ON THE CAUCHY-PROBLEM FOR GENERALIZED KADOMTSEV-PETVIASHVILI-II EQUATIONS

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1 Electronic Journal of Differential Equations, Vol. 009(009), No. 8, pp ISSN: URL: or ftp ejde.math.tstate.edu ON THE CAUCHY-PROBLEM FOR GENERALIZED KADOMTSEV-PETVIASHVILI-II EQUATIONS AXEL GRÜNROCK Abstract. The Cauchy-problem for the generalized Kadomtsev-Petviashvili- II equation u t + u + 1 uyy = (ul ), l 3, is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maimal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in X s,b -spaces. 1. Introduction Inspired by the wor of Kenig and Ziesler [13, 14], we consider the Cauchy problem u(, y, 0) = u 0 (, y), (, y) R (1.1) for the generalized Kadomtsev-Petviashvili-II equation (for short: gkp-ii) u t + u + 1 u yy = (u l ), (1.) where l 3 is an integer. Concerning earlier results on related problems for this equation we refer to the wors of Saut [15], Iório and Nunes [8], and Hayashi, Naumin, and Saut [7]. For the Cauchy data we shall assume u 0 H (s), where for s = (s 1, s, ε) the Sobolev type space H (s) is defined by its norm in the following way: Let ξ := (, η) R denote the Fourier variables corresponding to (, y) R and D σ1 = F 1 σ1 F, D y σ = F 1 η σ F, as well as D 1 D y σ3 = F 1 1 η σ3 F, where F denotes the Fourier transform and σ = (1 + ) σ. Setting u 0 σ1,σ,σ 3 := D σ1 D y σ D 1 D y σ3 u 0 L y we define u 0 H (s) := u 0 s1+s +ε,0,0 + u 0 s1,s,ε. Almost the same data spaces are considered in [13], [14], the only new element here is the additional parameter s, which will play a major role only for powers l 4. Using the contraction mapping principle we will prove a local well-posedness result for (1.1), (1.) with regularity assumptions on the data as wea as possible. 000 Mathematics Subject Classification. 35Q53. Key words and phrases. Cauchy-problem; local well-posedness; generalized KP-II equations. c 009 Teas State University - San Marcos. Submitted April 9, 009. Published July 10, 009. Supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich

2 A. GRÜNROCK EJDE-009/8 Two types of estimates will be be involved in the proof: On the one hand there is the combination of local smoothing effect ([13], see also [10]) and maimal function estimate (proven in [13, 14]), which has been used in [13], a strategy that had been developed in [11] in the contet of generalized KdV equations. On the other hand we rely on Strichartz type estimates (cf. [15]) and especially on the bilinear refinement thereof taen from [5, 6], see also [9, 16, 18, 17]. A similar bilinear estimate involving y-derivatives (to be proven below) will serve to deal with the D y containing part of the norm. To mae these two elements meet, we use Bourgain s Fourier restriction norm method [1], especially the following function spaces of X s,b -type. The basic space X 0,b is given as usual by the norm u X0,b := τ φ(ξ) b Fu L ξτ, where φ(ξ) = 3 η is the phase function of the linearized KP-II equation. According to the data spaces chosen above we shall also use as well as u Xσ1,σ,σ 3 ;b := D σ1 D y σ D 1 D y σ3 u X0,b u X(s),b := u Xs1 +s +ε,0,0;b + u X s1,s,ε;b. Finally the time restriction norm spaces X (s),b (δ) constructed in the usual manner will become our solution spaces. Now we can state the main result of this note. Theorem 1.1. Let s 1 > 1/, s l 3 (l 1) and 0 < ε min (s 1, 1). Then for s = (s 1, s, ε) and u 0 H (s) there eist δ = δ( u 0 H (s)) > 0 and b > 1 such that there is a unique solution u X (s),b (δ) of (1.1), (1.). This solution is persistent and the flow map S : u 0 u, H (s) X (s),b (δ) is locally Lipschitz. The lower bounds on s 1, s, and ε are optimal (ecept for endpoints) in the sense that scaling considerations strongly suggest the necessity of the condition s 1 + s + ε 1 + l 3 l 1. Moreover, for l = 3 C -illposedness is nown for s 1 < 1 or ε < 0, see [13, Theorem 4.1]. The affirmative result in [13] concerning the cubic gkp-ii equation, that was local well-posedness for s 1 > 3 4, s = 0, and ε > 1 is improved here by 3 4 derivatives. Some effort was made to eep the number of y-derivatives as small as possible 1, but we shall not attempt to give evidence for the necessity of the individual lower bounds on s 1, s, and ε, respectively. By the general arguments concerning the Fourier restriction norm method introduced in [1] and further developed in [, 1], matters reduce to proving the following multilinear estimate. Theorem 1.. Let s 1 > 1, s l 3 (l 1) and 0 < ε min (s 1, 1). Then there eists b > 1, such that for all b > 1 and all u 1,..., u l X (s),b supported in { t 1} the estimate u j X(s),b u j X(s),b holds. 1 For l 5 the use of local smoothing effect and maimal function estimate alone yields LWP for s 1 > l 3 l 1, s = 0, and ε > 1, which is optimal from the scaling point of view, too. This result may be seen as essentially contained in [13, Theorem.1 and Lemma 3.] plus [11, proofs of Theorem.10 and Theorem.17]. This variant always requires 1 + derivatives in y.

