Existence for the α-patch model and the QG sharp front in Sobolev spaces

a Xi :ma h/0701447 1 [ma h.AP] 16 Jan 2007 Exis ence o he α-pa ch model and he QG sha p on in Sobole spaces F ancisco Gancedo Abs ac We conside a amily o con ou dynamics equa ions depending on a pa ame e α wi h 0 < α ≤1. The o ex pa ch p oblem o he 2-D Eule equa ion is ob ained aking α→0, and he case α= 1 co esponds o a sha p on o he QG equa ion. We p o e local-in- ime exis ence o he amily o equa ions in Sobole spaces. 1 In oduc ion The 2-D QG equa ion p o ides pa icula solu ions o he e olu ion o he empe a u e om a gene al quasi-geos ophic sys em o a mosphe ic and oceanic lows. This equa ion is de i ed conside ing small Rossby and Ekman numbe s and cons an po en ial o ici y (see [12] o mo e de ails). I eads θ (x, ) + u(x, )· ∇θ(x, ) = 0, x ∈R2, θ(x, 0) = θ0(x).(1) He e θis he empe a u e o he luid, he incomp essible eloci y uis exp essed by means o he s eam unc ion as ollows u=∇⊥ψ= (−∂x2ψ, ∂x1ψ), and he ela ion be ween he s eam unc ion and he empe a u e is gi en by θ=−(−∆)1/2ψ. This sys em ha e been conside ed in on ogenesis, whe e he dynamics o ho and cold luids is s udied oge he wi h he o ma ion and he e olu ion o on s (see [4], [5], [8], [11]). F om a ma hema ical poin o iew, his equa ion ha e been p esen ed as a wo dimen- sional model o he 3-D Eule equa ion due o hei s ong analogies (see [4]), being he o ma ion o singula i ies o a egula ini ial da a an open p oblem (see [4], [6], [7]). Ne e - heless he QG equa ion has global in ime weak solu ions due o an ex a cancella ion (see [13]). A ew spa se esul s a e known abou weak solu ions o he 2-D and 3-D Eule equa ion in i s p imi i e- a iable o m. 1 An ou s anding kind o weak solu ions o he QG equa ion a e hose in which he em- pe a u e akes wo di e en alues in complemen a y domains, modelling he e olu ion o a sha p on as ollows θ(x1, x2, ) = θ1,Ω( ) θ2,R2 Ω( ).(2) In his wo k we s udy a p oblem simila o he 2-D o ex pa ch p oblem, whe e he o ici y o he 2-D Eule equa ion is gi en by a cha ac e is ic unc ion o a domain, and i is conside ed he egula i y o he ee bounda y o such domain. Fo his equa ion he o ici y sa is ies w (x, ) + u(x, )· ∇w(x, ) = 0, x ∈R2, w(x, 0) = w0(x),(3) in a weak sense, and he eloci y is gi en by he Bio -Sa a law o analogously u=∇⊥ψ, and w= ∆ψ. Chemin [3] p o ed global-in- ime egula i y o he ee bounda y using pa adi e en ial cal- culus. A simple p oo can be ound in [1] due o Be ozzi and Cons an in. We poin ou ha in he QG equa ion, he eloci y is de e mined om he empe a u e by singula in eg al ope a o s (see [15]) as ollows u= (−R2θ, R1θ),(4) whe e R1and R2a e he Riesz ans o ms, making he sys em mo e singula han (3). Rod igo [14] p oposed he p oblem o he e olu ion o a sha p on o he QG equa ion. He de i ed he eloci y on he ee bounda y in he no mal di ec ion, and p o ed local- exis ence and uniqueness o a pe iodic C∞ on , i.e. θ(x1, x2, ) = θ1,{ (x1, )> x2} θ2,{ (x1, )≤x2}, wi h (x1, ) pe iodic, using he Nash-Mose i e a ion. In his pape we s udy a amily o con ou dynamics equa ion gi en by weak solu ions o he ollowing sys em θ +u· ∇θ= 0, x ∈R2, u=∇⊥ψ, θ =−(−∆)1−α/2ψ, 0< α ≤1, (5) whe e he ac i e scala