Generalized surface quasi-geostrophic equations with singular velocities

arXiv:1101.3537v1 [math.AP] 18 Jan 2011 GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS WITH SINGULAR VELOCITIES DONGHO CHAE1, PETER CONSTANTIN2, DIEGO C´ ORDOBA3, FRANCISCO GANCEDO2, JIAHONG WU4 Abstract. This 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. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field urelated to the scalar θby u=∇⊥Λβ−2θ, where 1< β ≤2 and Λ = (−∆)1/2is the Zygmund operator. The 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. The second family is a dissipative active scalar equation with u=∇⊥(log(I−∆))µθfor 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. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani [84]. 1. Introduction This 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 ∂tθ+u·∇θ= 0, u=∇⊥ψ≡(−∂x2, ∂x1)ψ, Λψ=θ, (1.1) where Λ = (−∆)1/2is the Zygmund operator, θ=θ(x, t) is a scalar function, udenotes the 2D velocity field and ψthe stream function. Clearly, ucan be represented in terms of the Riesz transforms of θ, namely u= (−R2,R1)θ≡(−∂x2Λ−1, ∂x1Λ−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], [21]-[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 2000 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, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU generalized SQG equations. The first one assumes the form ∂tθ+u·∇θ= 0, u=∇⊥ψ, ∆ψ= Λβθ, (1.2) where βis a real parameter satisfying 1 < β ≤2. Here the spatial domain is either the whole plane R2or the 2D periodic box T2and the fractional Laplacian operator (−∆)α is defined through the Fourier transform \ (−∆)αf(ξ) = |ξ|2αb f(ξ). The borderline case β= 1 of (1.2) is the SQG equation (1.1), while (1.2) with β= 0 is the well-known 2D Euler vorticity equation with θrepresenting the vorticity (see e.g. [71]). The second generalized SQG equation under study is the dissipative active scalar equation ∂tθ+u·∇θ+κ(−∆)αθ= 0, u=∇⊥ψ, ψ = (log(I−∆))µθ, (1.3) where κ > 0, α > 0 and µ > 0 are real parameters, and (log(I−∆))µdenotes the Fourier multiplier operator defined by \ (log(I−∆))µf(ξ) = log(1 + |ξ|2)µb f(ξ). (1.3) is closely related to (1.2). In fact , both (1.2) with β= 2 and (1.3) with κ= 0 and µ= 0 formally reduce to the trivial linear equation ∂tθ+∇⊥θ·∇θ= 0 or ∂tθ= 0. For µ > 0, the velocity field uin (1.3) is at least logarithmically more singular than those in (1.2). We establish four main results for the existence and uniqueness of solutions to the equations defined in (1.2) and in (1.3) with a given initial data θ(x, 0) = θ0(x). We now preview these results. Our first main result establishes the local existence and uniqueness of smooth solutions to (1.2) associated with any given smooth initial data. More precisely, we have the following theorem. Theorem 1.1. Consider (1.2) with 1< β ≤2. Assume that θ0∈Hm(R2)with m≥4. Then there exists T=T(kθ0kHm)>0such that (1.2) has a unique solution θon [0, T ]. In addition, θ∈C([0, T ]; Hm(R2)). Remark 1.2. As mentioned previously, when β= 2,ψ=θand u=∇⊥θand (1.2) reduces to the trivial equation ∂tθ= 0 or θ(x, t) = θ0(x). Therefore, (1.2) with β= 2 has a global steady-state solution. For 1 < β < 2, the velocity uis determined by a very singular integral of θand ∇u is not known to be bounded in L∞. As a consequence, the nonlinear term can not be directly bounded. To