; Ḃs+2α, ) We note that another global well-osedness result under small initial data is obtainable generalizing the work in [23] on the suercritical QG. In this aer, we obtain a smallness condition similar to that in [3]. + 2α We note that Ḃ, is critical Besov saces for the CCF model; i.e. θ λ (, θ(, λ t) Ḃ 2α + 2α., t) Ḃ + 2α, We state analogous results concerning global well-osedness of similar euations in critical Besov saces: in [2] two-dimensional critical QG in Ḃ,, in [7] one-dimensional critical Burgers euation in Ḃ,, [, ), in [9] N-dimensional Navier-Stokes euation articularly in Ḃ N 2 + 2α 2,. In contrast to the revious work in [2], the imrovement concerning the eriodicity is due to the satial decay of the solution. In comarison to that of [], our result is more general allowing all ossible regularity index s + 2α. In the critical case, in contrast to the case of [2] or [7], we obtain results allowing α (, 2 ] and in contrast to the case of [7] we will extend the local result to global in time under the smallness condition and consider the subcritical index as well. In the suercritical case, our result is similar to the work on [3] and [22] in which the authors constructed θ χ s = { B, s B, s Ḃ, if < otherwise for s N + 2α (N =2 in the case [3]) and obtained global wellosedness of the suercritical QG and Porous Media euation under small initial data condition resectively. The reason for the restriction on in our case while theirs was [, ] is the lack of divergence-free roerty. We mention that our result concerning the global well-osedness with the smallness condition alies for the Burgers euation with little modification. We make a remark that it is not clear if a similar result in the sace χ s constructed above works for the CCF model due to the lack of L maximum rincile for the solutions to the CCF model for in general.

4 4 KAZUO YAMAZAKI In the next section, we set notations and state reliminary results. We focus on the critical index s = + 2α and only mention key modification necessary to generalize the result for the subcritical in Aendix A. 2. Preliminary Notations and Results We denote by S(R N ) the Schwartz class functions and S (R N ), its dual, the sace of temered distributions. Define S to be the subsace of S by S = {φ S, φ(x)x γ dx =, γ =,, 2,...} R N Its dual S is given by S = S/S = S /P where P is the sace of olynomials. For j Z we define A j = {ξ R N : 2 j < ξ < 2 j+ }. Then there exists a seuence (Φ j ) S(R N ) such that suˆφ j A j, ˆΦj (ξ) = ˆΦ (2 j ξ) or Φ j (x) = 2 jn Φ (2 j x) and j= ˆΦ j (ξ) = Conseuently, for any f S, { if ξ R N \ {} if ξ = j= Φ j f = f To define the homogeneous Besov sace, we set j f = Φ j f, j =, ±, ±2,... We define for s R,, r, the homogeneous Besov sace Ḃs,r by where f Ḃs,r = Ḃ s,r = {f S : f Ḃs,r < } { ( j (2js j f L ) r ) /r for r < su j 2 js j f L for r = To define inhomogeneous Besov sace, let Ψ C (RN ) be even and satisfy i.e. for any f S, = ˆΨ(ξ) + ˆΦ j (ξ) j=

5 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI Ψ f + Φ j f = f j= We further set if j 2 j f = Ψ f, if j = Φ j f, if j =,, 2,... and define for any s R,, r, the inhomogeneous Besov sace where f B s,r = B s,r = {f S : f B s,r < } { ( j= (2js j f L ) r ) /r, if r <, su j< 2 js j f L, if r = We use the usual notations of low-freuency cut-off Ṡj k<j recall that j k = if j k 2 hence conseuently k, and j (S k f k f) = if j k 4 We now state some imortant results (cf. [7]): Lemma 2.. Bernstein s Lemma: Let α, r. (a)if f satisfies su ˆf {ξ R N : ξ K2 j } for some integer j and a constant K >, then Λ 2α f L r (R N ) C 2 2αj+jN( r ) f L (R N ) (b)if f satisfies su ˆf {ξ R N : K 2 j ξ K 2 2 j } for some integer j and constants < K K 2, then C 2 2αj f L r (R N ) Λ 2α f L r (R N ) C 2 2 2αj+jN( r ) f L (R N ), where C and C 2 are constants deending on α,, and r only. Proosition 2.2. The following facts are also well-known: (a). Generalized derivatives: Let σ R, then the oerator Λ σ is an isomorhism from Ḃs,r to Ḃs σ,r. ) (b). Sobolev embedding: If 2 and r r 2, then Ḃs,r Ḃs N( 2 2,r 2 (c). For (, r) [, ] 2 and s >, there exists a ositive constant C = C(N, s) such that uv Ḃs,r C( u L v Ḃs,r + v L u Ḃs,r )

