2002 Mathematics Subject Classification: 35G20, 47H30We introduce scales of Banach spaces of anisotropic Gevrey functions depending on multidimensional parameters. We prove estimates in such spaces for composition maps and nonlinearities in conservative forms. Applications for solvability and regularity of solutions of nonlinear PDEs are outlined.*The author is supported by a local research grant of the University of Cagliari
*The author is supported by a NATO grant PST.CLG.97934

Cadeddu, Lucio

Gramchev, Todor

We introduce scales of Banach spaces of anisotropic Gevrey functions depending on multidimensional parameters. We prove estimates in such spaces for composition maps and nonlinearities in conservative forms. Applications for solvability and regularity of solutions of nonlinear PDEs are outlined

CADEDDU, LUCIO

GRAMTCHEV, TODOR VASSILEV

Archivio istituzionale della ricerca - Università di Cagliari

Nonlinear estimates in anisotropic Gevrey spaces

Bulgarian Digital Mathematics Library at IMI-BAS

Provided for non-commercial research and educational use. Not for reproduction, distribution or commercial use.PLISKASTUDIA MATHEMATICA BULGARICAПЛИСКАБЪЛГАРСКИ МАТЕМАТИЧЕСКИ СТУДИИThe attached copy is furnished for non-commercial research and education use only. Authors are permitted to post this version of the article to their personal websites or institutional repositories and to share with other researchers in the form of electronic reprints. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to third party websites are prohibited.For further information on Pliska Studia Mathematica Bulgarica visit the website of the journal http://www.math.bas.bg/-pliska/ or contact: Editorial Office Pliska Studia Mathematica Bulgarica Institute of Mathematics and Informatics Bulgarian Academy of Sciences Telephone: (+359-2)9792818, FAX:(+359-2)971-36-49 e-mail: pliska@math.bas.bgPliska Stud. Math. Bulgar. 15 (2003), 149-160 PUSI<&S T U D I A  M A T H E M A T I C A  B U L G A R IC ANONLINEAR ESTIMATES IN ANISOTROPIC GEVREY  SPACESLucio Cadeddu*, Todor Gramchev*A b s t r a c t . We introduce scales of Banach spaces of anisotropic Gevrey functions depending on multidimensional parameters. We prove estimates in such spaces for composition maps and nonlinearities in conservative forms. Applications for solvability and regularity of solutions of nonlinear PDEs are outlined.1. Anisotropic Gevrey SpacesLet О С  W1 be an open domain and let a — ( a i , . . . ,  on) E  [1, + o o [ n . We define G°n(ft) - the spaces of the uniformly anisotropic Ga(ft) Gevrey functions - as the set of all ∕  E  C°°(∩) such that there exists С > 0 satisfying(1.1) sup \∂%f(x)\ < C '^ +1a!cr, ö E Z ” .where a\ =  a \ \ . . .  a n!, a\a =  a\ai . . .  an\an, a  =  ( a i , . . . ,  an) E  In particular, if o\ =  . . .  