GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS WITH SINGULAR VELOCITIES

Transcription

1 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS WIH SINGULAR VELOCIIES DONGHO CHAE 1, PEER CONSANIN, DIEGO CÓRDOBA3, FRANCISCO GANCEDO, JIAHONG WU 4 Abstract. his paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. he first family is a generalized surface quasi-geostrophic SQG) equation with the velocity field u related to the scalar θ by u = Λ β θ, where 1 < β and Λ = ) 1/ is the Zygmund operator. he borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions and the local existence of patch type solutions. he second family is a dissipative active scalar equation with u = logi )) µ θ for µ > 0, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. his result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani [84]. 1. Introduction his paper studies solutions of generalized surface quasi-geostrophic SQG) equations with velocity fields given by more singular integral operators than the Riesz transforms. Recall the inviscid SQG equation 1.1) t θ + u θ = 0, u = ψ x, x1 )ψ, Λψ = θ, where Λ = ) 1/ is the Zygmund operator, θ = θx, t) is a scalar function, u denotes the D velocity field and ψ the stream function. Clearly, u can be represented in terms of the Riesz transforms of θ, namely u = R, R 1 )θ x Λ 1, x1 Λ 1 )θ. 1.1), its counterpart with fractional dissipation and several closely related generalizations have recently been investigated very extensively and significant progress has been made on fundamental issues concerning solutions of these equations see, e.g. [1]-[18], [1]-[65], [67]-[96], [98]-[116]). Our goal here is to understand solutions of the SQG type equations with velocity fields determined by even more singular integral operators. Attention is focused on two 000 Mathematics Subject Classification. 35Q35, 35B45, 35B65, 76B03, 76D03. Key words and phrases. generalized surface quasi-geostrophic equation, active scalar equation, existence and uniqueness. 1

2 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU generalized SQG equations. he first one assumes the form 1.) t θ + u θ = 0, u = ψ, ψ = Λ β θ, where β is a real parameter satisfying 1 < β. Here the spatial domain is either the whole plane R or the D periodic box and the fractional Laplacian operator ) α is defined through the Fourier transform ) α fξ) = ξ α fξ). he borderline case β = 1 of 1.) is the SQG equation 1.1), while 1.) with β = 0 is the well-known D Euler vorticity equation with θ representing the vorticity see e.g. [71]). he second generalized SQG equation under study is the dissipative active scalar equation 1.3) t θ + u θ + κ ) α θ = 0, u = ψ, ψ = logi )) µ θ, where κ > 0, α > 0 and µ > 0 are real parameters, and logi )) µ denotes the Fourier multiplier operator defined by logi )) µ fξ) = log1 + ξ ) )µ fξ). 1.3) is closely related to 1.). In fact, both 1.) with β = and 1.3) with κ = 0 and µ = 0 formally reduce to the trivial linear equation t θ + θ θ = 0 or t θ = 0. For µ > 0, the velocity field u in 1.3) is at least logarithmically more singular than those in 1.). We establish four main results for the existence and uniqueness of solutions to the equations defined in 1.) and in 1.3) with a given initial data θx, 0) = θ 0 x). We now preview these results. Our first main result establishes the local existence and uniqueness of smooth solutions to 1.) associated with any given smooth initial data. More precisely, we have the following theorem. heorem 1.1. Consider 1.) with 1 < β. Assume that θ 0 H m R ) with m 4. hen there exists = θ 0 H m) > 0 such that 1.) has a unique solution θ on [0, ]. In addition, θ C[0, ]; H m R )). Remark 1.. As mentioned previously, when β =, ψ = θ and u = θ and 1.) reduces to the trivial equation t θ = 0 or θx, t) = θ 0 x). herefore, 1.) with β = has a global steady-state solution. For 1 < β <, the velocity u is determined by a very singular integral of θ and u is not known to be bounded in L. As a consequence, the nonlinear term can not be directly bounded. o deal with this difficulty, we rewrite the nonlinear term in the form

3 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 3 of a commutator to explore the extra cancellation. In order to prove heorem 1.1, we need to derive a suitable commutator estimate see Proposition.1 in Section ). Our second main result proves the local existence and uniqueness of smooth solutions to 1.3). In fact, the following theorem holds. heorem 1.3. Consider the active scalar equation 1.3) with κ > 0, α > 0 and µ > 0. Assume the initial data θ 0 H 4 R ). hen there exists > 0 such that 1.3) has a unique solution θ C[0, ]; H 4 R )). We remark that the velocity field u in 1.3) is determined by u = logi )) µ θ with µ > 0 which is even logarithmically more singular than that in 1.) with β =, namely the trivial steady-state case. In a recent lecture [84], K. Ohkitani argued that 1.3) with κ = 0 may be globally well-posed based on numerical computations. heorem 1.3 is a first step towards positively confirming his prediction. Again the difficulty arises from the nonlinear term. In order to obtain a local in time) bound for θ H 4, we need to rewrite the most singular part in the nonlinear term as a commutator. his commutator involves the logarithm of Laplacian and it appears that no L -bound for such commutator is currently available. By applying Besov space techniques, we are able to prove the following bound for such commutators. Proposition 1.4. Let µ 0. Let x denote a partial derivative, either x1 or x. hen, for any δ > 0 and ɛ > 0, [lni )) µ x, g] f L C µ,ɛ,δ 1 + ln 1 + f )) µ ) Ḣ δ f L g H +3ɛ, f L where C µ,ɛ,δ is a constant depending on µ, ɛ and δ only, Ḣ δ denotes the standard homogeneous Sobolev space and the brackets denote the commutator, namely [lni )) µ x, g] f = lni )) µ x fg) lni )) µ x f) g. Our third main result assesses the global existence of weak solutions to 1.). Our consideration is restricted to the setting of periodic boundary conditions. he weak solution is essentially in the distributional sense and its precise definition is as follows. in the definition denotes the D periodic box. Definition 1.5. Let > 0. A function θ L [0, ]; L )) is a weak solution of 1.) if, for any test function φ Cc [0, ) ), the following integral equation holds, 1.4) θ t φ + u φ) dx dt = θ 0 x) φx, 0) dx. 0 Although the velocity u is more singular than the scalar θ and the nonlinear term above could not make sense, it is well defined due to a commutator hidden in the equation see Section 4). We prove that any mean-zero L data leads to a global in time) weak solution. hat is, we have the following theorem.