deal with this difficulty, we rewrite the nonlinear term in the form GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 3 of a commutator to explore the extra cancellation. In order to prove Theorem 1.1, we need to derive a suitable commutator estimate (see Proposition 2.1 in Section 2). Our second main result proves the local existence and uniqueness of smooth solutions to (1.3). In fact, the following theorem holds. Theorem 1.3. Consider the active scalar equation (1.3) with κ > 0,α > 0and µ > 0. Assume the initial data θ0∈H4(R2). Then there exists T > 0such that (1.3) has a unique solution θ∈C([0, T ]; H4(R2)). We remark that the velocity field uin (1.3) is determined by u=∇⊥(log(I−∆))µθwith µ > 0 which is even logarithmically more singular than that in (1.2) with β= 2, 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. Theorem 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 kθkH4, we need to rewrite the most singular part in the nonlinear term as a commutator. This commutator involves the logarithm of Laplacian and it appears that no L2-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 ∂xdenote a partial derivative, either ∂x1or ∂x2. Then, for any δ > 0and ǫ > 0, k[(ln(I−∆))µ∂x, g]fkL2≤Cµ,ǫ,δ 1 + ln 1 + kfk˙ Hδ kfkL2µkfkL2kgkH2+3ǫ, where Cµ,ǫ,δ is a constant depending on µ,ǫand δonly, ˙ Hδdenotes the standard homogeneous Sobolev space and the brackets denote the commutator, namely [(ln(I−∆))µ∂x, g]f= (ln(I−∆))µ∂x(fg)−((ln(I−∆))µ∂xf)g. Our third main result assesses the global existence of weak solutions to (1.2). Our consideration is restricted to the setting of periodic boundary conditions. The weak solution is essentially in the distributional sense and its precise definition is as follows. T2in the definition denotes the 2D periodic box. Definition 1.5. Let T > 0. A function θ∈L∞([0, T ]; L2(T2)) is a weak solution of (1.2) if, for any test function φ∈C∞ c([0, T )×T2), the following integral equation holds, (1.4) ZT 0ZT2 θ(∂tφ+u·∇φ)dx dt =ZT2 θ0(x)φ(x, 0) dx. Although the velocity uis more singular than the scalar θand 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 L2data leads to a global (in time) weak solution. That is, we have the following theorem. 4 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU Theorem 1.6. Assume that θ0∈L2(T2)has mean zero, namely ZT2 θ0(x)dx = 0. Then (1.2) has a global weak solution in the sense of Definition 1.5. This result is an extension of Resnick’s work [88] on the inviscid SQG equation (1.1). However, for 1 < β < 2, 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 Theorem 1.6 in Section 4. Our last main result establishes the local well-posedness of the patch problem associated with the active scalar equation (1.2). This result extends Gancedo’s previous work for (1.2) with 0 < β ≤1 [50]. Since βis now in the range (1,2), uis given by a more singular integral and demands regular function and more sophisticated manipulation. The initial data is given by (1.5) θ0(x) = θ1, x ∈Ω; θ2, x ∈R2\Ω, where Ω ⊂R2is a bounded domain. We parameterize the boundary of Ω by x=x0(γ) with γ∈T= [−π, π] so that |∂γx0(γ)|2=A0, where 2π√A0is the length of the contour. In addition, we assume that the curve x0(γ) does not cross itself and there is a lower bound on |∂γx0(γ)|, namely (1.6) |x0(γ)−x0(γ−η)| |η|>0,∀γ, η ∈T. Alternatively, if we define (1.7) F(x)(γ, η, t) =          |η| |x(γ, t)−x(γ−η, t)|,if η6= 0, 1 |∂γx(γ, t)|,if η= 0, then (1.6) is equivalent to (1.8) F(x0)(γ, η, 0) <∞ ∀γ, η ∈T. The solution of (1.2) 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 x(γ, t) of the boundary ∂Ω(t) satisfies (1.9) ∂tx(γ, t) = Cβ(θ1−θ2)ZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|βdη, where Cβis a constant depending on βonly. For β∈(1,2), the integral on the right of (1.9) is singular. Since the velocity in the tangential direction does not change the GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 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 (1.10) ∂tx(γ, t) = Cβ(θ1−θ2)ZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|βdη +λ(γ, t)∂γx(γ, t) with λ(γ, t) so chosen that ∂γx(γ, t)·∂2 γx(γ, t) = 0 or |∂γx(γ, t)|2=A(t), where A(t) denotes a function of tonly. A similar calculation as in [50] leads to the following explicit formula for λ(γ, t), λ(γ, t) = Cγ+π 2πZT ∂γx(γ, t) |∂γx(γ, t)|2·∂γZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|βdηdγ(1.11) −CZγ −π ∂γx(η, t) |∂γx(η, t)|2·∂ηZT ∂γx(η, t)−∂γx(η−ξ, t) |x(η, t)−x(η−ξ, t)|βdξdη, where C=Cβ(θ1−θ2). We establish the local well-posedness of the contour dynamics equation (CDE) given by (1.10) and (1.11) corresponding to an initial contour x(γ, 0) = x0(γ) satisfying (1.8). More precisely, we have the following theorem. Theorem 1.7. Let x0(γ)∈Hk(T)for k≥4and F(x0)(γ, η, 0) <∞for any γ, η ∈T. Then there exists T > 0such that the CDE given by (1.10) and (1.11) has a solution x(γ, t)∈C([0, T ]; Hk(T)) with x(γ, 0) = x0(γ). This theorem is proven by obtaining an inequality of the form d dt (kxkH4+kF(x)kL∞)≤C(kxkH4+kF(x)kL∞)9+β. The ingredients involved in the proof include appropriate combination and cancellation of terms. The detailed proof is provided in Section 5. 2. Local smooth solutions This section proves Theorem 1.1, which assesses the local (in time) existence and uniqueness of solutions to (1.2) in Hmwith m≥4. For 1 < β ≤2, the velocity uis determined by a very singular integral of θand the nonlinear term can not be directly bounded. To deal with this difficulty, we rewrite the nonlinear term in the form of a commutator to explore the extra cancellation. The following proposition provides a L2-bound for the commutator Proposition 2.1. Let sbe a real number. Let ∂xdenote a partial derivative, either ∂x1 or ∂x2. Then, k[Λs∂x, g]fkL2(R2)≤CkΛsfkL2kc Λg(η)kL1+CkfkL2k\ Λ1+sg(η)kL1, 6 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU where Cis a constant depending on sonly. In particular, by Sobolev embedding, for any ǫ > 0, there exists Cǫsuch that k[Λs∂x, g]fkL2(R2)≤Cǫ(kΛsfkL2kgkH2+ǫ+kfkL2kgkH2+s+ǫ). Since this commutator estimate itself appears to be interesting, we provide a proof for this proposition. Proof. The Fourier transform of [Λs∂x, g]fis given by (2.1) \ [Λs∂x, g]f(ξ) = ZR2 (|ξ|sξj−|ξ−η|s(ξ−η)j)b f(ξ−η)bg(η)dη. where j= 1 or 2. It is easy to verify that, for any real number s, (2.2) ||ξ|sξj−|ξ−η|s(ξ−η)j| ≤ Cmax{|ξ|s,|ξ−η|s}|η|. In fact, we can write |ξ|sξj−|ξ−η|s(ξ−η)j=Z1 0 d dρ (|A|sAj)(2.3) =Z1 0 (|A|sηj+s|A|s−2(A·η)Aj)dρ, where A(ρ, ξ, η) = ρξ + (1 −ρ)(ξ−η). Therefore, ||ξ|sξj−|ξ−η|s(ξ−η)j| ≤ (1 + |s|)|η|Z1 0|A|sdρ. For s≥0, it is clear that |A|s≤max{|ξ|s,|ξ−η|s}. When s < 0, F(x) = |x|sis convex and |A|s=|ρξ + (1 −ρ)(ξ−η)|s≤ρ|ξ|s+ (1 −ρ)|ξ−η|s≤max{|ξ|s,|ξ−η|s}. To obtain the bound