θ(x, ) sa is ies (2). We no ice ha he case α= 0 is he 2-D o ex pa ch p oblem, and α= 1 co espond o he sha p on o he QG equa ion. This sys em was in oduced by C´o doba, Fon elos, Mancho and Rod igo in [9], whe e hey p esen a p oo o local-exis ence o a pe iodic C∞ on , and show e idence o singula i ies in ini e ime. The singula scena io is due o wo pa ches collapse poin -wise. 2 He e we gi e a p oo o local-exis ence o he sys em (5) whe e he solu ion sa is ies (2), wi h he bounda y ∂Ω( ) gi en by he cu e ∂Ω( ) = {x(γ, ) = (x1(γ, ), x2(γ, )) : γ∈[−π, π]}, and x(γ, ) belongs o a Sobole space. In he cases 0 < α < 1 we show uniqueness. I is well-known (see [10] and [14]) ha in his kind o con ou dynamics equa ions, he eloci y in he angen ial di ec ion only mo es he pa icles on he bounda y. The e o e we do no al e he shape o he con ou i we change he angen ial componen o he eloci y; i.e., we a e making a change on he pa ame iza ion. In he mos singula case, α= 1 o he QG equa ion, we need o change he eloci y in he angen ial di ec ion in o de o ge exis ence in he Sobole spaces. We ake a angen ial eloci y in such a way ha |∂γx(γ, )| sa is ies |∂γx(γ, )|2=A( ), and does no depend on γ. We would like o ci e he wo k o Hou, Loweng ub and Shelley [10] in which his idea was used o s udy a con ou dynamics p oblem. We no ice ha in o de o ge a non-singula no mal eloci y o he cu e o 0 < α ≤1 (see [9] and [14]), we need a one o one cu e, and pa ame e ized in such a way ha |∂γx(γ, )|2>0. Rigo ously, we need ha |x(γ, )−x(γ−η, )| |η|>0,∀γ, η ∈[−π, π],(6) he e o e we gi e an ini ial da a sa is ying his p ope y, and we p o e ha his condi ion is sa is ied locally in ime. We poin ou he impo ance o ake in o accoun he e olu ion o his quan i y due o he nume ical simula ions in [9]. Finally, I wish o hank An onio C´o doba and my hesis ad iso Diego C´o doba o hei s ong in luence in his wo k, hei ad ices and sugges ions. The au ho was pa ially sup- po ed by he g an s PAC-05-005-2 o he JCLM (Spain) and MTM2005-05980 o he MEC (Spain). 2 The Con ou Equa ion In his sec ion we deduce he amily o con ou equa ions in e m o he ee bounda y x(γ, ). We conside he equa ions gi en by he sys em (1), wi h a eloci y sa is ying u(x, ) = ∇⊥ψ(x, ),(7) o he s eam unc ion i ollows θ=−(−∆)1−α/2ψ, (8) and he ac i e scala ul ills 3 θ(x1, x2, ) = θ1,Ω( ) θ2,R2 Ω( ).(9) The bounda y o Ω( ) is gi en by he cu e ∂Ω( ) = {x(γ, ) = (x1(γ, ), x2(γ, )) : γ∈[−π, π] = T}, wi h x(γ, ) one o one. Due o he iden i y (9), we ind ha ∇⊥θ= (θ1−θ2)∂γx(γ, )δ(x−x(γ, )), whe e δis he Di ac dis ibu ion. Using (7) and (8), we go ha u=−(−∆)α/2−1∇⊥θ. Due o he in eg al ope a o s −(−∆)α/2−1a e Riesz po en ials (see [15]), using he las o iden i ies we ob ain ha u(x, ) = −Θα 2πZT ∂γx(γ−η, ) |x−x(γ−η, )|αdη, (10) o x6=x(γ, ), and Θα= (θ1−θ2)Γ(α/2)/21−αΓ(2 −α/2). We no ice ha o α= 1, i x→x(γ, ) he in eg al in (10) is di e gen . As we ha e showed be o e, we a e in e es ed in he no mal eloci y o he sys ems. Then we ha e ha using he iden i y (10), and aking he limi as ollows u(x, )·∂⊥ γx(γ, ), x →x(γ, ),(11) we ob ain u(x(γ, ), )·∂⊥ γx(γ, ) = −Θα 2πZT ∂γx(γ−η, )·∂⊥ γx(γ, ) |x(γ, )−x(γ−η, )|αdη. (12) This iden i y is well de ined o 0 < α ≤1 and a one o one cu e x(γ, ). Due o he ac ha angen ial eloci y does no change he shape o he bounda y, we ix he con ou α-pa ch equa ions as ollows x (γ, ) = Θα 2πZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|αdη, 0< α ≤1, x(γ, 0) = x0(γ). (13) Seeing he equa ion (10), we show ha he eloci y in QG p esen s a loga i hmic di e gence in he angen ial di ec ion on he bounda y. Ne e heless i belongs o Lp(R2) o 1 < p < ∞, and o he bounded mean oscilla ion space (see [15] o he de ini ion o he BMO space). In QG he eloci y is gi en by (4), and w i ing he empe a u e in he ollowing way θ(x, ) = (θ1−θ2)XΩ( )(x) + θ2, we ind ha u(x, ) = (θ1−θ2)(−R2(XΩ( )), R1(XΩ( ))). Using ha XΩ( )∈Lp(R2) o 1 ≤p≤ ∞, we conclude de a gumen . In pa icula he ene gy o he sys em is conse ed due o kukL2( ) = |θ1−θ2| |Ω( )|1/2, and he a ea o Ω( ) is cons an in ime. 4 3 Weak solu ions o he α-sys em In his sec ion we show ha i θ(x, ) is de ined by (9) and he cu e x(γ, ) is con ec ed by he no mal eloci y (12), hen θ(x, ) is a weak solu ion o he sys em (5) and con e sely. We gi e he de ini ion o weak solu ion below. De ini ion 3.1 The ac i e scala θis a weak solu ion o he α-sys em i o any unc ion ϕ∈C∞ c(R2×(0, T)), we ha e ZT 0ZR2 θ(x, )(∂ ϕ(x, ) + u(x, )· ∇ϕ(x, ))dxd = 0,(14) whe e he incomp essible eloci y uis gi en by (7), and he s eam unc ion sa is ies (8). Then P oposi ion 3.2 I θ(x, )is de ined by (9), and he cu e x(γ, )sa is ies (6) and (12), hen θ(x, )is a weak solu ion o he α-sys em. Fu he mo e, i θ(x, )is a weak solu ion o he α-sys em gi en by (9), and x(γ, )sa is ies (6), hen x(γ, ) e i ies (12). P oo : Le θ(x, ) be a weak solu ion o he α-sys em de ined by (9). In eg a ing by pa s we ha e I=ZT 0ZR2 θ(x, )∂ ϕ(x, )dxd =θ1ZT 0ZΩ( ) ∂ ϕ(x, )dxd +θ2ZT 0ZΩ( ) R2 ∂ ϕ(x, )dxd =−(θ1−θ2)ZT 0ZT ϕ(x(γ, ), )x (γ, )·∂⊥ γx(γ, )dγd . On he o he hand, we ob ain J=ZT 0ZR2 θ u · ∇ϕ dxd =θ1ZT 0ZΩ u· ∇ϕ dxd +θ2ZT 0ZR2 Ω u· ∇ϕ dxd . Taking Ωε 1( ) = {x∈Ω : dis (x, Ω( )) ≥ε}, and Ωε 2( ) = {x∈R2 Ω : dis (x, R2 Ω( )) ≥ε}, we ha e ha Jε→Ji ε→0, whe e Jεis gi en by Jε=θ1ZT 0ZΩε 1( ) u· ∇ϕ dxd +θ2ZT 0ZΩε 2( ) u· ∇ϕ dxd . In eg a ing by pa in Jε, using ha he eloci y is di e gence ee, and aking he limi as in (11), we ob ain J= (θ1−θ2)ZT 0ZT ϕ(x(γ, ), )u(x(γ, ), )·∂⊥ γx(γ, )dγd =−(θ1−θ2)Θα 2πZT 0ZT ϕ(x(γ, ), )ZT ∂γx(γ−η, )·∂⊥ γx(γ, ) |x(γ, )−x(γ−η, )|αdηdγd . 5 We ha e ha I+J= 0 using (14), and i ollows ZT 0ZT (γ, )x (γ, )·∂⊥ γx(γ, ) + Θα 2πZT ∂γx(γ−η, )·∂⊥ γx(γ, ) |x(γ, )−x(γ−η, )|αdηdγd = 0, o (γ, ) pe iodic in γ. We ind ha (12) is sa is ied. Following he same a gumen s i is easy o check ha i x(γ, ) sa is ies (12), hen θis a weak solu ion gi en by (9). 4 Local well-posedness o 0< α < 1 In his sec ion we p o e exis ence and uniqueness o he con ou equa ion in he cases 0< α < 1. We deno e he Sobole spaces by Hk(T), wi h no ms kxk2 Hk=kxk2 L2+k∂k γxk2 L2, and he spaces Ck(T) wi h kxkCk= max j≤kk∂j γxkL∞. We need ha he cu e sa is ies |x(γ, )−x(γ−η, )| |η|>0,∀γ, η ∈[−π, π],(15) hen we de ine F(x)(γ, η, ) = |η| |x(γ, )−x(γ−η, )|∀γ, η ∈[−π, π],(16) wi h F(x)(γ, 0, ) = 1 |∂γx(γ, )|. The main heo em in his sec ion is he ollowing Theo em 4.1 Le x0(γ)∈Hk(T) o k≥3wi h F(x0)(γ, η)<∞. Then he e exis