4 4 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU heorem 1.6. Assume that θ 0 L ) has mean zero, namely θ 0 x) dx = 0. hen 1.) has a global weak solution in the sense of Definition 1.5. his result is an extension of Resnick s work [88] on the inviscid SQG equation 1.1). However, for 1 < β <, the velocity is more singular and we need to write the nonlinear term as a commutator in terms of the stream function ψ. More details can be found in the proof of heorem 1.6 in Section 4. Our last main result establishes the local well-posedness of the patch problem associated with the active scalar equation 1.). his result extends Gancedo s previous work for 1.) with 0 < β 1 [50]. Since β is now in the range 1, ), u is given by a more singular integral and demands regular function and more sophisticated manipulation. he initial data is given by { θ1, x Ω; 1.5) θ 0 x) = θ, x R \ Ω, where Ω R is a bounded domain. We parameterize the boundary of Ω by x = x 0 γ) with γ = [ π, π] so that γ x 0 γ) = A 0, where π A 0 is the length of the contour. In addition, we assume that the curve x 0 γ) does not cross itself and there is a lower bound on γ x 0 γ), namely 1.6) Alternatively, if we define x 0 γ) x 0 γ η) η 1.7) F x)γ, η, t) = then 1.6) is equivalent to > 0, γ, η. η, if η 0, xγ, t) xγ η, t) 1, if η = 0, γ xγ, t) 1.8) F x 0 )γ, η, 0) < γ, η. he solution of 1.) corresponding to the initial data in 1.5) can be determined by studying the evolution of the boundary of the patch. As derived in [50], the parameterization xγ, t) of the boundary Ωt) satisfies 1.9) t xγ, t) = C β θ 1 θ ) γ xγ, t) γ xγ η, t) xγ, t) xγ η, t) β where C β is a constant depending on β only. For β 1, ), the integral on the right of 1.9) is singular. Since the velocity in the tangential direction does not change the dη,

5 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 5 shape of the curve, we can modify 1.9) in the tangential direction so that we get an extra cancellation. More precisely, we consider the modified equation γ xγ, t) γ xγ η, t) 1.10) t xγ, t) = C β θ 1 θ ) xγ, t) xγ η, t) dη + λγ, t) γxγ, t) β with λγ, t) so chosen that γ xγ, t) γxγ, t) = 0 or γ xγ, t) = At), where At) denotes a function of t only. A similar calculation as in [50] leads to the following explicit formula for λγ, t), λγ, t) = C γ + π ) γ xγ, t) π γ xγ, t) γ xγ, t) γ xγ η, t) 1.11) γ xγ, t) xγ η, t) dη dγ β γ ) γ xη, t) C γ xη, t) γ xη, t) γ xη ξ, t) η xη, t) xη ξ, t) dξ dη, β where C = C β θ 1 θ ). π We establish the local well-posedness of the contour dynamics equation CDE) given by 1.10) and 1.11) corresponding to an initial contour xγ, 0) = x 0 γ) satisfying 1.8). More precisely, we have the following theorem. heorem 1.7. Let x 0 γ) H k ) for k 4 and F x 0 )γ, η, 0) < for any γ, η. hen there exists > 0 such that the CDE given by 1.10) and 1.11) has a solution xγ, t) C[0, ]; H k )) with xγ, 0) = x 0 γ). his theorem is proven by obtaining an inequality of the form d dt x H 4 + F x) L ) C x H 4 + F x) L )9+β. he ingredients involved in the proof include appropriate combination and cancellation of terms. he detailed proof is provided in Section 5.. Local smooth solutions his section proves heorem 1.1, which assesses the local in time) existence and uniqueness of solutions to 1.) in H m with m 4. For 1 < β, the velocity u is determined by a very singular integral of θ and the nonlinear term can not be directly bounded. o deal with this difficulty, we rewrite the nonlinear term in the form of a commutator to explore the extra cancellation. he following proposition provides a L -bound for the commutator Proposition.1. Let s be a real number. Let x denote a partial derivative, either x1 or x. hen, ) [Λ s x, g] f L R ) C Λ s f L Λgη) L 1 + C f L Λ 1+s gη) L 1,

6 6 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU where C is a constant depending on s only. In particular, by Sobolev embedding, for any ɛ > 0, there exists C ɛ such that [Λ s x, g] f L R ) C ɛ Λ s f L g H +ɛ + f L g H +s+ɛ). Since this commutator estimate itself appears to be interesting, we provide a proof for this proposition. Proof. he Fourier transform of [Λ s x, g] f is given by.1) [Λs x, g] fξ) = ξ s ξ j ξ η s ξ η) j ) fξ η) ĝη) dη. R where j = 1 or. It is easy to verify that, for any real number s,.) ξ s ξ j ξ η s ξ η) j C max{ ξ s, ξ η s } η. In fact, we can write.3) ξ s ξ j ξ η s ξ η) j = = where Aρ, ξ, η) = ρξ + 1 ρ)ξ η). herefore, For s 0, it is clear that 0 d dρ A s A j ) ξ s ξ j ξ η s ξ η) j 1 + s ) η When s < 0, F x) = x s is convex and A s max{ ξ s, ξ η s }. A s η j + s A s A η)a j )dρ, 1 0 A s dρ. A s = ρξ + 1 ρ)ξ η) s ρ ξ s + 1 ρ) ξ η s max{ ξ s, ξ η s }. o obtain the bound in Proposition.1, we first consider the case when s 0. Inserting.) in.1) and using the basic inequality ξ s s 1 ξ η s + η s ), we have [Λs.4) x, g] fξ) C ξ R s fξ η) ηĝη) dη + C ξ η s fξ η) ηĝη) dη R C ξ η s fξ η) ηĝη) dη R + C fξ η) η 1+s ĝη) dη. R By Plancherel s heorem and Young s inequality for convolution, [Λ s x, g] f L C Λ s f L Λgη) L 1 + C f L Λ 1+s gη) L 1. Applying the embedding inequality η 1+s ĝη) L 1 R ) C ɛ g H +s+ɛ R ),