6 6 KAZUO YAMAZAKI We use couled sace-time Besov saces. Firstly, for T >, ρ [, ], L ρ ([, T ], Ḃs,r) = L ρ T Ḃs,r = {f S : f L ρ = ( 2 sr T Ḃs,r f r L ) r L ρ < } T Z and the second mixed sace is L ρ ([, T ], Ḃs,r) = L ρ T Ḃs,r = {f S : f = ( Lρ 2 sr T Ḃs,r f r L ρ ) T L r < } Z Due to Minkowski s ineuality, we have f f Lρ T Ḃs,r L ρ T Ḃs,r f f Lρ T Ḃs,r L ρ T Ḃs,r if ρ r if ρ r The following is due to [3]: Lemma 2.3. Let φ be a smooth function suorted in the shell {ξ R N : R ξ R 2, < R < R 2 }. There exist two ositive constants κ and C deending only on φ such that for all, τ and λ >, we have φ(λ D )e τλ2α u L Ce κτλ2α φ(λ D )u L Next result is due to [2]: Lemma 2.4. Let v be a smooth vector field. Let ψ t be a solution to ψ t (x) = x + v(τ, ψ τ (x))dτ Then for all t R, the flow ψ t is a C diffeomorhism over R N and ψ t ± L e V (t) ψ t ± Id L e V (t) 2 ψ t ± L e V (t) 2 v(τ) L e V (τ) dτ, where V (t) = v(τ) L dτ. The following is due to [8]: Lemma 2.5. Let v be a given vector field in L loc (R+ ; Li(R N ). For Z, we set u = u and denote by ψ the flow of the regularized vector field Ṡ v. Then for u Ḃα, with α [, ) and [, ] for some C = C(α, ) >, Λ 2α (u ψ ) (Λ 2α u ) ψ L Ce CV (t) V α (t)2 2α u L

7 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI where V(t) is same as that in the revious Lemma. The following two results from [7] and [3] concern the Transort- Diffusion euation defined as (T D) ν,α t θ + v θ + νλ 2α θ = f, θ(x, ) = θ (x) where v does not need to be divergence-free: Lemma 2.6. Let σ R and. Let R = (Ṡ v v) θ [, v ]θ. There exists a constant C = C(N, σ) such that 2 σ R L C[ v L 2 σ θ L 4, 2 > 3, Ṡ v L 2 σ θ L 2 ( )(σ N ) 2 N v L 2 σ θ L 2 ( )(σ+nmin(, )) 2 N (2 v L + div(v) L )2 σ θ L ] Lemma 2.7. Let ρ ρ, and r. Let s R satisfy s < + N (or s + N if r = ), s > Nmin(, )(or s > Nmin(, )if div(v) = ), There exists a constant C > deending only on N, α, s,, and r such that for any smooth solution θ of (T D) ν,α, we have the following a riori estimate: ν ρ θ 2α Lρ T Ḃs+ ρ,r where Z(T ) = T Ce CZ(T ) ( θ Ḃs,r + ν ρ f Lρ T 2α ρ ) Ḃs 2α+,r v(t) N dt. Besides, if v = Hθ, we have above Ḃ, L estimate valid for all s > with Z(T ) = T xv L dt. Finally, the following is a Lemma A. in [2]: Lemma 2.8. Let χ S(R N ). There exists a constant C = C(χ, N) such that for all C 2 diffeomorhisms ψ over R N and ψ = φ, for all θ S (R N ), for all [, + ] and (, ) Z 2, we have χ(2 D)( θ ψ) L J φ L θ L (2 J φ L J ψ L +2 φ L )