in Proposition 2.1, we first consider the case when s≥0. Inserting (2.2) in (2.1) and using the basic inequality |ξ|s≤2s−1(|ξ−η|s+|η|s), we have \ [Λs∂x, g]f(ξ)≤C|ξ|sZR2|b f(ξ−η)||ηbg(η)|dη(2.4) +CZR2||ξ−η|sb f(ξ−η)||ηbg(η)|dη ≤CZR2||ξ−η|sb f(ξ−η)||ηbg(η)|dη +CZR2|b f(ξ−η)|||η|1+sbg(η)|dη. By Plancherel’s Theorem and Young’s inequality for convolution, k[Λs∂x, g]fkL2≤CkΛsfkL2kc Λg(η)kL1+CkfkL2k\ Λ1+sg(η)kL1. Applying the embedding inequality k|η|1+sbg(η)kL1(R2)≤Cǫ||gkH2+s+ǫ(R2), GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 7 we have, for s≥0, k[Λs∂x, g]fkL2(R2)≤CǫkΛsfkL2(R2)kgkH2+ǫ(R2)+kfkL2(R2)kgkH2+s+ǫ(R2). The case when s < 0 is handled differently. We insert (2.3) in (2.1) and change the order of integration to obtain \ [Λs∂x, g]f(ξ) = H1+H2, where H1=Z1 0ZR2|A|sb f(ξ−η)ηjbg(η)dη dρ,(2.5) H2=sZ1 0ZR2|A|s−2(A·η)Ajb f(ξ−η)bg(η)dη dρ.(2.6) Using the fact that F(x) = |x|swith s < 0 is convex, we have |A|s=|(ξ−η) + ρη|s= (1 + ρ)s 1 1 + ρ(ξ−η) + ρ 1 + ρη s ≤(1 + ρ)s1 1 + ρ|ξ−η|s+ρ 1 + ρ|η|s = (1 + ρ)s−1|ξ−η|s+ρ(1 + ρ)s−1|η|s. Inserting this inequality in (2.5), we obtain |H1| ≤ Z1 0 (1 + ρ)s−1dρ ZR2||ξ−η|sb f(ξ−η)||ηbg(η)|dη +Z1 0 ρ(1 + ρ)s−1dρ ZR2|b f(ξ−η)|||η|1+s|bg(η)||η. Applying Young’s inequality for convolution, Plancherel’s theorem and Sobolev’s inequality, we have kH1kL2≤CkΛsfkL2kc Λg(η)kL1+CkfkL2k\ Λ1+sg(η)kL1 ≤CǫkΛsfkL2kgkH2+ǫ+CǫkfkL2kgkH2+s+ǫ. To bound H2, it suffices to notice that |H2| ≤ |s|Z1 0ZR2|A|s|b f(ξ−η)||ηbg(η)|dη dρ Therefore, kH2kL2admits the same bound as kH1kL2. This completes the proof of Proposition 2.1.  With this commutator estimate at our disposal, we are ready to prove Theorem 1.1. Proof of Theorem 1.1. This proof provides a local (in time) a priori bound for kθkHm. 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. 8 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU We consider the case when m= 4. The general case can be dealt with in a similar manner. By ∇·u= 0, 1 2 d dtkθ(·, t)k2 L2= 0 or kθ(·, t)kL2=kθ0kL2. Let σbe a multi-index with |σ|= 4. Then, 1 2 d dtkDσθk2 L2=−ZDσθ Dσ(u·∇θ)dx, where Rmeans the integral over R2and we shall omit dx when there is no confusion. Clearly, the right-hand side can be decomposed into I1+I2+I3+I4+I5with I1=−ZDσθ Dσu·∇θ dx, I2=−X |σ1|=3,σ1+σ2=σZDσθ Dσ1u·Dσ2∇θ dx, I3=−X |σ1|=2,σ1+σ2=σZDσθ Dσ1u·Dσ2∇θ dx, I4=−X |σ1|=1,σ1+σ2=σZDσθ Dσ1u·Dσ2∇θ dx, I5=ZDσθ u ·∇Dσθ dx. The divergence-free condition ∇·u= 0 yields I5= 0. We now estimate I1. For 1 < β < 2, Dσu=∇⊥Λ−2+βDσθwith |σ|= 4 can not bounded directly in terms of kθkH4. We rewrite I1as a commutator. For this we observe that for any skew-adjoint operator Ain L2(i.e. (Af, g)L2=−(f, Ag)L2for all f, g ∈L2) we have RfA(f)g dx =−RfA(gf)dx, and therefore (2.7) ZfA(f)g dx =−1 2Z{fA(gf)−fgA(f)}dx =−1 2Zf[A, g]f dx. Applying this fact to I1with A:= Λ−2+β∇⊥,f:= Dσθand g:= ∇θ, one obtains I1=1 2ZDσθΛ−2+β∇⊥·,∇θDσθ dx. By H¨older’s inequality and Proposition 2.1 with s=−2 + β < 0, we have |I1| ≤ CǫkDσθkL2kDσθkL2+kΛ−2+βDσθkL2kθkH3+ǫ≤CkDσθkL2kθk2 H4. The estimate for I2is easy. By H¨older’s and Sobolev’s inequalities, |I2| ≤ CkDσθkL2kθkH2+βkθkH4. By H´older’s inequality and the Gagliardo-Nirenberg inequality, |I3| ≤ CX |σ1|=2,σ1+σ2=4 kDσθkL2kDσ1ukL4kDσ2∇θkL4 ≤CkDσθkL2kθk1/2 Hβ+1 kθk1/2 Hβ+2 kθk1/2 H3kθk1/2 H4 ≤CkDσθkL2kθkH3kθkH4. GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 9 By H¨older’s and Sobolev’s inequalities, |I4| ≤ CX |σ1|=1,σ1+σ2=4 kDσθkL2kDσ1ukL∞kDσ2∇θkL2 ≤CkDσθkL2kθkHβ+2 kθkH4. For 1 < β < 2, the bounds above yields d dtkθk2 H4≤Ckθk3 H4. This inequality allows us to obtain a local (in time) bound for kθkH4. 