7 we have, for s 0, GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 7 [Λ s x, g] f L R ) C ɛ Λ s f L R ) g H +ɛ R ) + f L R ) g H +s+ɛ R )). he case when s < 0 is handled differently. We insert.3) in.1) and change the order of integration to obtain where.5).6) H 1 = H = s [Λ s x, g] fξ) = H 1 + H, 0 A s fξ η) ηj ĝη) dη dρ, R R A s A η)a j fξ η) ĝη) dη dρ. Using the fact that F x) = x s with s < 0 is convex, we have A s = ξ η) + ρη s = 1 + ρ) s 1 ρ ξ η) ρ 1 + ρ η ρ) s 1 + ρ ξ η s + ρ ) 1 + ρ η s = 1 + ρ) s 1 ξ η s + ρ1 + ρ) s 1 η s. Inserting this inequality in.5), we obtain 1 H ρ) s 1 dρ ξ η s fξ η) ηĝη) dη 0 R 1 + ρ1 + ρ) s 1 dρ fξ η) η 1+s ĝη) η. R 0 Applying Young s inequality for convolution, Plancherel s theorem and Sobolev s inequality, we have H 1 L C Λ s f L Λgη) L 1 + C f L Λ 1+s gη) L 1 C ɛ Λ s f L g H +ɛ + C ɛ f L g H +s+ɛ. o bound H, it suffices to notice that 1 H s A s fξ η) ηĝη) dη dρ 0 R herefore, H L admits the same bound as H 1 L. his completes the proof of Proposition.1. With this commutator estimate at our disposal, we are ready to prove heorem 1.1. Proof of heorem 1.1. his proof provides a local in time) a priori bound for θ H m. Once the local bound is established, the construction of a local solution can be obtained through standard procedure such as the successive approximation. We shall omit the construction part to avoid redundancy. s

8 8 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU We consider the case when m = 4. he general case can be dealt with in a similar manner. By u = 0, 1 d dt θ, t) L = 0 or θ, t) L = θ 0 L. Let σ be a multi-index with σ = 4. hen, 1 d dt Dσ θ L = D σ θ D σ u θ) dx, where means the integral over R and we shall omit dx when there is no confusion. Clearly, the right-hand side can be decomposed into I 1 + I + I 3 + I 4 + I 5 with I 1 = D σ θ D σ u θ dx, I = D σ θ D σ 1 u D σ θ dx, I 3 = I 4 = I 5 = σ 1 =3,σ 1 +σ =σ σ 1 =,σ 1 +σ =σ σ 1 =1,σ 1 +σ =σ D σ θ u D σ θ dx. D σ θ D σ 1 u D σ θ dx, D σ θ D σ 1 u D σ θ dx, he divergence-free condition u = 0 yields I 5 = 0. We now estimate I 1. For 1 < β <, D σ u = Λ +β D σ θ with σ = 4 can not bounded directly in terms of θ H 4. We rewrite I 1 as a commutator. For this we observe that for any skew-adjoint operator A in L i.e. Af, g) L = f, Ag) L for all f, g L ) we have faf)g dx = fagf) dx, and therefore.7) faf)g dx = 1 {fagf) fgaf)} dx = 1 f[a, g]f dx. Applying this fact to I 1 with A := Λ +β, f := D σ θ and g := θ, one obtains I 1 = 1 D σ θ [ Λ +β, θ ] D σ θ dx. By Hölder s inequality and Proposition.1 with s = + β < 0, we have ) I 1 C ɛ D σ θ L D σ θ L + Λ +β D σ θ L θ H 3+ɛ C D σ θ L θ H 4. he estimate for I is easy. By Hölder s and Sobolev s inequalities, I C D σ θ L θ H +β θ H 4. By Hólder s inequality and the Gagliardo-Nirenberg inequality, I 3 C D σ θ L D σ 1 u L 4 D σ θ L 4 σ 1 =,σ 1 +σ =4 C D σ θ L θ 1/ H β+1 θ 1/ H β+ θ 1/ H 3 θ 1/ H 4 C D σ θ L θ H 3 θ H 4.