3 EJDE-009/8 CAUCHY-PROBLEM FOR GKP-II EQUATIONS 3 To prepare for the proof of Theorem 1. let us first recall those estimates for free solutions W (t)u 0 of the linearized KP-II equation, which we tae over from the literature. First we have the local smoothing estimate from [13, Lemma 3.]: For 0 λ 1, D (1 λ) (D 1 D y ) λ W (t)u 0 L L u 0 L y (1.3) and hence by the transfer principle [, Lemma.3] for b > 1, D (1 λ) (D 1 D y ) λ u L L u X 0,b. (1.4) Interpolation with the trivial case L = X 0,0 and duality give for 0 θ 1, 1 p θ = 1 θ, [D (1 λ) (D 1 D y ) λ ] θ u p L θ L u X 0,θb (1.5) as well as [D (1 λ) (D 1 D y ) λ ] θ u X0, θb u L p θ. (1.6) L (For Hölder eponents p we will always have 1 p + 1 p = 1.) To complement the local smoothing effect, we shall use the maimal function estimate W (t)u 0 L 4 L yt D D 1 D y 1 + u 0 L y (1.7) due to Kenig and Ziesler [13, Theorem.1], which is probably the hardest part of the whole story. The capital T here indicates, that this estimate is only valid local in time, the +-signs at the eponents on the right denote positive numbers, which can be made arbitrarily small at the cost of the implicit constant but independent of other parameters. (This notation will be used repeatedly below.) The transfer principle implies for u supported in { t 1} u L 4 L where b > 1. The Strichartz type estimate taen from [15, Proposition.3], becomes For its bilinear refinement and the dualized version thereof D D 1 D y 1 + u X0,b, (1.8) W (t)u 0 L 4 u 0 L y, (1.9) u L 4 u X0,b, b > 1. (1.10) uv L u X0,b D 1 v X0,b, b > 1, (1.11) (uv) X0, b u X0,b v L, b > 1, (1.1) we refer to [6, Theorem 3.3 and Proposition 3.5]. In order to estimate the X s1,s,ε;b - norm of the nonlinearity the following bilinear estimate involving y-derivatives will be useful. We introduce the bilinear pseudodifferential operator M(u, v) in terms of its Fourier transform M(u, v)(ξ) := ξ=ξ 1+ξ 1 η η 1 1/ û(ξ 1 ) v(ξ )dξ 1