8 8 KAZUO YAMAZAKI where J φ is the Jacobian determinant of φ. 3. Proof of Local Result In this section we rove Theorem. in stes of contraction argument: 3.. Ste : Aroximating Solution. Let θ (x, t) = e tλ2α θ (x), u = Hθ and θ n+ solve the system of linear euations: (2) for n =,, 2,... We have θ L (R +,Ḃ +, ) { t θ + Λ 2α θ = θ (, x) = θ (x) { t θ n+ + u n x θ n+ + Λ 2α θ n+ = θ n+ (, x) = θ (x), u n = Hθ n R + j Z 2 j( + 2α) 2 j2α e κτ2j2α θ L dτ θ Ḃ + 2α, by Lemma 2.3. Thus, θ (t, x) L (R +, Ḃ +, ). By definition of θ and the fact that θ Ḃ + 2α,, we have θ L (R +, Ḃ + 2α, ) and thus, θ = e tλ2α θ L (R +, Ḃ + 2α, ) L (R +, Ḃ +, ) Suose this holds for n; we will rove it true for n + using Lemma 2.7. For Lemma 2.7 we take =, r =, ρ = and s = + 2α so reuire < + 2α. Hence, { [, ), α = /2 [ 2α, ], α (, 2 ) With such choice, by Lemma 2.7 we have (3) θ n+ Ce C R T L T Ḃ + 2α, un x Ḃ, L dt θ Ḃ + 2α, Moreover, Ḃ, Ḃ, Ḃ, L by Proosition 2.2(b). Therefore, C θ n θ n+ Ce L T Ḃ + 2α, L T Ḃ +, θ Ḃ + 2α, < by the induction hyothesis and continuity of Hilbert transform in homogeneous Besov saces. On the other hand,

9 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI θ n+ L T Ḃ +, θ n+ e c R T L T Ḃ +, c θ n e L T Ḃ +, θ Ḃ + 2α, xun Ḃ, L dt θ Ḃ < + 2α, by Lemma 2.7 with ρ =, r =, s = + 2α, =. This imlies for all n N, θ n L (R +, Ḃ + 2α, ) L (R +, Ḃ +, ) Ste 2: Uniform Bounds. Here we obtain uniform bounds with resect to arameter n for some T > indeendent of n. Let θ = θ and alying to (2) we obtain t θ n+ + (Ṡ u n )(θ n+ ) x + Λ 2α θ n+ = R := (Ṡ u n u n )(θ n+ ) x [, u n x ]θ n+ = (Ṡ u n u n )(θ n+ ) x u n x θ n+ + u n x θ n+ Let ψ be the flow of the regularized Ṡ u n and denote θ = θ ψ, R = R ψ and G = Λ 2α (θ n+ ψ ) (Λ 2α θ n+ ) ψ. Then we have t θ n+ + Λ 2α θ n+ = R + Λ 2α (θ n+ ψ ) (Λ 2α θ n+ ) ψ = R + G Alying j to this we get j θ n+ (t) = e Λ2α t j θ, + which gives j θ n+ (t) L e Λ2α t j θ, L + By Lemma 2.3 we have j θ n+ (t) L e κt2j2α j θ, L + e (t τ)λ2α ( j R + j G )dτ for some κ >. Now by Lemma 2.5 we have e (t τ)λ2α ( j R + j G ) L dτ e κ(t τ)2j2α [ j R L + j G L ]dτ (4) j G (t) L Ce CV (t) V α (t)2 2α θ n+ L where V (t) = xu n (τ) L dτ. On the other hand,

10 KAZUO YAMAZAKI (5) (6) j R (t) L 2 j x j R (t) L 2 j ( x R ) ψ L x ψ L 2 j x R L J ψ 2 j R L J ψ L x ψ L e CV (t) 2 j R L by Bernstein s Lemma and Lemma 2.4. This gives + j θ n+ (t) L e κt2j2α j θ, L L x ψ L e κ(t τ)2j2α e CV (τ) [V α (τ)2 2α θ n+ (τ) L + 2 j R (τ) L ]dτ We take L 2 norm over [, t] and multily by 2 (s+α) to obtain (7) 2 (s+α) j θ n+ (t) L 2 t L 2( j)α ( e κt2j2α+ ) 2 2 s j θ, L (8) Fix M Z to be secified later. Decomose + 2 (s+α) 2 ( j)2α e CV (t) V α (t) θ n+ L 2 t L t + 2 ( j)(+α)+s e CV (τ) R L dτ θ n+ = Ṡ M θ n+ ψ + j θ n+ ψ j M Then, for all t [, T ], θ n+ L 2 t L Ṡ M θ n+ ψ L 2 t L + j θ n+ j M ψ L 2 t L e CV (t) ( Ṡ M θ n+ L 2 t L + j θ n+ L 2 t L ) j M by Lemma 2.4. Now, by Lemma 2.8 we have Ṡ M θ n+ L J ψ L (2 x J ψ L J ψ L +2 M x ψ L ) θ n+ By Lemma 2.4 and Bernstein s ineuality, we estimate L Ṡ M θ n+ L J ψ L ( J ψ Taking L 2 norm over [, t] of this, we get L J ψ L + 2 M e V (t) ) θ n+ L (9) Ṡ M θ n+ L 2 t L ecv (t) (e CV (t) + 2 M ) θ n+ L 2 t L