9 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 9 By Hölder s and Sobolev s inequalities, I 4 C D σ θ L D σ 1 u L D σ θ L σ 1 =1,σ 1 +σ =4 C D σ θ L θ H β+ θ H 4. For 1 < β <, the bounds above yields d dt θ H 4 C θ 3 H 4. his inequality allows us to obtain a local in time) bound for θ H 4. In order to get uniqueness, one could check the evolution of two solution with the same initial data. With a similar approach, we find d dt θ θ 1 H 1 C θ H 4 + θ 1 H 4) θ θ 1 H 1. An easy application of the Gronwall inequality provides θ = θ 1. his concludes the proof of heorem he case that is logarithmically beyond β = his section focuses on the dissipative active scalar equation defined in 1.3) and the goal is to prove heorem 1.3. As mentioned in the introduction, the major difficulty in proving this theorem is due to the fact that the velocity u is determined by a very singular integral of θ. o overcome this difficulty, we rewrite the nonlinear term in the form of a commutator to explore the extra cancellation. he commutator involves the logarithm of the Laplacian and we need a suitable bound for this type of commutator. he bound is stated in Proposition 1.4, but we restated here. Proposition 3.1. Let µ 0. Let x denote a first partial, i.e., either x1 or x. hen, for any δ > 0 and ɛ > 0, [lni )) µ x, g] f L C µ,ɛ,δ 1 + ln 1 + f )) µ ) Ḣ δ f L g H +3ɛ, f L where C µ,ɛ,δ is a constant depending on µ, ɛ and δ only and Ḣδ denotes the standard homogeneous Sobolev space. Remark 3.. he constant C µ,ɛ,δ approaches as δ 0 or ɛ 0. When µ = 0, the constant depends on ɛ only. We shall also make use of the following lemma that bounds the L -norm of the logarithm of function. Lemma 3.3. Let µ 0 be a real number. hen, for any δ > 0, 3.1) lni )) µ f L C µ,δ f L ln 1 + f )) µ Ḣ δ. f L where C µ,δ is a constant depending on µ and δ only.

10 10 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU In the rest of this section, we first prove heorem 1.3, then Proposition 3.1 and finally Lemma 3.3. Proof of heorem 1.3. he proof obtains a local a priori bound for θ H 4. Once the local bound is at our disposal, a standard approach such as the successive approximation can be employed to provide a complete proof for the local existence and uniqueness. Since this portion involves no essential difficulties, the details will be omitted. o establish the local H 4 bound, we start with the L -bound. By u = 0, 1 d dt θ L + κ Λα θ L = 0 or θ, t) L θ 0 L. Now let σ be a multi-index with σ = 4. hen, 1 d 3.) dt Dσ θ L + κ Λα D σ θ L = D σ θ D σ u θ) dx = J 1 + J + J 3 + J 4 + J 5, where J 1 = J = J 3 = J 4 = J 5 = D σ θ D σ u θ dx, D σ θ D σ 1 u D σ θ dx, σ 1 =3,σ 1 +σ =σ σ 1 =,σ 1 +σ =σ σ 1 =1,σ 1 +σ =σ D σ θ u D σ θ dx. D σ θ D σ 1 u D σ θ dx, D σ θ D σ 1 u D σ θ dx, By u = 0, J 5 = 0. o bound J 1, we write it as a commutator integral. Applying.7) with A := logi )) µ, f := D σ θ and g := θ, we have J 1 = 1 D σ θ [ logi )) µ, θ ] D σ θ dx. By Hölder s inequality and Proposition 3.1, J 1 C D σ θ L [ logi )) µ, θ ] D σ θ L C D σ θ L θ H +ɛ 1 + ln1 + Dσ θ H δ)) µ ) C ɛ D σ θ L θ H 3+ɛ ln1 + θ H 4+δ))µ. Applying Hölder s inequality, Lemma 3.3 and the Sobolev embedding 3.3) H 1+ɛ R ) L R ), ɛ > 0 we obtain J C σ 1 =3,σ 1 +σ =4 D σ θ L D σ 1 u L D σ θ L C ɛ D σ θ L ln1 + θ H 4+δ))µ θ H 3+ɛ.

11 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 11 o bound J 3, we first apply Hölder s inequality to obtain J 3 C D σ θ L D σ 1 u L 4 D σ θ L 4. By the Sobolev inequality σ 1 =,σ 1 +σ =4 f L 4 R ) C f 1/ L R ) f 1/ L R ) and applying Lemma 3.3, we have J 3 C D σ θ L D σ 1 u 1/ L D σ 1 u 1/ L D σ θ 1/ L D σ θ 1/ L σ 1 =,σ 1 +σ =4 C D σ θ L θ H 4 ln1 + θ H 4+δ))µ. By Hölder s inequality, 3.3) and Lemma 3.3, J 4 C D σ θ L D σ 1 u L D σ θ L C σ 1 =1,σ 1 +σ =4 σ 1 =1,σ 1 +σ =4 D σ θ L D σ 1 u H 1+ɛ D σ θ L C D σ θ L θ H 4 θ H 3+ɛ ln1 + θ H 3+ɛ+δ)) µ. Let 0 < ɛ 1 and 0 < δ < α. he estimates above on the right-hand side of 3.) then implies that 1 d dt Dσ θ L + κ Λα D σ θ L C θ 3 H ln1 + θ 4 H 4+α))µ. his inequality is obtained for σ = 4. Obviously, for σ = 1, and 3, the bound on the right remains valid. herefore, if we sum the inequalities for α = 1,, 3 and 4 and recalling, we have 1 d dt θ H + 4 κ θ H C 4+α θ 3 H ln1 + θ 4 H 4+α))µ. he local in time) a priori bound for θ H 4 then follows if we notice the simple inequality ln1 + a)) µ a for large a > 0. his completes the proof of heorem 1.3. We now present the proof of Proposition 3.1. Proof of Proposition 3.1. he proof involves Besov spaces and related concepts such as the Fourier localization operator j for j = 1, 0, 1, and the operator S j. hese tools are now standard and can be found in several books, say [0], [66] and [91]. A self-contained quick introduction to the notation used in this proof can be found in [14]. We start by identifying L with the inhomogeneous Besov space B 0,, namely f L = j= 1 j f L. Let N 1 be an integer to be determined later. We write 3.4) [lni )) µ x, g] f L = K 1 + K,