s a ime T > 0so ha he e is a unique solu ion o (13) o 0< α < 1in C1([0, T]; Hk(T)) wi h x(γ, 0) = x0(γ). P oo : We can choose Θα= 2πwi hou loss o gene ali y, ob aining he ollowing equa ion x (γ, ) = ZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|αdη, 0< α < 1, x(γ, 0) = x0(γ). (17) We p esen he p oo o k= 3, being analogous o k > 3, using ene gy es ima es (see [2] o mo e de ails). We igno e he ime dependence o simpli y he no a ion in some e ms. Conside ing he quan i y 6 ZT x(γ)·x (γ)dγ =ZTZT x(γ)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|αdηdγ =−ZTZT x(η)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|αdηdγ =1 2ZTZT (x(γ)−x(η)) ·(∂γx(γ)−∂γx(η)) |x(γ)−x(η)|αdηdγ =1 2(2 −α)ZTZT ∂γ|x(γ)−x(γ−η)|2−αdγdη = 0, (18) we ob ain d d kxkL2( ) = 0.(19) We decompose as ollows ZT ∂3 γx(γ)·∂3 γx (γ)dγ =I1+I2+I3+I4, whe e I1=ZTZT ∂3 γx(γ)·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|αdηdγ, I2= 3 ZTZT ∂3 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η))∂γ(|x(γ)−x(γ−η)|−α)dηdγ, I3= 3 ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−α)dηdγ, I4=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))∂3 γ(|x(γ)−x(γ−η)|−α)dηdγ. Ope a ing as in (18), he e m I1becomes I1=1 2ZTZT (∂3 γx(γ)−∂3 γx(γ−η)) ·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|αdηdγ =1 4ZTZT ∂γ|∂3 γx(γ)−∂3 γx(γ−η)|2 |x(γ)−x(γ−η)|αdηdγ =α 4ZTZT |∂3 γx(γ)−∂3 γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|α+2 dηdγ. 7 One inds ha I1≤α 4ZTZT |∂3 γx(γ)−∂3 γx(γ−η)|2|∂γx(γ)−∂γx(γ−η)| |x(γ)−x(γ−η)|α+1 dηdγ, and due o he inequali y |∂γx(γ)−∂γx(γ−η)||η|−1≤ kxkC2,i ollows I1≤α 4kxkC2ZTZT |η|−α|F(x)(γ, η)|1+α|∂3 γx(γ)−∂3 γx(γ−η)|2dηdγ ≤1 2kF(x)k1+α L∞kxkC2ZT |η|−αZT (|∂3 γx(γ)|2+|∂3 γx(γ−η)|2)dγdη ≤ kF(x)k1+α L∞kxkC2k∂3 γxk2 L2ZT |η|−αdη ≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2. (20) As be o e, we can ob ain I2=−6I1, and i yields I2≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2.(21) In o de o es ima e he e m I3, we conside I3=J1+J2+J3, whe e J1=−3αZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) A(γ, η) |x(γ)−x(γ−η)|α+2 dηdγ, J2=−3α ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|α+2 dηdγ, J3= 3α(2 + α) ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) (B(γ, η))2 |x(γ)−x(γ−η)|α+4 dηdγ, wi h A(γ, η) = (x(γ)−x(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)), and B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)). The iden i y ∂2 γx(γ)−∂2 γx(γ−η) = ηZ1 0 ∂3 γx(γ+ (s−1)η)ds, (22) yields J1≤3Z1 0ZTZT |η|(|∂2 γx(γ)|+|∂2 γx(γ−η)|)|∂3 γx(γ)||∂3 γx(γ+ (s−1)η)| |x(γ)−x(γ−η)|α+1 dγdηds ≤3kF(x)k1+α L∞kxkC2Z1 0ZT |η|−αZT (|∂3 γx(γ)|2+|∂3 γx(γ+ (s−1)η)|2)dγdηds ≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2. 8 Using (22), we ha e o J2 J2=−3α Z1 0ZTZT |F(x)(γ, η)|2+α|∂γx(γ)−∂γx(γ−η)|2 η ∂3 γx(γ)·∂3 γx(γ+(s−1)η) |η|αdγdηds ≤3kF(x)k2+α L∞kxk2 C2Z1 0ZT |η|−αZT (|∂3 γx(γ)|2+|∂3 γx(γ+ (s−1)η)|2)dγdηds ≤CαkF(x)k2+α L∞kxk2 C2k∂3 γxk2 L2. The e m J3is es ima ed by J3≤9 Z1 0ZTZT |η||∂γx(γ)−∂γx(γ−η)|2|∂3 γx(γ)||∂3 γx(γ+(s−1)η)| |x(γ)−x(γ−η)|α+2 dγdηds ≤CαkF(x)k2+α L∞kxk2 C2k∂3 γxk2 L2. We ge inally I3≤Cα(kF(x)k1+α L∞kxkC2+kF(x)k2+α L∞kxk2 C2)k∂3 γxk2 L2.