11 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI For j >, we have due to comact suort, j θ, = ; hence, from (7) taking sum over M j, we have () 2 (s+α) j θ n+ L 2 t L 2M α ( e κt22α+ ) 2 2 s θ, L j M + 2 (s+α) 2 M 2α e CV (t) V α (t) θ n+ + 2 M (+α) 2 s e CV (τ) R L dτ Plugging (9) and () into (8) multilied by 2 (s+α) gives L 2 t L 2 (s+α) θ n+ L 2 t L c( e κt22α+ ) 2 2 s θ, L + ce CV (t) [(2 M + 2 M2α V α (t))2 (s+α) θ n+ L 2 t L + 2M (+α) We choose M and C > such that V (t) C imlies 2 s R L dτ] ce CV (t) (2 M + 2 M 2α V α (t)) 2 and cecv (t) 2 M (+α) < Then we have for t [, T ], there exists C > such that 2 (s+α) θ n+ L 2 t L C [( e κt22α+ ) 2 2 s θ, L + As reviously stated we take s = + 2α, 2 s R L dτ] 2 ( + α) θ n+ L 2 t L C [( e κt22α+ ) 2 2 ( + 2α) θ, L + We sum over and by definition we have 2 ( + 2α) R L dτ] θ n+ C + α [( e κt22α+ ) ( α) θ, L +, Z 2 ( + 2α) R L dτ] On 2 ( + 2α) R L we use Lemma 2.6 with σ = + 2α, N =, = to estimate; recall R := (Ṡ u n u n )(θ n+ ) x [, u n x ]θ n+,

12 2 KAZUO YAMAZAKI 2 ( + 2α) R L 2 ( + 2α)+ + Ṡ u n L θ n+ L ( +2 2α)+ u n L θ n+ L 4, 2 3, (2α) ( +) u n L θ n+ L 2 ( )( + 2α+min(, ))+ + ( + 2α) (2 u n L + 2 u n L ) θ n+ L Relacing the sum over where 4 by constant multiles of the sum over leads to 2 ( + α) R (τ) L dτ C 2 ( + α) u n L 2 t L 2 ( + α) θ n+ L 2 t L = C u n + α, θ n+ + α, C θ n + α, θ n+ + α, by Hölder s ineuality and continuity of Hilbert transform. Thus, θ n+ + α, Z C ( e κt22α+ ) 2 2 ( + 2α) θ, L +C 2 θ n We go back to (6) and take L norm over [,t] and multily by 2 (s+2α) to obtain () 2 (s+2α) j θ n+ (t) L t L 2( j)2α ( e κt2j2α )2 s j θ, L A very similar rocedure as before leads us to + 2 s 2 ( j)4α e CV (t) V α (t) θ n+ L t L t + 2 ( j)(+2α)+s e CV (τ) R L dτ + α, θ n+ + α, (2) θ n+ + θ n+ + α, L t Ḃ +, C ( e κt22α+ ) ( α) θ, L Z + C 2 θ n + α, θ n+ + α,

13 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI We now estimate the last term for each n. For n =, we certainly have θ + α, = e tλ2α θ + α, On the other hand, for θ + α, θ < Ḃ + 2α,, we aly Lemma 2.7 to t θ + u x θ + Λ 2α θ = ; θ (x, ) = θ (x), u = Hθ i.e. (2) with n =, with ρ = 2, r =, s = + 2α, = to obtain and θ e C R T L2 T Ḃ + α, xhθ Ḃ, L dt θ Ḃ + 2α, x Hθ L T (Ḃ, L ) θ Ḃ L T Ḃ +, < by Proosition 2.2(b) giving, Ḃ, Ḃ, L. Therefore, C 2 θ θ + α is bounded. On the other hand, by Lebesgue, + α, dominated convergence theorem, we have lim ( e κt 22α+ ) ( α) θ, L = T + Z Thus, we see that there exists some ɛ > sufficiently small such that for all t T where T = su{t > : C ( e κt22α+ ) ( α) θ, L ɛ } we have Z θ + θ + α 2ɛ, L t Ḃ +, by (2). Thus, inductively for all n Z + {}, t T, we have if (3) then ( e κt 22α+ ) ( α) θ, L ɛ Z (4) θ n+ + θ n+ L2 T Ḃ + α, L T Ḃ +, 2ɛ for all α (, 2 ). We use Ḃ, Ḃ, Ḃ, L and obtain below as we did on (3),