12 1 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU where 3.5) 3.6) K 1 = K = N 1 j= 1 j [lni )) µ x, g] f L, j [lni )) µ x, g] f L. j=n Following Bony s notion of paraproducts, F G = k S k 1 F k G + k k F S k 1 G + k k F k G with k = k 1 + k + k+1, we have the decomposition 3.7) [lni )) µ x, g] f = lni )) µ x fg) lni )) µ x f) g = L 1 + L + L 3, where L 1 = lni )) µ x S k 1 f k g) S k 1 lni )) µ x f) k g, k L = k L 3 = k lni )) µ x k f S k 1 g) k lni )) µ x f) S k 1 g, lni )) µ x k f ) k g k lni )) µ x f) k g. Inserting the decomposition 3.7) in 3.5) and 3.6) yields the following corresponding decompositions in K 1 and K, with K 11 = K 1 = K 1 K 11 + K 1 + K 13, K K 1 + K + K 3 N 1 j= 1 j L 1 L, K 1 = j L 1 L, K = j=n N 1 j= 1 j L L, K 13 = j L L, K 3 = j=n N 1 j= 1 j L 3 L, j L 3 L. j=n Attention is now focused on bounding these terms and we start with K 11. When j is applied to L 1, the summation over k in L 1 becomes a finite summation for k satisfying k j 3, namely j L 1 = j lni )) µ x S k 1 f k g) S k 1 lni )) µ x f) k g). k j 3 For the sake of brevity, we shall just estimate the representative term with k = j in j L 1. he treatment of the rest of the terms satisfying k j 3 is similar and yields the same bound. herefore, j L 1 L C j lni )) µ x S j 1 f j g) S j 1 lni )) µ x f) j g) L.

13 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 13 Without loss of generality, we set x = x1. By Plancherel s theorem, j L 1 L C Φ j ξ) Hξ) Hξ η)) Ŝ j 1 fξ η) j gη) dη R where Φ j denotes the symbol of j, namely j fξ) = Φ j ξ) fξ) and Hξ) = ln1 + ξ ) ) µ ξ1. o further the estimate, we first invoke the inequality Hξ) Hξ η) ln1 η + max{ ξ, ξ η }) ) µ + µ ln1 + max{ ξ, ξ η }) ) ) µ 1. Clearly, the first term on the right-hand side dominates. We assume, without loss of generality, that 3.8) Hξ) Hξ η) C η ln1 + max{ ξ, ξ η }) ) µ. Noticing that supp Φ j, supp j g {ξ R : j 1 ξ < j+1 }, we have, for 1 j N 1, j L 1 L C Φ j ξ) ln1 + max{ ξ, ξ η }) ) µ 3.9) R Ŝ j 1 fξ η) η j gη) dη L C ln1 + N ) ) µ Φ jξ) Ŝ j 1 fξ η) η j gη) dη R C ln1 + N ) ) µ Ŝ j 1 fξ η) η j gη) dη R By Young s inequality for convolution, j L 1 L C ln1 + N ) ) µ Ŝj 1 f L η j gη) L 1. By Plancherel s theorem and Hölder s inequality, for any ɛ > 0, herefore, 3.10) Ŝ j 1 f L = S j 1 f L f L, η j gη) L 1 C ɛ Λ +ɛ j g L K 11 C ɛ ln1 + N ) ) N 1 µ f L j= 1 C ɛ ln1 + N ) ) µ f L g H +ɛ. Λ +ɛ j g L We now estimate K 1. As in j L 1, we have j L = j lni )) µ x k f S k 1 g) k lni )) µ x f) S k 1 g). k j 3 L. L, L

14 14 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU It suffices to estimate the representative term with k = j. As in the estimate of j L 1, we have j L L C ln1 + N ) ) µ j fξ η) ηŝ j 1 gη) dη R herefore, C ln1 + N ) ) µ j f L ηŝ j 1 gη) L 1 C ln1 + N ) ) µ j f L g H +ɛ. K 1 C ln1 + N ) ) µ N 1 j= 1 j f L g H +ɛ 3.11) C ln1 + N ) ) µ f L g H +ɛ. K 13 involves the interaction between high frequencies of f and g and the estimate is slightly more complicated. First we notice that j L 3 = j lni )) µ x k f k g) k lni )) µ x f) ) k g. k j 1 Applying Plancherel s theorem and invoking 3.8), we find j L 3 L 3.1) j lni )) µ x k f k g) k j 1 k lni )) µ x f) ) k g L C Φ j ξ) ln1 + max{ ξ, ξ η }) ) µ k j 1 R k fξ η) η k gη) dη. L Since Φ j is supported on {ξ R : j 1 ξ < j+1 } and k f is on {ξ R : k 1 ξ < k+1 }, we have, for k j 1, ln1 + max{ ξ, ξ η }) ) µ ln1 + max{ j+, k+1) } ) µ k j 1 ln1 + k+4 ) ) µ. herefore, j L 3 L C ln1 + k+4 ) ) µ Φj ξ) k fξ η) η k gη) dη. R L When η is in the support of k g, η is comparable to k and η ɛ ɛk. Using this fact and Young s inequality for convolution, we have j L 3 L C ln1 + k+4 ) ) µ ɛk k fξ η) η 1+ɛ k gη) dη k j 1 R L C ln1 + k+4 ) ) µ ɛk k f L η 1+ɛ k gη) L 1. k j 1 L

15 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 15 Using the fact that ln1 + k+4 ) ) µ ɛk C ɛ, η 1+ɛ k gη) L 1 C ɛ g H +3ɛ, we obtain herefore, 3.13) K 13 = j L 3 L C ɛ g H +3ɛ N 1 j= 1 j L 3 L N 1 C ɛ g H +3ɛ j= 1 C ɛ g H +3ɛ f L. k j 1 ɛj k j 1 ɛk k f L. ɛk j) k f L We now turn to K 1. j L 1 is bounded differently. As in 3.9), we have j L 1 L C Φ j ξ) ln1 + max{ ξ, ξ η }) ) µ R Ŝ j 1 fξ η) η j gη) dη. L Since supp Φ j, supp j g {ξ R : j 1 ξ < j+1 }, we have ln1 + max{ ξ, ξ η }) ) µ C ln1 + j ) ) µ and η supp j g indicates that η is comparable with j. herefore, j L 1 L C ln1 + j ) ) µ ɛj Ŝ j 1 fξ η) η 1+ɛ j gη) dη R L herefore, C ln1 + j ) ) µ ɛj Ŝ j 1 f L η 1+ɛ j gη) L 1 C ln1 + j ) ) µ ɛj f L Λ+ɛ j g L. K 1 = j L 1 L j=n C f L ln1 + j ) ) µ ɛj Λ +ɛ j g L j=n C f L ln1 + N ) ) µ ɛn g H +ɛ 3.14) C f L g H +ɛ. We now bound K. j L admits the following bound j L L C Φ j ξ) ln1 + max{ ξ, ξ η }) ) µ R j fξ η) ηŝ j 1 gη) dη. L