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 dtkθ2−θ1kH1≤C(kθ2kH4+kθ1kH4)kθ2−θ1kH1. An easy application of the Gronwall inequality provides θ2=θ1. This concludes the proof of Theorem 1.1.  3. The case that is logarithmically beyond β= 2 This section focuses on the dissipative active scalar equation defined in (1.3) and the goal is to prove Theorem 1.3. As mentioned in the introduction, the major difficulty in proving this theorem is due to the fact that the velocity uis determined by a very singular integral of θ. To overcome this difficulty, we rewrite the nonlinear term in the form of a commutator to explore the extra cancellation. The commutator involves the logarithm of the Laplacian and we need a suitable bound for this type of commutator. The bound is stated in Proposition 1.4, but we restated here. Proposition 3.1. Let µ≥0. Let ∂xdenote a first partial, i.e., either ∂x1or ∂x2. Then, for any δ > 0and ǫ > 0, k[(ln(I−∆))µ∂x, g]fkL2≤Cµ,ǫ,δ 1 + ln 1 + kfk˙ Hδ kfkL2µkfkL2kgkH2+3ǫ, where Cµ,ǫ,δ is a constant depending on µ,ǫand δonly and ˙ Hδdenotes the standard homogeneous Sobolev space. Remark 3.2. The constant Cµ,ǫ,δ approaches ∞as δ→0or ǫ→0. When µ= 0, the constant depends on ǫonly. We shall also make use of the following lemma that bounds the L2-norm of the logarithm of function. Lemma 3.3. Let µ≥0be a real number. Then, for any δ > 0, (3.1) k(ln(I−∆))µfkL2≤Cµ,δkfkL2ln 1 + kfk˙ Hδ kfkL2µ . where Cµ,δ is a constant depending on µand δonly. 16 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU Since supp Φj⊂ {ξ∈R2: 2j−1≤ |ξ|<2j+1}and supp \ Sj−1g⊂ {ξ∈R2:|ξ|<2j}, we still have ln(1 + max{|ξ|2,|ξ−η|2})µ≤Cln(1 + 22j)µ. In contrast to the previous estimate on ∆jL1,η∈\ Sj−1gno longer implies that |η|is comparable to 2j. However, any ξ∈supp d ∆jfmust have |ξ|comparable to 2j. Therefore, for any δ > 0, k∆jL2k2 L2≤Cln(1 + 22j)2µ2−2δj ZR2||ξ−η|δd ∆jf(ξ−η)||η\ Sj−1g(η)|dη 2 L2 ≤Cln(1 + 22j)2µ2−2δjk|ξ−η|δd ∆jf(ξ−η)k2 L2kη\ Sj−1g(η)k2 L1 ≤Cln(1 + 22j)2µ2−2δjk∆jΛδfk2 L2kgk2 H2+ǫ. Thus, K22 ≤C ∞ X j=Nln(1 + 22j)2µ2−2δjk∆jΛδfk2 L2kgk2 H2+ǫ ≤Cln(1 + 22N)2µ2−2δN kgk2 H2+ǫ ∞ X j=Nk∆jΛδfk2 L2 ≤Cln(1 + 22N)2µ2−2δN kgk2 H2+ǫkfk2 Hδ.(3.15) The last term K23 can be dealt with exactly as K13. The bound for K23 is (3.16) K23 ≤Cǫkgk2 H2+3ǫkfk2 L2. 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, k[(ln(I−∆))µ∂x, g]fk2 L2≤Cǫln(1 + 22N)2µkfk2 L2kgk2 H2+ǫ +Cǫkfk2 L2kgk2 H2+3ǫ +Cǫln(1 + 22N)2µ2−2δN kfk2 Hδkgk2 H2+ǫ. We now choose Nsuch that 2−2δN kfk2 Hδ≤Ckfk2 L2. In fact, we can choose (3.17) N=1 δlog2kfkHδ kfkL2. It then follows that k[(ln(I−∆))µ∂x, g]fkL2≤Cµ,ǫ,δ 1 + ln 1 + kfkHδ kfkL2µkfkL2kgkH2+3ǫ, where Cµ,ǫ,δ is a constant depending on µ,ǫand δonly. It is easy to see that the inhomogeneous Sobolev norm kfkHδcan be replaced by the homogeneous norm kfk˙ Hδ. This completes the proof of Proposition 3.1.  