(23) We decompose he e m I4=J4+J5+J6+J7+J8as ollows J4=−αZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) C(γ, η) |x(γ)−x(γ−η)|α+2 dηdγ, J5=−3α ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) D(γ, η) |x(γ)−x(γ−η)|α+2 dηdγ, J6= 5α(α+ 2) ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) A(γ, η)B(γ, η) |x(γ)−x(γ−η)|α+4 dηdγ, J7= 5α(α+ 2) ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))B(γ, η)|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|α+4 dηdγ, J8=−2α(α+ 2)(α+ 4) ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) (B(γ, η))3 |x(γ)−x(γ−η)|α+6 dηdγ, wi h C(γ, η) = (x(γ)−x(γ−η)) ·(∂3 γx(γ)−∂3 γx(γ−η)), D(γ, η) = (∂γx(γ)−∂γx(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). The mos singula e m is J4, in such a way ha J4≤ kF(x)k1+α L∞kxkC2ZT |η|−αZT |∂3 γx(γ)||∂3 γx(γ)−∂3 γx(γ−η)|dγdη ≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2. Fo J5, we ha e J5≤3kF(x)k2+α L∞kxk2 C2ZT |η|−αZT |∂3 γx(γ)||∂2 γx(γ)−∂2 γx(γ−η)|dγdη ≤CαkF(x)k2+α L∞kxk2 C2k∂2 γxkL2k∂3 γxkL2. 9 Theo em 5.1 Le x0(γ)∈Hk(T) o k≥3wi h F(x0)(γ, η)<∞. Then he e exis s a ime T > 0so ha he e is a solu ion o (31) in C1([0, T]; Hk(T)) wi h x(γ, 0) = x0(γ)and λ(γ, ) gi en by (36). P oo : Being analogous o k > 3, we gi e he p oo o k= 3. We ha e showed be o e ha (33) is sa is ied i x(γ, ) is a solu ion o (31). Then we can ew i e λ(γ, ) as ollows λ(γ, ) = γ+π 2πA( )ZT ∂γx(γ, )·∂γZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|dηdγ −1 A( )Zγ −π ∂γx(η, )·∂ηZT ∂γx(η, )−∂γx(η−ξ, ) |x(η, )−x(η−ξ, )|dξdη. (37) We ob ain ZT x(γ)·x (γ)dγ =ZTZT x(γ)·∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ +ZT λ(γ)x(γ)·∂γx(γ)dγ =I1+I2, One inds ha I1= 0, since I1=ZTZT x(γ)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|dηdγ =−ZTZT x(η)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|dηdγ =1 2ZTZT (x(γ)−x(η)) ·(∂γx(γ)−∂γx(η)) |x(γ)−x(η)|dηdγ =1 2ZTZT ∂γ|x(γ)−x(γ−η)|dγdη = 0. Fo he e m I2, one ob ains ha I2≤ kλkL∞kxkL2k∂γxkL2, and kλkL∞≤2 A( )ZT |∂γx(γ)|∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ ≤2 A( )ZT |∂γx(γ)|ZT |∂2 γx(γ)−∂2 γx(γ−η)| |x(γ)−x(γ−η)|dηdγ +2 A( )ZT |∂γx(γ)|ZT |∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|2dηdγ =J1+J2. Due o 1/A( )≤ kF(x)k2 L∞( ), we ha e J1≤2kF(x)k3 L∞Z1 0ZTZT |∂3 γx(γ+ (s−1)η)||∂γx(γ)|dγdηds ≤2kF(x)k3 L∞kxk2 H3, and J2≤2kF(x)k4 L∞kxkC1Z1 0ZTZT |∂2 γx(γ+ (s−1)η)|2dγdηds ≤2kF(x)k4 L∞kxk3 H3. 16 The e o e we ob ain ha d d kxk2 L2( )≤CkF(x)k4 L∞( )kxk5 H3( ).(38) We decompose as ollows ZT ∂3 γx(γ)·∂3 γx (γ)dγ =ZT ∂3 γx(γ)·∂3 γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ +ZT ∂3 γx(γ)·∂3 γ(λ(γ)∂γx(γ))dγ =I3+I4. We ake I3=J3+J4+J5+J6whe e J3=ZTZT ∂3 γx(γ)·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|dηdγ, J4= 3 ZTZT ∂3 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η))∂γ(|x(γ)−x(γ−η)|−1)dηdγ, J5= 3 ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−1)dηdγ, J6=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))∂3 γ(|x(γ)−x(γ−η)|−1)dηdγ. The e m J3can be w i en as J3=1 2ZTZT (∂3 γx(γ)−∂3 γx(γ−η)) ·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|dηdγ =1 4ZTZT ∂γ|∂3 γx(γ)−∂3 γx(γ−η)|2 |x(γ)−x(γ−η)|dηdγ =1 4ZTZT |∂3 γx(γ)−∂3 γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|3dηdγ. I we de ine B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)), due o (32), we ob ain ha J3=1 4ZTZT |F(x)(γ, η)|3|∂3 γx(γ)−∂3 γx(γ−η)|2B(γ, η)η−2−∂γx(γ)·∂2 γx(γ) |η|dηdγ. 17 Using ha  B(γ, η)η−2−∂γx(γ)·∂2 γx(γ) η≤2kxk2 C2,1 2|η|−1/2, we ind J3≤ kF(x)k3 L∞kxk2 C2,1 2ZT |η|−1/2ZT (|∂3 γx(γ)|2+|∂3 γx(γ−η)|2)dγdη ≤CkF(x)k3 L∞kxk2 C2,1 2k∂3 γxk2 L2 ≤CkF(x)k3 L∞kxk4 H3. (39) We ob ain ha J4=−6J3, and i yields J4≤CkF(x)k3 L∞kxk4 H3.