an =  5 we recover the well known space G^n(0) of uniformly Gevrey functions of index «s. Local Gevrey spaces Gs(0) are defined in a natural way cf. [18], [16] for more details on Gevrey spaces. In particular, note that ∂x : G°n(tt) I-» G°n(tt) for all ß E  and G°n(tt) ^  G^n(fl) provided aj < Tj, j  = l , . . . , n .2002 Mathematics Subject Classification: 35G20, 47H30Key words: anisotropic Gevrey classes, nonlinear composition estimates, evolution PDE *The author is supported by a local research grant of the University of Cagliari*The author is supported by a NATO grant PST.CLG.979347150 L. Cadeddu, T. GramchevWe point out that the Gevrey spaces are a natural framework for the study of PDEs with multiple characteristics and questions of regularity of solutions to evolution PDEs of Mathematical Physics. Typically, in the applications, one introduces scales of Banach spaces with norms depending on one parameter. The first type of norms is(!∙2) I NU t  := ∑  47l|∂£«ll < +∞>where || ∙ || stands for the sup-norm in О or for some LP based Sobolev normsII ∙ ||я *, 1 < p < 00 , к > 0 . cf. [1], [10], [11] and the references therein. Another approach relies on the use of exponential norms by means of the Fourier transform cf. [2], [4], [5], [6], [7], [8], [10], [12], [14], [17]).Given ∕  E G^n(Rn), in view of (1.1), we can define(1.3) Ps(f) — sup{T > 0 : such that (1.1) holds for o\ =  .. .  =  on =  s.}and if s =  1, we obtain that every ∕  E G^n(Rn) is extended to a holomor­phic function in {z =  (^i , . . . ,  zn) E Cn ; maxj= i ri.5n \Imz j I < Ps(f)}∙ Clearly the definition of ps(f) does not take into account the different behaviour in the multidimensional case.We introduce scales of Banach spaces G^(0;T), T  =  (Ti , .. .  ,Tn) e]1, +oo[n defined as follows(1.4) := ∑  T \ m \ ∞ < + 00 ,where | ∙ |∞ stands for the sup-norm in O. If a — (s , . . . ,  s) we writeII I^Up := N s,(p,...,p)’ ^  (^5p) := ^  (^5 (z9’ ∙ ∙ ∙ ’рУ)'One readily gets that Ga(ft;T) are Banach algebras (because oj > 1 for all j  =  1, . . . ,  n). We stress that, broadly speaking, when T\ =  . . .  =  Tn, such type of spaces have been used for showing local Gevrey solvability and/or Gevrey well-posedness of the Cauchy problem for nonlinear evolution PDEs wih multiple characteristics cf. [2], [10], [5], [6], [12], [15]. In particular, [2], [10], [6] deal with some nonanalytic Gevrey perturbations.For more details and geometrical features of Ga(Q) type spaces, as well as applications to linear PDEs with multiple characteristics cf. [3] and the references therein (see also [13] where a rather complete theory of microlocal inhomogeneous Gevrey spaces, closely related to Ga(Q), is developed).Anisotropic Gevrey spaces 151We illustrate the advantage of the multi-scale type norms (1.4) even for the usual isotropic Gevrey space G^n(Rn) when n > 1. Let ∕  E Gs (R;po) for some 5 > 1, Po > 0. Consider a multi-rescaling action g \ ( x  1,^ 2) = / ( ^ l / A i ) / ^ / ^ ) ,  where A =  (Ai,A2) E]0, + o o [2 . Then g \  E Gs(R2;p) for p < po min{Ai, A2}. On the other hand, g E G^s,s\ R2 ; (pcAi,/>0^2))- Next, consider the Cauchy problemn(1.5) ∂tu +  ajXj∂Xju =  0, i / (0 ,ж) =  щ ( х )  E Gs (Rn ;po), £ > 0 , ж Е К . п ,3=1where aj E M, j  = l , . . . , n ,  5 > 1, po > 0. The unique solution is given by u(t,x) =  uo(xie~ait, .. .  , x ne~ant). If we use the one parameter scale of Ba- nach spaces Gs(∩;p) the solution u(t,∙) of (1.5) satisfies \\u(t, ∙)lls,p(t) < + 00, where p(t) =  po exp(am n^t) and amin =  min{ai , . . . ,  an} while setting T(t) = (po exp(ait), . . . ,  po exp(a∏t)) we get |u(t, ∙)|s < +00  for t > 0. Moreover, thefollowing estimate holds: ||u(t, *) IU,p(t) ≤ \u (t, *)ls f(t) o^r  ^ > 0∙The first goal of our paper is motivated by results on nonlinear estimates in the framework of the isotropic Gs classes and their applications to PDEs cf. [2],[10]. Our first main result - see section 2 - is concerned with estimates of the action of composition maps in Ga(∩;T). It may be also viewed as an analogue in Ga classes of the Schauder type estimates for inhomogeneous Sobolev spaces in [9].Secondly, we propose abstract energy estimates for nonlinearities in conser­vative forms in scales of suitable Hilbert spaces of L2(0) based Ga anisotropic functions, for О = W1 and О being the n-dimensional torus. As particular cases we recover, when o\ =  . . .  =  on =  «s, energy estimates in [7], [8], [14] for «s =  1 (see also [11] for s > 1).Our results will be applied for the study of solvability and regularity proper­ties of solutions of Cauchy problems for abstract systems of nonlinear evolution PDEs in anisotropic Gevrey spaces. This will be done in another paper.2. Nonlinear superposition estimatesGiven 0  =  (01 , . . . ,  an) E]l, + 0 0 [n we set 00 =  min{0 i , . . .  crn}. Clearly, a smooth function g preserves Ga(∩ : M) iff g E Ga°(B). Our goal is to investigate the action of (g о f )(x)  := g(f(x))  for ∕  E G^(0;T), f  real valued.First, we show the following identity152 L. Cadeddu, T. GramchevLem m a 2.1. Let p E N, а  =  ( a i , . . . ,  a n) E Z+ with \а\ > 1. ТЛеп(2 .6) X ∏ ^ ∙ ∙ ∙ ^ M * )Pr oof .  We shall proceed by induction with respect to n. The detailed proof of the case n =  1 can be found in [GR], [GB]. We observe that for n > 2where a f =  ( a i , . . . ,  a n_i), ж' =  (x i , . . . ,  £n-i)∙  The inductive assumption for dimension n — 1 and the Leibnitz rule for the action of ∂^, on the products of p functions complete the proof. □Next we propose a refined version of Faä di Bruno type formulasP roposition  2.2. Let a =  (cri,. . . ,  a∏) e]0, +oo[n. Then for every g E C°°(IR : Ш), f  E C°°(Wl : Ш), a  =  ( a i , . . . ,  a n) Е Z+ with \a\ > 1 йе  following identity is trueVwhere =  (af , . . . ,  a^) E Z+ and°k~ 1Pr oof .  We havej=iAnisotropic Gevrey spaces 153where(2.11) M ^ [ f ]  = (j l∏ “ 1 ∑a 1!...  аЛ ∂ f  f (x )  . . . ∂ g  f(x).и >  ! / = ! , . . . JDividing by a'!ö’(j!)ö’° 1 in (2.11) we obtainal + ...+a∙?=a ^—1∑  ∏n11 гу1]аwhich implies (2.9). □Set \ f \ t f f  =  l^l^f — l/l∞∙ The main result on the action of nonlinear maps in G *(ft,f) isT heorem  2.3. Let a E  [ l , + o o [ n and let g E  G^°(M), #(0) = 0; with сто = m in{ai,.. .  ,a n}. Smce </ E  G^° (M) as well, we get by (1.3) that po := рао(д') >0. Then the following estimate holdsfor T  e ]0, +oo[n; ∕  E Ga(tt; T), f  being real valued, provided p := na° 1| / | - f  < Po∙P ro o f . We note that by g ( 0) =  0 we get | g о / | ∞  <  \ gr \∞ \ f \ oo ∙  I∏ view of(2.8), (2.9), given a  E Z™, \a\ > 1, we have(2.13)(2.14)~ ^ \ ∂ x W ( x ) ) \  : a r154 L. Cadeddu, T. GramchevTherefore, summing (2.14) over а Е  Z+, 0  /  0, we getb ' f b f  ≤ ∑  ∑ ^ T ^  ∑  ^Ia<l^i/=i>∙∙∙>iX ∏ ^ l ^ l -(2.15) < ∑  la>l ° J ! ∞ N ,l f ∙S ,T ]3=1wherea < E Z ™ ,  | a | > j  a l + . . . + a J = a  ^ = 1\a£\> I j(2.16)The final step depends on subtle combinatorial estimates. Lemma 2.4. Let ao = minjcri,. . . ,  cr∏}. Then<2∙17> -  ∑  ∏ ^ l / (∕, гч 1 «'er) I_ о^\и |o°a-*-+...+a^ " =0; £—1for all j  E  N.P ro o f . Fix a 1, . . . ,  E  Z™ with |c/| > 0 for  ^ =  1 , . . . ,  j. Let к E  {1,... , n} and we fix ^  to be a nonnegative integer such that at least components of (a^ ,. . . ,  Gf£) are nonzero. Recall the combinatorial inequalities (cf. [10])Anisotropic Gevrey spaces 155Hence, by (2.9), we getk l ,  -(ol^O-1 ∕  j} X^o-l(2.3.9) < , ,4 V  ∕ .4--г <°k~1(si!)^-1... (s^ !)^-1 V5i!∙∙∙5n!Clearly, if =  j  for at least one fc, the last inequality yields a<7∙ < 1. Letnow Sfc < j  for к — 1 ,... , j. We note that the inequalities |c/| > 1,  ^ =  1 , . . . ,  jimply si + .. .  + sn > j. If si +  .. .  +  sn > j, standard combinatorial arguments and the fact that the RHS of (2.19) increases if some S£ decreases reduce to thecase si + .. .  + sn = j. The final step consists of bounding∕  ∙| \ 1 Jb∙∙∙Jn∑  ≤ ∑ _ . b b )  ∑ .  -+ . . . + a i  =a S i  “b ∙ ∙ ∙ + S n—J al ,-\-a3 = a\ae\>l,£=l,...,j \ae\>l,£=l,...,jwhere the internal summation in the RHS is over a 1,. . .« -7 with s i , . . . , s j  as above. The proof of (2.17) is complete. □Next, we plug (2.17) into (2.15) and obtain3 =1 ^  j l + - + j n = j  ^ ' 1' ' ' 'J  rpct1∑  П ^ | / (“ ’|a^+...-(-aJ=a —^1H > i/= i,∙∙∙j<<У ' 1»Ш ° /1 ~  У '  (  3'- Г ^ Г Т  У 'h w  lLh«a-Y |9<J> 0/100 ад -у V  ( й (2.20) < n'J° - 1| / | ?)f||3/ ||(T0,pprovided p := f  < po∙ The proof of (2.13) is complete. □156 L. Cadeddu, T. Gramchev3. Gevrey energy estimates for nonlinear conservative termsThe main goal of this section is to derive energy estimates in anisotropic Gevrey spaces for nonlinear terms in a conservative form of the type V ∙ (К[ф\ф) where typically К[ф] =  (К\[ф\,. . . ,  К п[ф]), Kj, j  =  1 , . . . , n are linear continuous oper­ators in Sobolev spaces (in some cases homogeneous Sobolev spaces). We assume that и =  K[v\ is divergence free, i.e.,n(3.1) у . в д  =  £ ^ я - > ]  =  0.3 = 1Such nonlinear terms appear in evolution PDEs of Navier-Stokes type, Euler type or Kirchhoff type (see [8], [14], [11] and the references therein).We shall treat (simultaneously) two cases: О = W1 and О = Tn =  W1 ∕ (27rZ)n. The choice of the lattice (27rZn) instead of Z n is not restrictive but it allows us to use unified notations. More precisely, both the continuous (on W1) and the discrete (on Tn) Fourier transform are defined byf ( 0 = F f ( 0 =  [  exP(-ix£) f (x )dx ,  x£ = X &  +  . . .  +  xn£n,Jnidentifying Tn = [0,27r]n in the case О = Tn. The inverse Fourier transform iswritten as T 7 \ xf  =  f  exp (гж£) /(£) = (27x)~nd^  with IK = M if О = W1JKnwhile in the case of О = Tn К  =  Z and ∕  h(£) Щ stands for (2тг)~п h(£)J%nc∙ ∙i i . TVtvmx f  LP(^n) ifK = R _  ^Similarly, we set I f  (K ) =  j if к  = z  ’ 1 ≤ P ≤ ∞∙Let а E [1, +oo[n, г E [0, +oo[n, and r > 0. We define:(3.2) < f,9> ∏ r  =  [  ( Н |{ |Г)2Ш ^JKn10 1(3.3) exp(f < D >l/a)v(x) = )«(£))3 =  1(3-4) < / ,  g = < exp(f < D >1/<T) f , exp(f < D >1/a g > Hr(3.5) \\f\\ff,r-,Hr = \J < exp(r < D > 1/l?)/, exp(r < D > l / ff f  > HrAnisotropic Gevrey spaces 157where S = 1 when ft =  W1 while S = 0 if ft = Tn, and in that case we consider functions on Tn with mean value zero, i.e. /(0) =  0. In view of (3.4), (3.5) we introduce in a natural way the Hilbert space Ga(f t ;r ,H r) as the set of all functions ∕  E G°n(ft) such that \\f\\^^∙Hr < +oo.We are interested in estimating the commutatorС[ф\ =  < ехр(т < D >1^ (7)К[ф\ ∙ V0), ехр(т < D >1^ )ф  >∏r(3.6) — < V ∙ К[ф\ ехр(т < D > ^  )</>), exp (т < D >1/а)ф >∏rwhere ф E Ga(ft; r, H r ) for some f  > r.We set, as in section 2, сто = minjcri,. . . ,  crn} > 1.T heorem  3.1. Suppose that r > tq := n / 2 + 1 + 1/(2сго). ТЛеп for every £ > 0 there exist two positive constants C\ =  Ci(e), C2 =  C2(e) such thatm \  ≤ С г т н г Щ и Ц н г Щ ^  + т H r 1М1яго+0n+ c 2 , т ; Я г + 1 / ( 2(7о )  ( U h  ^ . H r +  l / ( 2 a o)  I M I ^ , f ; # r o + £3 =  1(3-7) + ||'u||^^;^r+1/(2cr0 ||(/)||öJ,r;i7ro+£r)-P ro o f . Set и := К[Ф]. Note that V ∙ и =  0 iff й(£) ∙ ( ≡ 0. Setn(3.8) /<?(£;tO = < С > r exp ( ∑ Tj < ii  >l,(Ti),3 =  1(3-9) =  f a i &T)  -  foil]-, f )By the Parseval identity, С[ф\ can be rewritten as follows(3.10) С[ф] = i f  f  Qff(Z,rr,T)(ü(£-ri)-г])Ы^г)ф(ОФ(г])йСйг]JR™ JR™We have\Qs{€,rfiT)\ ≤ <?!£ -  ??!(< £ -  ?? > r_1 + < v > r~r)n+  C \ i  -  rj\ ∑ T j ( <  Z -  ц >r~1 + < ц >r_1) < Щ > 1/ai fff(r/;T)3 =  1∏(3.11) +  c ∑ T j \ i j  - v j \ 1/aj{< £ - я  >r + < я  >r) h ( t  ~ v;t ) fa(v, t ).3 =  1158 L. Cadeddu, T. GramchevIndeed, in view of the inequalities exp(|^|) < 1+|г|ехр(|я|), \t+s\P < \t\p+\s\p, \\t\p — |s|p| < 1t — s\p, provided 0 < p < 1, we getI <  I <  £  > r  -  <  V > r \ f * ( V i T ) +  <  £  > T \f#(£∙,T) -  fa{r)]T)\n≤ ∑  ri(<  С “  »7 >r_1 + < ^ >r_1) < Vj >1/aj h i m  T)3=1n+  ∑  тМз -  Vj\1/aj (<  С -  Я >r +  <  V >r)fc?(£ -  »7 ; T)U{ri; t )3 =  1which implies (3.11).Set∏[</>]r(0 = <  £ > r |0(€)U н§,?[ФШ) = <  £ > p exp ( ∑ Tj  <  > 1/aj) \ k o \ -3 = 14In view of (3.10) and (3.11) we have \С[ф\\ < ^^&к[Ф]^ wherek= l0 i[0] = [  [  [Ф]г(0[и]га-л)[Ф\JKn JKn©2[ф\ = [  [  M f ( 0 N l( f  -  ^/)[0]r(r/) Зт/,n Г Г0 3  [ 0]  =  ∑ T j  ∕  i V £ T# ]  ( £ ) А Й 1 / ( 2 ^ } [« ] (e  -  r ? ) iV j+ 1 /(2 ^ } [0 ] f a )  3 £  Щ ,  ть Г Ге 4 [ф] =  ∑ T 3 /  А Г :+ 1/(2^ }[0]( ^ ) ^ + 1/(2^ }[ г г ] -77)АГ^+1/(2^ }[0](^) Щ  Щ .Jep JepWe needLem m a 3.2. Let gj : K.n i— M, j  =  1,2,3 satisfy the following property: there exist j  I E {1,2,3} and ц > n / 2 swc/a that(3.12) II < ∙ 5 j J L2(K„) ~  (  [  < C > 2ß 1Ьл(С)1|2^ )  < + ∞wk™ ∕(3.13) 119jp \ \l  ^(K.n ) <  + ∞  for jp e  {1,2,3} \  { j i} ,  p =  2,3.Anisotropic Gevrey spaces 159Then1 \9 ъ9 2 ,9 ъ \  =  Ы£-»7)32(£)0з(»7)1<№JKd JKd(3-14) < Afy|| < ∙ IIlH I^IIlHI^JsIIl2where =  ( f Kn < £ >~2p d£ )^ 2 < +oo.Pr oof .  