Finally we prove Lemma 3.3. GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 17 Proof of Lemma 3.3. Let N≥1 be an integer to be specified later. We write k(ln(I−∆))µfk2 L2=L1+L2 where L1= N−1 X j=−1k∆j(ln(I−∆))µfk2 L2, L2= ∞ X j=Nk∆j(ln(I−∆))µfk2 L2. According to Theorem 1.2 in [14], we have, for j≥0, k∆j(ln(I−∆))µfkL2≤Cln(1 + 22j)µk∆jfkL2. Clearly, for j=−1, k∆−1(ln(I−∆))µfkL2≤Ck∆−1fkL2. Therefore, L1≤Cln(1 + 22N)2µN−1 X j=−1k∆jfk2 L2≤Cln(1 + 22N)2µkfk2 L2. For any δ > 0, L2≤ ∞ X j=Nln(1 + 22j)2µ2−2δj 22δjk∆jfk2 L2 ≤ln(1 + 22N)2µ2−2δN kfk2 Hδ. Therefore, k(ln(I−∆))µfk2 L2≤Cln(1 + 22N)2µkfk2 L2+ln(1 + 22N)2µ2−2δN kfk2 Hδ. If we choose Nin a similar fashion as in (3.17), we obtain the desired inequality (3.1). This completes the proof of Lemma 3.3.  4. Global weak solutions This section establishes the global existence of weak solutions to (1.2), namely Theorem 1.6. The following commutator estimate will be used. Lemma 4.1. Let s≥0. Let j= 1 or 2. Then, for any ǫ > 0, there exists a constant C depending on sand ǫsuch that (4.1) k[Λs∂xj, g]hkL2(T2)≤C(khkL2kgkH2+s+ǫ+kΛshkL2kgkH2+ǫ). Although the lemma is for the periodic setting, it can be proven in a similar manner as Proposition 2.1 and we thus omit its proof. Proof of Theorem 1.6. The proof follows a standard approach, the Galerkin approximation. Let n > 0 be an integer and let Kndenotes the subspace of L2(T2), Kn=eim·x:m6= 0 and |m| ≤ n. 18 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU Let Pnbe the projection onto Kn. For each fixed n, we consider the solution of the projected equation, ∂tθn+Pn(un·∇θn) = 0, un=∇⊥Λ−2+βθn, θn(x, 0) = Pnθ0(x). This equation has a unique global solution θn. Clearly, θnobeys the L2global bound (4.2) kθn(·, t)kL2=kPnθ0kL2≤ kθ0kL2. In addition, let ψnbe the corresponding stream function, namely ∆ψn= Λβθn. Then we have 1 2 d dt Λ1−β 2ψn2 L2=−ZψnPn(un·∇θn)dx =−Zψnun·∇θndx. Noticing that un=∇⊥ψn, we integrate by parts in the last term to obtain −Zψnun·∇θndx =Zψnun·∇θndx. Therefore, (4.3) d dt Λ1−β 2ψn2 L2= 0 or Λ1−β 2ψnL2≤Λ1−β 2ψ0L2. Furthermore, for any φ∈H3+ǫwith ǫ > 0, we have (4.4) Z∂tθn(x, t)φ(x)dx =−Z(un·∇θn)Pnφ dx =Zθnun·∇Pnφ dx. On the one hand, θn= Λ2−βψnand Zθnun·∇Pnφ dx =ZψnΛ2−β(un·∇Pnφ)dx =ZψnΛ2−β∇⊥ψn·∇Pnφdx. On the other hand, un=∇⊥ψnand Zθnun·∇Pnφ dx =Zθn∇⊥·(ψn∇Pnφ)dx =−Zψn∇⊥Λ2−βψn·∇Pnφ dx. Thus, Zθnun·∇Pnφ dx =1 2ZψnΛ2−β∇⊥·,∇Pnφψndx. It then follows from H¨older’s inequality and Lemma 4.1 that Zθnun·∇Pnφ dx≤CkψnkL2kψnkH2−βkPnφkH3+ǫ (4.5) ≤CkΛ−2+βθnkL2kθnkL2kφkH3+ǫ ≤Ckθ0k2 L2kφkH3+ǫ where we have used the fact that mean-zero functions in L2(T2) are also in H−2+β(T2). Therefore, by (4.4), (4.6) k∂tθnkH−3−ǫ≤Ckθ0k2 L2. GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 19 The bounds in (4.2), (4.3) and (4.6), together with the compact embedding relation L2(T2)֒→H−2+β(T2) for 1 < β < 2, imply that there exists θ∈C([0, T ]; L2(T2)) such that (4.7) θn⇀ θ in L2, ψn→ψin L2. In addition, because of the uniform boundedness of kθnkL2and the embedding L2(T2)֒→ H−3−ǫ(T2), the Arzel`a-Ascoli Theorem implies (4.8) lim n→∞ sup t∈[0,T ]Z(θn(x, t)−θ(x, t))φ(x)dx→0, where φ∈H3+ǫ(T2). The convergence in (4.7) and (4.8) allows us to prove that θsatisfies (1.4). Clearly, θnsatisfies the integral equation ZT 0ZT2 θn(∂tφ+un·∇Pnφ)dx dt =ZT2 Pnθ0(x)φ(x, 0) dx. It is easy to check that ZT2 Pnθ0(x)φ(x, 0) dx →ZT2 θ0(x)φ(x, 0) dx, and (4.8) implies that, as n→ ∞, ZT 0ZT2 θn∂tφ dxdt →ZT 0ZT2 θ ∂tφ dxdt. To show the convergence in the nonlinear term, we write ZT 0ZT2 θnun·∇Pnφ dx dt −ZT 0ZT2 θ u ·∇φ dx dt =1 2ZT 0ZT2 ψnΛ2−β∇⊥·,∇Pnφψndx dt −1 2ZT 0ZψΛ2−β∇⊥·,∇φψ dx dt =1 2ZT 0ZψnΛ2−β∇⊥·,∇(Pnφ−φ)ψndx dt +1 2ZT 0ZT2 (ψn−ψ)Λ2−β∇⊥·,∇φψndx