(40) In o de o es ima e he e m J5, we conside J5=K1+K2+K3, whe e K1=−3ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) C(γ, η) |x(γ)−x(γ−η)|3dηdγ, K2=−3ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|3dηdγ, K3= 9 ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) (B(γ, η))2 |x(γ)−x(γ−η)|5dηdγ, wi h C(γ, η) = (x(γ)−x(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). The inequali y |∂2 γx(γ)−∂2 γx(γ−η)||η|−1/2≤ kxkC2,1 2,(41) yields K1≤3kF(x)k2 L∞kxkC2,1 2Z1 0ZT |η|−1/2ZT |∂3 γx(γ)||∂3 γx(γ+ (s−1)η)|dγdηds ≤CkF(x)k2 L∞kxk3 H3. As be o e, we ha e o K2 ha K2≤CkF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤CkF(x)k3 L∞kxk4 H3. The e m K3is es ima ed by K3≤CkF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤CkF(x)k3 L∞kxk4 H3. We ge inally J5≤CkF(x)k3 L∞kxk4 H3.(42) 18 We decompose he e m J6=K4+K5+K6+K7+K8as ollows K4=−ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) D(γ, η) |x(γ)−x(γ−η)|3dηdγ, K5=−3ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) E(γ, η) |x(γ)−x(γ−η)|3dηdγ, K6= 15 ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) B(γ, η)C(γ, η) |x(γ)−x(γ−η)|5dηdγ, K7= 15 ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))B(γ, η)|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|5dηdγ, K8=−30 ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) (B(γ, η))3 |x(γ)−x(γ−η)|7dηdγ, wi h D(γ, η) = (x(γ)−x(γ−η)) ·(∂3 γx(γ)−∂3 γx(γ−η)), E(γ, η) = (∂γx(γ)−∂γx(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). We ob ain K5≤3kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤3kF(x)k3 L∞kxk4 H3, K6≤15kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤15kF(x)k3 L∞kxk4 H3, K7≤15kF(x)k4 L∞kxk3 C2k∂3 γxkL2k∂2 γxkL2≤15kF(x)k4 L∞kxk5 H3, and K8≤30kF(x)k4 L∞kxk3 C2k∂3 γxkL2k∂2 γxkL2≤30kF(x)k4 L∞kxk5 H3. Fo he mos singula e m, we ha e K4=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))η ∂γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η)) −D(γ, η) |x(γ)−x(γ−η)|3dηdγ −ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))η ∂γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η)) |x(γ)−x(γ−η)|3dηdγ =L1+L2. One inds ha L1≤ kF(x)k3 L∞kxk2 C2ZTZT |∂3 γx(γ)||∂3 γx(γ)−∂3 γx(γ−η)|dγdη ≤CkF(x)k3 L∞kxk4 H3. The e m L2is decomposed, and i yields 19 L2=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))η(∂γx(γ)−∂γx(γ−η)) ·∂3 γx(γ−η) |x(γ)−x(γ−η)|3dηdγ −ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) η∂γx(γ)·∂3 γx(γ)−∂γx(γ−η)·∂3 γx(γ−η) |x(γ)−x(γ−η)|3dηdγ =M1+M2. We es ima e he e m M1as ollows M1≤ kF(x)k3 L∞kxk2 C2ZTZT |∂3 γx(γ)||∂3 γx(γ−η)|dγdη ≤ kF(x)k3 L∞kxk4 H3. Taking he de i a i e in (32), we ind ha ∂γx(γ)·∂3 γx(γ) = −|∂2 γx(γ)|2, and we ew i e M2=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) η|∂2 γx(γ)|2− |∂2 γx(γ−η)|2 |x(γ)−x(γ−η)|3dηdγ. The inequali y ||∂2 γx(γ)|2− |∂2 γx(γ−η)|2| ≤ 2kxkC2|η|Z1 0 |∂3 γx(γ+ (s−1)η)|ds, (43) yields M2≤2kF(x)k3 L∞kxk2 C2Z1 0ZTZT |∂3 γx(γ)||∂3 γx(γ+ (s−1)η)|dγdηds ≤CkF(x)k3 L∞kxk4 H3. We ecall ha K4=L1+L2=L1+M1+M2≤CkF(x)k3 L∞kxk4 H3,and inally i ollows J6≤CkF(x)k4 L∞kxk5 H3.