14 4 KAZUO YAMAZAKI θ n+ Ce C R T L T Ḃ + 2α, We bound above by C R T un (5) Ce xun Ḃ, L dτ θ Ḃ Ḃ +, dτ θ Ḃ + 2α, + 2α, C θ Ḃ + 2α, C R T xun Ce by Proosition 2.2 (a) of Λ σ being an isomorhism from Ḃs,m to Ḃs σ and by continuity of Hilbert transform in Ḃ +, we conclude that (θ n ) n N is uniformly bounded in L T Ḃ,,m. Combining this and (4), Ḃ + 2α, L T Ḃ +,. + 2α 3.3. Ste 3: Strong Convergence. We rove (θ n ) n N is a Cauchy seuence in L T Ḃ,. Let (n, m) N 2, n > m and θ n,m = θ n θ m. Now { t θ n+,m+ = u n x θ n+,m+ u n,m x θ m+ Λ 2α θ n+,m+ θ n+,m+ (x) = θ n+,m+ (x, ) = θ n+ (x, ) θ m+ (x, ) = We rely on Lemma 2.7 with =, ρ =, s = + 2α, ρ = r = dτ θ Ḃ + 2α, (6) θ n+,m+ L T Ḃ + 2α, C θ n Ce C θ n Ce L T Ḃ +2 2α, ( θ n+,m+ L T Ḃ +2 2α T, + 2α, + u n,m x θ m+ L T Ḃ u n,m x θ m+ (τ) Ḃ Using Proosition 2.2 (c) now, we get + 2α, dτ + 2α, ) u n,m x θ m+ Ḃ + 2α, u n,m L θ m+ Ḃ +2 2α, + x θ m+ L u n,m Ḃ + 2α, θ n,m Ḃ Substitute this into (6) and obtain θ n+,m+ L T Ḃ + 2α, C θ n,m L T Ḃ + 2α, + 2α, θ m+ Ḃ +2 2α, C θ n e L T Ḃ +2 2α T, By (3) and (4), we can choose ɛ small enough such that θ m+ (τ) Ḃ +2 2α, dτ

15 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI θ n+,m+ L T Ḃ + 2α, with ɛ <. By induction ɛ θ n,m L T Ḃ + 2α, θ n+,m+ L T Ḃ + 2α, ɛ m+ θ n, L T Ḃ + 2α, Cɛ m+ θ Ḃ + 2α, Thus, {θ n } n N is a Cauchy seuence in L T θ n converges strongly to θ in L T obtain θ L T Ḃ +, get a solution to () in L T Ḃ + 2α,. By comleteness, Ḃ + 2α,. By Fatou s Lemma and (5), we. Passing to the limit into the aroximate euation, we Ḃ + 2α, L T Ḃ +, Ste 4: Uniueness. Let θ and θ 2 be two solutions of () with same + 2α initial data in L T Ḃ, L T Ḃ +,. We can consider t θ,2 + u x θ,2 + Λ 2α θ,2 = u,2 x θ 2, θ,2 (x) = θ,2 (, x) = We slit two different cases. First, assume α = 2. With u,2 x θ 2 = f, ρ =, ρ =, =, r =, s = to obtain θ,2 L T Ḃ, C R T xu (t) Ḃ Ce C R T θ (t) Ḃ + Ce,, L dt( θ,2 dt T Ḃ, + u,2 x θ 2 u,2 x θ 2 dt (t) Ḃ, L t Ḃ, ) By continuity of Hilbert transform in Ḃ,, we have c θ θ,2 ce L T Ḃ, L T Ḃ + T, θ,2 L t Ḃ, θ 2 (t) Ḃ +, + 2α By Gronwall s ineuality, we obtain θ,2 (t) for all t T. L t Ḃ, Therefore, θ = θ 2. Now, for α (, 2 ), suose we have θ, θ 2 as two solutions with initial data in L T Ḃ, L T Ḃ,. Note we have L T Ḃ, L T Ḃ, L T Ḃ, L T Ḃ, for all [ 2α, ]. We aly Lemma 2.7 with = ρ =, r =, s =, =, ρ = : dt +