16 16 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU Since supp Φ j {ξ R : j 1 ξ < j+1 } and supp Ŝ j 1 g {ξ R : ξ < j }, we still have ln1 + max{ ξ, ξ η }) ) µ C ln1 + j ) ) µ. In contrast to the previous estimate on j L 1, η Ŝ j 1 g no longer implies that η is comparable to j. However, any ξ supp j f must have ξ comparable to j. herefore, for any δ > 0, j L L C ln1 + j ) ) µ δj ξ η δ j fξ η) ηŝ j 1 gη) dη R hus, 3.15) C ln1 + j ) ) µ δj ξ η δ j fξ η) L ηŝ j 1 gη) L 1 C ln1 + j ) ) µ δj j Λ δ f L g H +ɛ. K C ln1 + j ) ) µ δj j Λ δ f L g H +ɛ j=n C ln1 + N ) ) µ δn g H +ɛ j Λ δ f L j=n C ln1 + N ) ) µ δn g H +ɛ f H δ. he last term K 3 can be dealt with exactly as K 13. he bound for K 3 is 3.16) K 3 C ɛ g H +3ɛ f L. Collecting the estimates in 3.10), 3.11), 3.13), 3.14), 3.15) and 3.16), and inserting them in 3.4), we obtain, for any integer N > 1, L [lni )) µ x, g] f L C ɛ ln1 + N ) ) µ f L g H +ɛ We now choose N such that δn f H δ 3.17) N = It then follows that + C ɛ f L g H +3ɛ + C ɛ ln1 + N ) ) µ δn f H δ g H +ɛ. C f L. In fact, we can choose ]. [ 1 δ log [lni )) µ x, g] f L C µ,ɛ,δ 1 + f H δ f L ln 1 + f )) µ ) H δ f L g H +3ɛ, f L where C µ,ɛ,δ is a constant depending on µ, ɛ and δ only. It is easy to see that the inhomogeneous Sobolev norm f H δ can be replaced by the homogeneous norm f Ḣδ. his completes the proof of Proposition 3.1. Finally we prove Lemma 3.3.

17 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 17 Proof of Lemma 3.3. Let N 1 be an integer to be specified later. We write where L 1 = N 1 j= 1 lni )) µ f L = L 1 + L j lni )) µ f L, L = j lni )) µ f L. j=n According to heorem 1. in [14], we have, for j 0, Clearly, for j = 1, herefore, For any δ > 0, herefore, j lni )) µ f L C ln1 + j ) ) µ j f L. 1 lni )) µ f L C 1 f L. L 1 C ln1 + N ) ) µ N 1 L j= 1 j f L C ln1 + N ) ) µ f L. ln1 + j ) ) µ δj δj j f L j=n ln1 + N ) ) µ δn f H δ. lni )) µ f L C ln1 + N ) ) µ f L + ln1 + N ) ) µ δn f H δ. If we choose N in a similar fashion as in 3.17), we obtain the desired inequality 3.1). his completes the proof of Lemma Global weak solutions his section establishes the global existence of weak solutions to 1.), namely heorem 1.6. he following commutator estimate will be used. Lemma 4.1. Let s 0. Let j = 1 or. hen, for any ɛ > 0, there exists a constant C depending on s and ɛ such that 4.1) [Λ s xj, g]h L ) C h L g H +s+ɛ + Λ s h L g H +ɛ). Although the lemma is for the periodic setting, it can be proven in a similar manner as Proposition.1 and we thus omit its proof. Proof of heorem 1.6. he proof follows a standard approach, the Galerkin approximation. Let n > 0 be an integer and let K n denotes the subspace of L ), K n = { e im x : m 0 and m n }.

18 18 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU Let P n be the projection onto K n. For each fixed n, we consider the solution of the projected equation, t θ n + P n u n θ n ) = 0, u n = Λ +β θ n, θ n x, 0) = P n θ 0 x). his equation has a unique global solution θ n. Clearly, θ n obeys the L global bound 4.) θ n, t) L = P n θ 0 L θ 0 L. In addition, let ψ n be the corresponding stream function, namely ψ n = Λ β θ n. hen we have 1 d Λ 1 β ψn = ψ n P n u n θ n ) dx dt L = ψ n u n θ n dx. Noticing that u n = ψ n, we integrate by parts in the last term to obtain ψ n u n θ n dx = ψ n u n θ n dx. herefore, d 4.3) Λ 1 β ψn = 0 or Λ 1 β L ψn Λ 1 β L ψ0. dt L Furthermore, for any φ H 3+ɛ with ɛ > 0, we have 4.4) t θ n x, t) φx) dx = u n θ n )P n φ dx = θ n u n P n φ dx. On the one hand, θ n = Λ β ψ n and θ n u n P n φ dx = ψ n Λ β u n P n φ) dx = ψ n Λ β ψ n P n φ ) dx. On the other hand, u n = ψ n and θ n u n P n φ dx = θ n ψ n P n φ) dx = ψ n Λ β ψ n P n φ dx. hus, θ n u n P n φ dx = 1 [ ψ n Λ β, P n φ ] ψ n dx. It then follows from Hölder s inequality and Lemma 4.1 that 4.5) θ n u n P n φ dx C ψ n L ψ n H β P n φ H 3+ɛ C Λ +β θ n L θ n L φ H 3+ɛ C θ 0 L φ H 3+ɛ where we have used the fact that mean-zero functions in L ) are also in H +β ). herefore, by 4.4), 4.6) t θ n H 3 ɛ C θ 0 L.