(44) Due o (39), (40), (42) and (44), we ob ain I3≤CkF(x)k4 L∞kxk5 H3.(45) We ake I4=J7+J8+J9+J10, whe e J7=ZT λ(γ)∂3 γx(γ)·∂4 γx(γ)dγ, J8= 3 ZT ∂γλ(γ)|∂3 γx(γ)|2dγ, J9= 3 ZT ∂2 γλ(γ)∂3 γx(γ)·∂2 γx(γ)dγ, J10 =ZT ∂3 γλ(γ)∂3 γx(γ)·∂γx(γ)dγ. We in eg a e by pa s in he e m J7, and we ge 20 J7=−1 2ZT ∂γλ(γ)|∂3 γx(γ)|2dγ ≤1 2k∂γλkL∞k∂3 γxk2 L2. Using (37), we ind ha ∂γλ(γ, ) = 1 2πA( )ZT ∂γx(γ, )·∂γZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|dηdγ −1 A( )∂γx(γ, )·∂γZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|dη =K9+K10. (46) The e m K9is es ima ed as J1and J2, ob aining K9≤ kF(x)k4 L∞kxk3 H3. We ha e o K10 ha K10 ≤kxkC2 A( )ZT|∂2 γx(γ, )−∂2 γx(γ−η, )| |x(γ, )−x(γ−η, )|+|∂γx(γ, )−∂γx(γ−η, )|2 |x(γ, )−x(γ−η, )|2dη ≤2kF(x)k4 L∞kxk3 C2,1 2ZT |η|−1/2dη ≤CkF(x)k4 L∞kxk3 H3, and he e o e J7≤CkF(x)k4 L∞kxk5 H3.(47) Due o he iden i y J8=−6J7, one inds ha J8≤CkF(x)k4 L∞kxk5 H3.(48) Using ha ∂2 γλ(γ, ) = −1 A( )∂2 γx(γ, )·∂γZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|dη −1 A( )∂γx(γ, )·∂2 γZT ∂γx(γ, )−∂γx(γ−η, ) |x(γ, )−x(γ−η, )|dη, one ge s J9=−1 A( )ZT ∂3 γx(γ)·∂2 γx(γ)∂2 γx(γ)·∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ −1 A( )ZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·∂2 γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ =L3+L4. 21 The e o e L3≤kxk2 C2 A( )ZTZT |∂3 γx(γ)||∂2 γx(γ, )−∂2 γx(γ−η, )| |x(γ, )−x(γ−η, )|+|∂γx(γ, )−∂γx(γ−η, )|2 |x(γ, )−x(γ−η, )|2dηdγ ≤ kF(x)k4 L∞kxk3 C2Z1 0ZTZT |∂3 γx(γ)|(|∂3 γx(γ+ ( −1)η)|+|∂2 γx(γ+ ( −1)η)|)dγdηds ≤CkF(x)k4 L∞kxk5 H3. Mo eo e L4=−1 A( )ZTZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·∂3 γx(γ)−∂3 γx(γ−η) |x(γ)−x(γ−η)|dηdγ +2 A( )ZTZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))B(γ, η) |x(γ)−x(γ−η)|3dηdγ −1 A( )ZTZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·(∂γx(γ)−∂γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−1)dηdγ =M3+M4+M5. The e ms M4and M5a e es ima ed as be o e, and we ob ain M4+M5≤CkF(x)k5 L∞kxk6 H3. The mos singula e m is M3, bu we ind ha M3=1 A( )ZTZT ∂3 γx(γ)·∂2 γx(γ)∂3 γx(γ−η)·∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ −1 A( )ZTZT ∂3 γx(γ)·∂2 γx(γ)∂3 γx(γ)·∂γx(γ)−∂3 γx(γ−η)·∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ =N1+N2. We ob ain N1≤ kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤ kF(x)k3 L∞kxk4 H3, and using (32) N2=1 A( )ZTZT ∂3 γx(γ)·∂2 γx(γ)|∂2 γx(γ)|2− |∂2 γx(γ−η)|2 |x(γ)−x(γ−η)|dηdγ. Due o (43), we conclude ha N2≤2kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤2kF(x)k3 L∞kxk4 H3. We ha e J9=L3+L4=L3+M3+M4+M5=L3+N1+N2+M4+M5, and he e o e J9≤ kF(x)k5 L∞kxk6 H3.(49) 22 The iden i y (32) yields J10 =−ZT ∂3 γλ(γ)|∂2 γx(γ)|2dγ = 2 ZT ∂2 γλ(γ)∂3 γx(γ)·∂2 γx(γ)dγ =2 3J9, and he e o e J10 ≤ kF(x)k5 L∞kxk6 H3.(50) Due o he inequali ies (47), (48), (49), and (50), we ge I4≤CkF(x)k5 L∞kxk6 H3. Using (45) and he las es ima e, we ha e d d k∂3 γxk2 L2( )≤CkF(x)k5 L∞( )kxk6 H3( ). This inequali y and (38) bound he e olu ion o he Sobole no ms o he cu e as ollows d d kxkH3( )≤CkF(x)k5 L∞( )kxk5 H3( ).