16 6 KAZUO YAMAZAKI θ,2 L T Ḃ, Ce C R T xu (t) Ḃ, L dt ( θ,2 Ḃ, + u,2 x θ 2 L T Ḃ,) Ce C R T θ (t) Ḃ, dt T θ,2 L t Ḃ, θ 2 (t) Ḃ, dt due to Proosition 2.2(b). By Gronwall s ineuality, we obtain θ,2 (t) L T Ḃ, for all t [, T ] which gives the desired result Ste 5: Smoothing Effect. For the case α = 2, we show that for all β R +, C β = C(β), [, ), Note C(β+) θ t β θ C L T Ḃ +β β e, L T Ḃ +, θ L T Ḃ, t (t β θ) = βt β θ + t β t θ = u x (t β θ) Λ(t β θ) + βt β θ; (t β θ)(x, ) = When β =, by Lemma 2.7 again with ρ = ρ = =, r =, s = +, C R T θ Ḃ tθ(t) Ce L T Ḃ +, Assume true for n. Let β = n + : on +, dt θ L T Ḃ, t (t n+ θ) = u x (t n+ θ) Λ(t n+ θ) + βt n θ with Lemma 2.7 again with ρ = ρ = =, r =, s = + n +, =, ρ =, we get C θ t n+ θ(t) Ce L T Ḃ +n+, C θ C(n + )e L T Ḃ +, t n θ L T L T Ḃ + Ḃ +n,, βt n θ L T c(n+2) θ Ce Ḃ +n, L T Ḃ +, θ L T Ḃ, by Proosition 2.2(b) and the induction hyothesis. For general β R +, interolation comletes the roof Blowu Criterion. Let T be the maximum local existence time of θ in L T Ḃ + 2α, L T Ḃ +,. If T <, then we show T x Hθ(t) L dt =. Suose T x Hθ(t) L dt <. By Lemma 2.7 with s = + 2α, r =, = ρ =, for all t [, T ) we have

17 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI (7) M θ(t) Ḃ + 2α T = Ce C R T xhθ(t) L dt θ <, Ḃ + 2α, (8) Using ɛ from (3), we take T > such that ( e κ T 2 2α+) ) 2 MT ɛ Z By (7) and (8) we have for all t [, T ), ( e κ T 2 2α+ ) ( α) θ(t) L ɛ Z Thus, by local existence result already obtained, we see that there exists solution θ(t) on [, T ) to () with initial data θ(t T 2 ). By uniueness of local solution, θ(t) = θ(t + T T 2 ) on [, T 2 ). Thus, θ extends the solution θ beyond T. This comletes the roof. 4. Proof of Global Result 4.. Critical Case. We extend local results reviously obtained globally in time via the method of modulus of continuity (MOC) introduced in [5] in the context of the critical QG. As this is discussed in [] and [4] in detail, we only rovide a sketch of the roof. A MOC is a function ω : R + R + that is continuous, increasing and concave such that ω() =. We say a function θ has a MOC ω if for all x, y R, θ(x) θ(y) ω( x y ) = ω(ξ). It is easy to show that the su norm of the derivative of θ is uniformly bounded if θ has a MOC ω for all time t >. The blow-u criterion which alies to our case, namely that if T is the first finite blow-u time, then T θ x (, t) L dx = was shown in []. This imlies that in order to extend our local result, it suffices to show that θ has a MOC ω that is unbounded for all time t > ; we note that in the critical case the rescaling of ω would be ω λ (ξ) = ω(λξ). The following result is from [5]: Lemma 4.. If θ(x, t) has a strict MOC ω u to time T > but not after T, then at time T, there exists x y such that θ(x, T ) θ(y, T ) = ω( x y ). Thus, we may assume that θ loses a MOC ω and show t (θ(x, T ) θ(y, T )) < to reach a contradiction. The estimate on the dissiation term modulus some constants is as follows: ξ 2 ω(ξ + 2η) + ω(ξ 2η) 2ω(ξ) η 2 dη+ ξ 2 ω(ξ + 2η) ω(2η ξ) 2ω(ξ) η 2 dη