19 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 19 he bounds in 4.), 4.3) and 4.6), together with the compact embedding relation L ) H +β ) for 1 < β <, imply that there exists θ C[0, ]; L )) such that 4.7) θ n θ in L, ψ n ψ in L. In addition, because of the uniform boundedness of θ n L and the embedding L ) H 3 ɛ ), the Arzelà-Ascoli heorem implies 4.8) lim sup θ n x, t) θx, t))φx) dx 0, where φ H 3+ɛ ). n t [0, ] he convergence in 4.7) and 4.8) allows us to prove that θ satisfies 1.4). Clearly, θ n satisfies the integral equation θ n t φ + u n P n φ) dx dt = P n θ 0 x) φx, 0) dx. 0 It is easy to check that P n θ 0 x) φx, 0) dx θ 0 x) φx, 0) dx, and 4.8) implies that, as n, 0 θ n t φ dxdt 0 θ t φ dxdt. o show the convergence in the nonlinear term, we write θ n u n P n φ dx dt θ u φ dx dt 0 0 = 1 [ ψ n Λ β, P n φ ] ψ n dx dt 0 1 ψ [ Λ β, φ ] ψ dx dt 0 = 1 [ ψ n Λ β, P n φ φ) ] ψ n dx dt ψ n ψ) [ Λ β, φ ] ψ n dx dt ψ [ Λ β, φ ] ψ n ψ) dx dt. 0 In order to get the convergence for the first two terms above, we appeal to Lemma 4.1 and the strong convergence of ψ n in L. Let us point out that in the last term for Λ β ψ n we only have weak convergence in L so we have to proceed in a different manner. We

20 0 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU consider the following integral Q n t) = ψ [ Λ β, φ ] ψ n ψ) dx = k 0 ψ k) [ Λ β, φ ] ψ n ψ)) k), which is bounded by ) 1/ Q n t) k β ψ k) k β [ Λ β, φ ] ψ n ψ)) k) ) 1/. k 0 k 0 he first sum above is controlled by θ 0 L. Using a similar notation as before, the coefficients in the second sum have the form k β [ Λ β x, ϕ ] ψ n ψ)) k) where x is either x1 or x and ϕ is x φ. Since [ Λ β x, ϕ ] ψ n ψ)) k) = j ik a k β k j) a k j β )ψ n ψ) k j) ϕj) for a = 1,, following the bounds in Section we obtain [ Λ β x, ϕ ] ψ n ψ)) k) C k β + k j β ) ψ n ψ) k j) j ϕj) j C k β + j β ) ψ n ψ) k j) j ϕj). j For k 0, it yields k [ β Λ β x, ϕ ] ψ n ψ)) k) C ψ n ψ) k j) j 1 + j β ) ϕj). he above bound provides Q n t) C ɛ θ 0 L φ H 5 β+ɛ ψ n ψ L for any ɛ > 0. It then follows from 4.7) that lim n Q n t) = 0. he Dominated Convergence heorem then leads to the desired convergence of the third term. herefore, θ is a weak solution of 1. in the sense of Definition 1.5. his completes the proof of heorem 1.6. j 5. Local existence for smooth patches his section is devoted to proving heorem 1.7. Proof of heorem 1.7. Since β = corresponds to the trivial steady-state solution, it suffices to consider the case when 1 < β <. he major efforts are devoted to establishing a priori local in time) bound for x, t) H 4 + F x) L t) for x satisfying the contour dynamics equation 1.10) and F x)γ, η, t) defined in 1.7).

21 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 1 his proof follows the ideas in Gancedo [50]. he difference here is that the kernel in 1.10) is more singular but the function space concerned here is H 4 ), which is more regular than in [50] and compensates for the singularity of the kernel. For notational convenience, we shall omit the coefficient C β θ 1 θ ) in the contour dynamics equation 1.10). In addition, the t-variable will sometimes be suppressed. We start with the L -norm. Dotting 1.10) by xγ, t) and integrating over, we have 1 d xγ, t) dx = I 1 + I, dt where I 1 = I = xγ, t) γxγ, t) γ xγ η, t) xγ, t) xγ η, t) β λγ) xγ, t) γ xγ, t) dγ. dη dγ, I 1 is actually zero. In fact, by the symmetrizing process, I 1 = 1 xγ) xγ η)) γ xγ) γ xγ η)) dη dγ xγ) xγ η) β 1 = ) γ xγ) xγ η) β dγdη β) = 0. o bound I, we first apply Hölder s inequality to obtain I λ L x L γ x L. By the representation of λ in 1.11) and using the fact that 1 γ x F x) L t), we have where λ L C F x) L t) γ x γ = C F x) L t) I 1 + I ), I 1 = I = γ x γ x γ xγ) γ xγ η) xγ) xγ η) β γxγ) γxγ η) dη dγ, xγ) xγ η) β γ xγ) γ xγ η) dη dγ. xγ) xγ η) β+1 It is not hard to see that I 1 and I can be bounded as follows. herefore, I 1 C F x) β L t) γx L 3 γx L, I C F x) 1+β L t) γx L γx L. d dt x L C F x) 3+β L t) x 5 H 3. dη dγ