(51) We con inue he a gumen conside ing he e olu ion o he quan i y kF(x)kL∞( ). Taking p > 2, i yields d d kF(x)kp Lp( )≤pZTZT|η| |x(γ, )−x(γ−η, )|p+1 |x (γ, )−x (γ−η, )| |η|dγdη. We ha e x (γ)−x (γ−η) = ZT (∂γx(γ)−∂γx(γ−ξ) |x(γ)−x(γ−ξ)|−∂γx(γ)−∂γx(γ−ξ) |x(γ−η)−x(γ−η−ξ)|)dξ +ZT ∂γx(γ)−∂γx(γ−η) + ∂γx(γ−η−ξ)−∂γx(γ−ξ) |x(γ−η)−x(γ−η−ξ)|dξ + (λ(γ)−λ(γ−η))∂γx(γ) + λ(γ−η)(∂γx(γ)−∂γx(γ−η)) =I5+I6+I7+I8. The e m I5yields I5≤ZT |∂γx(γ)−∂γx(γ−ξ)||x(γ)−x(γ−ξ)| − |x(γ−η)−x(γ−η−ξ)| |x(γ)−x(γ−ξ)||x(γ−η)−x(γ−η−ξ)|dξ ≤ kF(x)k2 L∞kxkC2ZT |ξ|−1|x(γ)−x(γ−η)−(x(γ−ξ)−x(γ−η−ξ))|dξ ≤ kF(x)k2 L∞kxkC2|η|Z1 0ZT∂γx(γ+ (s−1)η)−∂γx(γ+ (s−1)η−ξ) |ξ|dξds ≤2πkF(x)k2 L∞kxk2 C2|η|. 23 Fo I6we ake I6≤ kF(x)kL∞|η|Z1 0ZT∂2 γx(γ+ (s−1)η)−∂2 γx(γ+ (s−1)η−ξ) |ξ|dξds ≤ kF(x)kL∞kxkC2,1 2|η|Z1 0ZT |ξ|−1/2dξds ≤CkF(x)kL∞kxkC2,1 2|η| We ha e o I7 I7≤2kxkC2 A( )|η|max γ|∂γx(γ)||∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dη| ≤2kF(x)k2 L∞kxk2 C2|η|max γZT |∂2 γx(γ)−∂2 γx(γ−η)| |x(γ)−x(γ−η)|dη+ZT |∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|2dη ≤4kF(x)k4 L∞kxk4 H3|η|. Es ima ing kλkL∞as be o e, easily we ge I8≤ kλkL∞kxkC2|η| ≤ 4kF(x)k4 L∞kxk4 H3|η|. The las ou es ima es show ha d d kF(x)kLp( )≤Ckxk4 H3( )kF(x)k5 L∞( )kF(x)kLp( ), by in eg a ing in ime and aking p→ ∞, we ob ain kF(x)kL∞( +h)≤ kF(x)kL∞( )exp CZ +h kxk4 H3(s)kF(x)k5 L∞(s)ds. As in he p e ious sec ion, i ollows d d kF(x)kL∞( )≤Ckxk4 H3( )kF(x)k6 L∞( ). Then, due o (51) and he abo e es ima e, we ind inally ha d d (kxkH3( ) + kF(x)kL∞( )) ≤C(kxkH3( ) + kF(x)kL∞( ))10. In eg a ing, we ha e kxkH3( ) + kF(x)kL∞( )≤kx0kH3+kF(x0)kL∞ 1− Ckx0kH3+kF(x0)kL∞91 9 , 24 whe e Cis a cons an . We ha e used he equali y (32) o ob ain he a p io i es ima es. In o de o ge he solu ion o (31), we ha e o choose an app op ia e egula ized p oblem p ese ing (32). We p opose he sys em xε,δ (γ, ) = φε∗ZT ∂γ(φε∗xε,δ(γ, )−φε∗xε,δ(γ−η, )) |xε,δ(γ, )−xε,δ(γ−η, )|+δdη +λε,δ(γ, )∂γxε,δ(γ, ), xε,δ(γ, 0) = x0(γ), (52) wi h λε,δ(γ, ) = γ+π 2πZT ∂γxε,δ(γ, ) |∂γxε,δ(γ, )|2·∂γφε∗ZT ∂γ(φε∗xε,δ(γ, )−φε∗xε,δ(γ−η, )) |xε,δ(γ, )−xε,δ(γ−η, )|+δdηdγ −Zγ −π ∂γxε,δ(η, ) |∂γxε,δ(η, )|2·∂ηφε∗ZT ∂γ(φε∗xε,δ(η, )−φε∗xε,δ(η−ξ, )) |xε,δ(η, )−xε,δ(η−ξ, )|+δdξdη. We can ob ain ene gy es ima es o he sys em (52) depending on εand δ, bu wi hou using (32), and he e o e we ob ain exis ence o (52). As long as he solu ion exis s, we ha e ha ∂γxε,δ(γ, )·∂2 γxε,δ(γ, ) = 0. Using his p ope y o he solu ion, we ob ain ene gy es ima es ha depend only on δ, and aking ε→0 we ge a solu ion o he ollowing equa ion xδ (γ, ) = ZT ∂γxδ(γ, )−∂γxδ(γ−η, )) |xδ(γ, )−xδ(γ−η, )|+δdη +λδ(γ, )∂γxδ(γ, ), xδ(γ, 0) = x0(γ), (53) wi h λδ(γ, ) = γ+π 2πZT ∂γxδ(γ, ) |∂γxδ(γ, )|2·∂γZT ∂γxδ(γ, )−∂γxδ(γ−η, ) |xδ(γ, )−xδ(γ−η, )|+δdηdγ −Zγ −π ∂γxδ(η, ) |∂γxδ(η, )|2·∂ηZT ∂γxδ(η, )−∂γxδ(η−ξ, )) |xδ(η, )−xδ(η−ξ, )|+δdξdη. Again we ha e ha he solu ions o his sys em sa is y ∂γxδ(γ, )·∂2 γxδ(γ, ) = 0, and aking ad an age o his, we ind ene gy es ima es independen o δ. I we end δ o 0, we conclude he exis ence esul . Re e ences [1] A. L. Be ozzi and P. Cons an in. Global egula i y o o ex pa ches. Comm. Ma h. Phys. 152 (1): 19–28, 1993. 25