dt +1 2ZT 0ZT2 ψΛ2−β∇⊥·,∇φ(ψn−ψ)dx dt. In order to get the convergence for the first two terms above, we appeal to Lemma 4.1 and the strong convergence of ψnin L2. Let us point out that in the last term for Λ2−βψn we only have weak convergence in L2so we have to proceed in a different manner. We 20 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU consider the following integral Qn(t) = ZT2 ψΛ2−β∇⊥·,∇φ(ψn−ψ)dx =X k6=0 b ψ(−k)(Λ2−β∇⊥·,∇φ(ψn−ψ))b(k), which is bounded by |Qn(t)| ≤ X k6=0 |k|2−βb ψ(−k)2!1/2 X k6=0 |k|β−2(Λ2−β∇⊥·,∇φ(ψn−ψ))b(k)2!1/2 . The first sum above is controlled by kθ0kL2. Using a similar notation as before, the coefficients in the second sum have the form |k|β−2(Λ2−β∂x, ϕ(ψn−ψ))b(k) where ∂xis either ∂x1or ∂x2and ϕis ∂xφ. Since (Λ2−β∂x, ϕ(ψn−ψ))b(k) = X j i(ka|k|2−β−(k−j)a|k−j|2−β)(ψn−ψ)b(k−j)bϕ(j) for a= 1,2, following the bounds in Section 2 we obtain (Λ2−β∂x, ϕ(ψn−ψ))b(k)≤CX j (|k|2−β+|k−j|2−β)|(ψn−ψ)b(k−j)||j||bϕ(j)| ≤CX j (|k|2−β+|j|2−β)|(ψn−ψ)b(k−j)||j||bϕ(j)|. For |k| 6= 0, it yields |k|β−2(Λ2−β∂x, ϕ(ψn−ψ))b(k)≤CX j|(ψn−ψ)b(k−j)||j|(1 + |j|2−β)|bϕ(j)|. The above bound provides |Qn(t)| ≤ Cǫkθ0kL2kφkH5−β+ǫkψn−ψkL2 for any ǫ > 0. It then follows from (4.7) that limn→∞ Qn(t) = 0. The Dominated Convergence Theorem then leads to the desired convergence of the third term. Therefore, θis a weak solution of 1.2 in the sense of Definition 1.5. This completes the proof of Theorem 1.6.  5. Local existence for smooth patches This section is devoted to proving Theorem 1.7. Proof of Theorem 1.7. Since β= 2 corresponds to the trivial steady-state solution, it suffices to consider the case when 1 < β < 2. The major efforts are devoted to establishing a priori local (in time) bound for kx(·, t)kH4+kF(x)kL∞(t) for xsatisfying the contour dynamics equation (1.10) and F(x)(γ, η, t) defined in (1.7). GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 21 This proof follows the ideas in Gancedo [50]. The difference here is that the kernel in (1.10) is more singular but the function space concerned here is H4(T), which is more regular than in [50] and compensates for the singularity of the kernel. For notational convenience, we shall omit the coefficient Cβ(θ1−θ2) in the contour dynamics equation (1.10). In addition, the t-variable will sometimes be suppressed. We start with the L2-norm. Dotting (1.10) by x(γ, t) and integrating over T, we have 1 2 d dt ZT|x(γ, t)|2dx =I1+I2, where I1=ZTZT x(γ, t)·∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|βdη dγ, I2=ZT λ(γ)x(γ, t)·∂γx(γ, t)dγ. I1is actually zero. In fact, by the symmetrizing process, I1=1 2ZTZT (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|βdη dγ =1 2(2 −β)ZTZT ∂γ|x(γ)−x(γ−η)|2−βdγdη = 0. To bound I2, we first apply H¨older’s inequality to obtain |I2| ≤ kλkL∞kxkL2k∂γxkL2. By the representation of λin (1.11) and using the fact that 1 |∂γx|2≤ kF(x)k2 L∞(t), we have kλkL∞≤CkF(x)k2 L∞(t)ZT|∂γx|∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|βdηdγ =CkF(x)k2 L∞(t) (I21 +I22), where I21 =ZT|∂γx|ZT |∂2 γx(γ)−∂2 γx(γ−η)| |x(γ)−x(γ−η)|βdη dγ, I22 =ZT|∂γx|ZT |∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|β+1 dη dγ. It is not hard to see that I21 and I22 can be bounded as follows. I21 ≤CkF(x)kβ L∞(t)k∂γxkL2k∂3 γxkL2, I22 ≤CkF(x)k1+β L∞(t)k∂2 γxk2 L2k∂γxkL2. Therefore, d dtkxk2 L2≤CkF(x)k3+β L∞(t)kxk5 H3. 