22 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU We now estimate 4 γx L. where 1 d dt 4 γx dγ = I 3 + I 4, I 3 = C γxγ) 4 γ 4 γ xγ) γ xγ η)) dη dγ, xγ) xγ η) β I 4 = γxγ) 4 γλ 4 γ x)γ) dγ. I 3 can be further decomposed into five terms, namely I 3 = I 31 + I 3 + I 33 + I 34 + I 35, where I 31 = γxγ) 4 5 γxγ) γxγ 5 η)) dη dγ, xγ) xγ η) β I 3 = 4 γxγ) 4 γxγ) 4 γxγ 4 η)) γ xγ) xγ η) β ) dη dγ, I 33 = 6 γxγ) 4 γxγ) 3 γxγ 3 η)) γ xγ) xγ η) β ) dη dγ, I 34 = 4 γxγ) 4 γxγ) γxγ η)) γ xγ) 3 xγ η) β ) dη dγ, I 35 = γxγ) 4 γ xγ) γ xγ η)) γ xγ) 4 xγ η) β ) dη dγ. By symmetrizing, I 31 can be written as I 31 = 1 γxγ) 4 γxγ 4 η)) γxγ) 5 γxγ 5 η)) dη dγ xγ) xγ η) β = 1 γ γxγ) 4 γxγ 4 η) ) dη dγ 4 xγ) xγ η) β = β γxγ) 4 γxγ 4 η) xγ) xγ η)) γ xγ) γ xγ η)) dη dγ. 4 xγ) xγ η) β+ Setting Bγ, η) = xγ) xγ η)) γ xγ) γ xγ η)) and using the fact that γ xγ) γxγ) = 0, we have I 31 C F x) +β L t) γxγ) 4 γxγ 4 η) Bγ, η)η γ xγ) γxγ) dη dγ. η β Using the bound that Bγ, η)η γ xγ) γxγ) C x C 3 η, we obtain I 31 C F x) +β L t) x C 3 x H 4.

23 GENERALIZED SURFACE QUASI-GEOSROPHIC EQUAIONS 3 o estimate of I 3, we realize that, after computing γ xγ) xγ η) β ), I 3 can be bounded in the same fashion as I 31. hat is, I 3 C F x) +β L t) x 4 H 4. In order to estimate I 33, we further decompose it into three terms, I 33 = I 331 +I 33 +I 333, where I 331 = C γxγ) 4 γxγ) 3 3 Dγ, η) γxγ η)) dη dγ, xγ) xγ η) +β I 33 = C γxγ) 4 γxγ) 3 γxγ 3 η)) γxγ) γ xγ η) dη dγ, xγ) xγ η) +β I 333 = C γxγ) 4 γxγ) 3 γxγ 3 B γ, η) η)) dη dγ xγ) xγ η) 4+β with It is not very difficult to see that Dγ, η) = xγ) xγ η)) γxγ) γxγ η)). I 331, I 33, I 333 C F x) +β L t) x 4 H 4. I 34 also admit similar bound. In I 35 one has to use identity γ xγ) 4 γxγ) = 3 γxγ) 3 γxγ) to find the same control. We shall not provide the detailed estimates since they can be obtained by modifying the lines in [50]. We also need to deal with I 4. o do so, we use the representation formula 1.11) and obtain In summary, we have 5.1) I 4 C F x) 4+β L t) x 5 H 4 d dt x H4 C F x) 4+β L t) x 5 H 4. We now derive the estimate for F x) L t). For any p >, we have ) p+1 d 5.) dt F x) p L pt) p η x t γ, t) x t γ η, t) dη dγ. xγ) xγ η) η Invoking the contour dynamics equation 1.10), we have x t γ) x t γ η) = I 5 + I 6 + I 7 + I 8 ) γ xγ) γ xγ ξ) γ xγ) γ xγ ξ) dξ xγ) xγ ξ) β xγ η) xγ η ξ) β γ xγ) γ xγ η) + γ xγ η ξ) γ xγ ξ) + dξ xγ η) xγ η ξ) β +λγ) λγ η)) γ xγ) + λγ η) γ xγ) γ xγ η)).

24 4 CHAE, CONSANIN, CÓRDOBA, GANCEDO AND WU Following the argument as in [50], we have Inserting these estimates in 5.), we find I 5 C F x) β L t) x 1+β C η, I 6 C F x) β L t) x C 3 η, I 7 C F x) 3+β L t) x 4 H η, 4 I 8 C F x) 3+β L t) x 4 H η. 4 d dt F x) L pt) C x 4 H4 F x) 4+β L t) F x) L pt). After integrating in time and taking the limit as p, we obtain d dt F x) L t) C x 4 H4 F x) 5+β L t). Combining with 5.1), we obtain d dt x H 4 + F x) L t)) C x 4 H4 F x) 5+β L t). his inequality would allow us to deduce a local in time) bound for x H 4. completes the proof of heorem 1.7. his Acknowledgements his work was partially completed when Chae, Constantin, Gancedo and Wu visited the Instituto de Ciencias Matemáticas ICMA), Madrid, Spain in November, 010 and they thank the ICMA for support and hospitality. Chae s research was partially supported by NRF grant No Constantin s research was partially supported by NSF grant DMS Cordoba and Gancedo were partially supported by the grant MM of the MCINN Spain) and the grant StG-03138CDSIF of the ERC. Gancedo was also partially supported by NSF grant DMS Wu s research was partially supported by NSF grant DMS and he thanks Professors Hongjie Dong, Susan Friedlander and Vlad Vicol for discussions. References [1] H. Abidi and. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal ), [] H. Bae, Global well-posedness of dissipative quasi-geostrophic equations in critical spaces. Proc. Amer. Math. Soc ), [3] L. Berselli, Vanishing viscosity limit and long-time behavior for D quasi-geostrophic equations, Indiana Univ. Math. J ), [4] W. Blumen, Uniform potential vorticity flow, Part I. heory of wave interactions and twodimensional turbulence, J. Atmos. Sci ), [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 3 007), [6] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math ),

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