We observe that by /i > n/2 and the Schwartz inequality we get(3.15) IIs jiIIli ≤ cll < ∙ 3j i IIl2 < +°°(3.16) \\ < - >^ OjpWb2 < +°°? for = 2,3where с = ( /K7l <  ^ > -2  ^ < +oo. We complete the proof of (3.14) byapplying the Schur lemma and the above estimates. □We apply the above lemma to ©&[</>], with the choice of gj1 being the term with Sobolev index 1 + l(2oj), к =  1,2, 3,4, and obtain the desired estimate(3.7). □Acknowledgements: The authors thank Petar Popivanov for the useful discussions and his valuable suggestions on the subject of the paper.R E F E R E N C E S[1] H.A. B iagioni, T. Gramchev. Fractional derivative estimates in Gevrey spaces, global regularity and decay for solutions to semilinear equations in W1. J. Differential Equations. 194 (2003), 140-165.[2] G. B ourdaud, M. R eissig, W. Sickel . Hyperbolic equations, function spaces with exponential weights and Nemytskij operators. Ann. Mat. Рига Appl. (4) 182 (2003), 409-455.[3] D. CALVO Multianisotropic Gevrey classes and Cauchy problem. Ph. D. Thesis, Universitä di Pisa (2003).[4] D. Calvo, P. P opivanov. Solvability in Gevrey classes for second powers of the Mizohata operator. 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В 2 (1999), 65-120.I l l  T. G r a m c h e v , Y.G. W a n g . Propagation of uniform Gevrey regularity of solutions to evolution equations. Evolution Equations. Banach Center Publ. 60 (2003), 279-293.121 K. K ajitani. Local solutions of Cauchy problem for nonlinear hyperbolic systems. Hokkaido Math. J., 12 (1983), 443-460.131 O. L iess, L. R odino. Inhomogeneous Gevrey classes and related pseu- dodiferential operators. Anal. Funz. Appl., Suppl. Boll. Un. Mat. Ital. 3 С (1984), 233-323.141 D. Le v e r m o r e , M. O liv e r . Analyticity of solutions for a generalized Euler equation. J. Differential Equations 133 (1997), 321-339.151 P∙ M a r c o l o n g o , A. O lia r o . Local solvability for semilinear anisotropic partial differential equations. Annali Mat. Рига Appl. 179, (4) (2001), 229- 262.161 M. M a s c a r e l l o , L. R o d in o . Partial differential equations with multiple characteristics. Wiley-VCH, Berlin, 1997.17] P. P opivanov. Local solvability of some classes of linear differential oper­ators with multiple characteristics. Ann. Univ. Ferrara, VII, Sc. Mat. 45 (1999), 263-274.18] L. R o d in o . Linear partial differential operators in Gevrey spaces. World Scientific, Singapore, 1993.Dipartimento di Matematica e InformaticaUniversitä di Cagliarivia Ospedale 72, 09124 CagliariItaliacadeddu@unica. it, todor@unica. it

Nonlinear Estimates in Anisotropic Gevrey Spaces

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Nonlinear Estimates in Anisotropic Gevrey Spaces

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