22 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU We now estimate k∂4 γxkL2. 1 2 d dt ZT|∂4 γx|2dγ =I3+I4, where I3=CZT ∂4 γx(γ)·∂4 γZT (∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|βdη dγ, I4=ZT ∂4 γx(γ)·∂4 γ(λ∂γx)(γ)dγ. I3can be further decomposed into five terms, namely I3=I31 +I32 +I33 +I34 +I35, where I31 =ZTZT ∂4 γx(γ)·(∂5 γx(γ)−∂5 γx(γ−η)) |x(γ)−x(γ−η)|βdη dγ, I32 = 4 ZTZT ∂4 γx(γ)·(∂4 γx(γ)−∂4 γx(γ−η))∂γ(|x(γ)−x(γ−η)|−β)dη dγ, I33 = 6 ZTZT ∂4 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−β)dη dγ, I34 = 4 ZTZT ∂4 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))∂3 γ(|x(γ)−x(γ−η)|−β)dη dγ, I35 =ZTZT ∂4 γx(γ)·(∂γx(γ)−∂γx(γ−η))∂4 γ(|x(γ)−x(γ−η)|−β)dη dγ. By symmetrizing, I31 can be written as I31 =1 2ZTZT (∂4 γx(γ)−∂4 γx(γ−η)) ·(∂5 γx(γ)−∂5 γx(γ−η)) |x(γ)−x(γ−η)|βdη dγ =1 4ZTZT ∂γ(|∂4 γx(γ)−∂4 γx(γ−η)|2) |x(γ)−x(γ−η)|βdη dγ =β 4ZTZT |∂4 γx(γ)−∂4 γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|β+2 dη dγ. Setting B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) and using the fact that ∂γx(γ)·∂2 γx(γ) = 0, we have |I31| ≤ CkF(x)k2+β L∞(t)ZTZT|∂4 γx(γ)−∂4 γx(γ−η)|2B(γ, η)η−2−∂γx(γ)·∂2 γx(γ) |η|βdη dγ. Using the bound thatB(γ, η)η−2−∂γx(γ)·∂2 γx(γ)≤Ckxk2 C3|η|, we obtain |I31| ≤ CkF(x)k2+β L∞(t)kxk2 C3kxk2 H4. GENERALIZED SURFACE QUASI-GEOSTROPHIC EQUATIONS 23 To estimate of I32, we realize that, after computing ∂γ(|x(γ)−x(γ−η)|−β), I32 can be bounded in the same fashion as I31. That is, |I32| ≤ CkF(x)k2+β L∞(t)kxk4 H4. In order to estimate I33, we further decompose it into three terms, I33 =I331 +I332 +I333, where I331 =CZTZT ∂4 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η)) D(γ, η) |x(γ)−x(γ−η)|2+βdη dγ, I332 =CZTZT ∂4 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|2+βdη dγ, I333 =CZTZT ∂4 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η)) B2(γ, η) |x(γ)−x(γ−η)|4+βdη dγ with D(γ, η) = (x(γ)−x(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). It is not very difficult to see that |I331|,|I332|,|I333| ≤ CkF(x)k2+β L∞(t)kxk4 H4. I34 also admit similar bound. In I35 one has to use identity ∂γx(γ)·∂4 γx(γ) = 3∂2 γx(γ)·∂3 γx(γ) 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 I4. To do so, we use the representation formula (1.11) and obtain |I4| ≤ CkF(x)k4+β L∞(t)kxk5 H4 In summary, we have (5.1) d dtkxk2 H4≤CkF(x)k4+β L∞(t)kxk5 H4. We now derive the estimate for kF(x)kL∞(t). For any p > 2, we have (5.2) d dtkF(x)kp Lp(t)≤pZTZT|η| |x(γ)−x(γ−η)|p+1 |xt(γ, t)−xt(γ−η, t)| |η|dη dγ. Invoking the contour dynamics equation (1.10), we have xt(γ)−xt(γ−η) = I5+I6+I7+I8 ≡ZT∂γx(γ)−∂γx(γ−ξ) |x(γ)−x(γ−ξ)|β−∂γx(γ)−∂γx(γ−ξ) |x(γ−η)−x(γ−η−ξ)|βdξ +ZT ∂γx(γ)−∂γx(γ−η) + ∂γx(γ−η−ξ)−∂γx(γ−ξ) |x(γ−η)−x(γ−η−ξ)|βdξ +(λ(γ)−λ(γ−η))∂γx(γ) + λ(γ−η)(∂γx(γ)−∂γx(γ−η)). 24 CHAE, CONSTANTIN, C ´ ORDOBA, GANCEDO AND WU Following the argument as in [50], we have |I5| ≤ CkF(x)k2β L∞(t)kxk1+β C2|η|, |I6| ≤ CkF(x)kβ L∞(t)kxkC3|η|, |I7| ≤ CkF(x)k3+β L∞(t)kxk4 H4|η|, |I8| ≤ CkF(x)k3+β L∞(t)kxk4 H4|η|. Inserting these estimates in (5.2), we find d dtkF(x)kLp(t)≤Ckxk4 H4kF(x)k4+β L∞(t)kF(x)kLp(t). After integrating in time and taking the limit as p→ ∞, we obtain d dtkF(x)kL∞(t)≤Ckxk4 H4kF(x)k5+β L∞(t). Combining with (5.1), we obtain d dt (kxkH4+kF(x)kL∞(t)) ≤Ckxk4 H4kF(x)k5+β L∞(t). This inequality would allow us to deduce a local (in time) bound for kxkH4. 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