18 8 KAZUO YAMAZAKI (cf. [5]) while the convection term Ω(ξ)ω (ξ) where Ω is the MOC of the Hilbert transform of θ which is identical to the case of the Riesz transform. This fact is also discussed in [4] Lemma 6. We omit further details Suercritical Case. By Lemma 2.7, we see that it suffices to bound Z(t) = T dt. We choose = and by continuity of xu(t) Ḃ, L Hilbert transform in homogeneous Besov saces we obtain Z(t) C T u(t) Ḃ, W, dt C T u(t) Ḃ +, dt c T θ(t) Ḃ by Proosition 2.. Alying lemma 2.7 on () with =, ρ = r =, s = + 2α and we satisfy s = + 2α + for all α (, 2 ), s > Nmin(, ) to obtain +, dt θ L T Ḃ +, Since ρ = r we have θ Z(t) C T θ(t) Ḃ L T Ḃ +, +, Ce CZ(T ) θ Ḃ + 2α, θ L T Ḃ +, dt C θ L T Ḃ +, and hence Ce CZ(t) θ Ḃ + 2α, Since the function Z(t) deends continuously in time and Z() =, we deduce that for small initial data Z does not blow u; i.e. there exists η > and C > such that θ < η Z(t) C θ Ḃ + 2α, Ḃ + 2α, 5. Aendix 5.. A: Subcritical Index. The key art of the roof that we must consider searately for the subindex is the uniform bound. Let s > + 2α and we see that it imlies the existence of k > such that s + 2α k. Using x u n L T (Ḃ, L ) θn, by Lemma 2.7 on (2) with n and L T Ḃ +, ρ = k, r =, s = + 2α k, =, we obtain (9) θ n L T Ḃ +, t k θ Ḃ + 2α k, t k θ Ḃ + 2α k, t k θ n L k T Ḃ +, e c R T xun Ḃ, L dτ c θ n L e T Ḃ +, where the first ineuality is Hölder s. Thus, there exists some ɛ, ɛ > such that for all n Z + {} the following holds:

19 GLOBAL WELL-POSEDNESS OF TRANSPORT EQUATION WITH NONLOCAL VELOCITY IN BESOV SPACES WI (2) t k θ Ḃ + 2α k, Now if n =, then ɛ θ n L T Ḃ +, ɛ θ L T Ḃ +, = e tλ2α θ L T Ḃ +, t k e tλ2α θ L k T Ḃ +, C t k θ Ḃ + 2α, C t k θ Ḃ + 2α k, C ɛ for t small enough. Now we iterate from (9) with n =: by Hölder s ineuality and Lemma 2.7 as before, θ and L T Ḃ +, C 2 t k θ Ḃ + 2α k, C 3 θ L e T Ḃ +, C 2 ɛ e C 3C ɛ 2C ɛ θ 2 L T Ḃ +, t k θ 2 L k T Ḃ +, C 2 t k θ Ḃ + 2α k, C 3 θ L e T Ḃ +, C ɛe C 32C ɛ 2C ɛ where C = max{c, C 2 } for ɛ small enough satisfying min{e C 3C ɛ, e 2C 3C ɛ } 2. Taking C = 2C η concludes the iteration. Using (2) in lace of (3) and (4) in the Strong Convergence roof, comletes the case for the subcritical index s > + 2α. 6. Acknowledgment The author exresses gratitude to Professor Jiahong Wu and David Ullrich for their teaching and the referees for helful comments and atience. References [] N. Alibaud, J. Droniou and J. Vovelle, Occurrence and non-aearance of shocks in fractal Burgers euations, J. Hyerbolic Diffe. E. 4, No. 3 (27) [2] H. Abidi and T. Hmidi, On the global well-osedness of the critical uasi-geostrohic euation, SIAM J. Math. Anal. 4 No. 3 (28) [3] A. Córdoba and D. Córdoba, A maximum rincile alied to uasi-geostrohic euations, Commun. Math. Phys. 249, No. 3 (24) [4] A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transort euation with nonlocal velocity, Annals of Math., 62 (25), [5] A. Castro, D. Córdoba and F. Gancedo, Singularity formation in a surface wave model, ArXiv. htt://arxiv.org/abs/ Accessed 7/5/2 [6] D. Chae, A. Córdoba, D. Córdoba and M.A. Fontelos, Finite time singularities in a D model of the uasi-geostrohic euation, Advances in Mathematics 94 (25) [7] J. Chemin Perfect incomressible fluids, Oxford Science Publications (998)

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