4 4 A. GRÜNROCK EJDE-009/8 and define the auiliary space L p L q by f bl p L q := F f L p, where F L q denotes the partial Fourier transform with respect to the first space variable only. Then we have: Lemma 1.3. Proof. We have M(W (t)u 0, W (t)v 0 ) bl 1 L u 0 L y v 0 L y. (1.13) M(W (t)u 0, W (t)v 0 )(ξ, τ) = 1 η η 1 1/ δ(τ η 1 + η )û 0 (ξ 1 ) v 0 (ξ )dξ 1. ξ=ξ 1+ξ 1 Because of η η as g(η 1 ) = 1 (η 1 1 = η + 1 (η 1 1 η) and with a = τ η η) a this equals ξ=ξ 1+ξ 1 η η 1 1/ δ(g(η 1 ))û 0 (ξ 1 ) v 0 (ξ )dη 1 d 1. as well The zero s of g are η 1 ± = 1 η ± 1 a and for the derivative we have g (η 1 ) = 1 1η η 1. So we get the two contributions I ± (ξ, τ) = 1 1+ = 1 1 η η ± 1 1/ û0( 1, η ± 1 ) v 0(, η η ± 1 )d 1. By Minowsi s integral inequality I ± (ξ, ) L τ 1 1 η η 1 ± 1 û0 ( 1, η 1 ± ) v 0(, η η 1 ± ) L d τ 1, 1+ = where, with λ := 1 a, the square of the last L τ -norm equals 1 η η 1 ± 1 û 0 ( 1, 1 η ± λ) v 0(, η λ) dτ û 0 ( 1, 1 1 η ± λ) v 0(, η λ) dλ, since dτ = λ dλ = 1 1 ( 1 η η 1 ± )dλ. This gives I ± (ξ, ) L τ 1 1/( û 0 ( 1, 1 η ± λ) v 0(, ) 1/d1 η λ) dλ. 1+ = It is not apparent from the proof, but there is a very simple idea behind Lemma 1.3. If we tae the partial Fourier transform F W (t)u 0 () = e it3 e i t y F u 0 () with respect to the -variable only, we obtain a free solution of the linear Schrödinger equation with rescaled time variable s = t, multiplied by a phase factor of size one. So any space time estimate for the Schrödinger equation should give a corresponding estimate for linearized KP-type equations. This idea was eploitet in [3], [4] to obtain suitable substitutes for Strichartz type estimates for semiperiodic and periodic problems. From this point of view Lemma 1.3 corresponds to the one-dimensional estimate y (e it y u 0 e it y v 0 ) L u 0 L y v 0 L y.

5 EJDE-009/8 CAUCHY-PROBLEM FOR GKP-II EQUATIONS 5 Using Parseval and again Minowsi s inequality for the L η-norm we arrive at F M(W (t)u 0, W (t)v 0 )() L 1 1/( û 0 ( 1, 1 1+ = η ± λ) v 0(, η λ) dλdη = 1 1/ û 0 ( 1, ) L η v 0 (, ) L η d 1 1+ = u 0 L y D 1/ v 0 L y by Cauchy-Schwarz and a second application of Parseval s identity. ) 1/d1 Corollary 1.4. Let b > 1/. Then M(u, v) bl 1 L u X0,b v X0,b (1.14) D 1 M(u, v) X0, b u bl L D 1/ v X0,b. (1.15) Proof. Lemma 1.3 implies (1.14) via the transfer principle, (1.15) is then obtained by duality. In fact, if we fi v and consider the linear map M v (u) := M(u, v), then its adjoint is given by M v = M v, and we have v X0,b = v X0,b. Now we are prepared for the proof of the central multilinear estimate. Proof of Theorem We use (1.6) with λ = 0, θ = 1 and Hölder to obtain for b 0 < 1 4 D 1/ u j X0,b0 u j 4 L 3 L u 1 L 4 L u L 4 L u 3 L 4 L u j L. For the first factor we have by (1.5), again with λ = 0, θ = 1, u 1 L 4 L D 1 u 1 X0, 1 4 +, while the second and third factor are estimated by (1.8) u,3 L 4 L D D 1 D y 1 + u,3 X0,b, where b > 1. It is here that the time support assumption is needed. For j 4 we use Sobolev embeddings in all variables to obtain for b > 1 u j L D 1 + D y 1 + u j X0,b. Summarizing we have for b 0 < 1/4, b > 1/, u j X0,b0 D 1 D D 1 u 1 X0,b D D 1 D y 1 + u X0,b D y 1 + u 3 X0,b D 1 + D y 1 + u j X0,b. (1.16)

6 6 A. GRÜNROCK EJDE-009/8. Combining the dual version (1.1) of the bilinear estimate with Hölder s inequality, (1.11) and Sobolev embeddings we obtain for b 1 < 1/, b > 1/, u j X0,b1 u 3 X0,b u 1 u L u j L D 1 u 1 X0,b u X0,b D 1/ u 3 X0,b D 1 + D y 1 + u j X0,b. 3. Bilinear interpolation involving u and u 3 gives u j X0,b D 1 u 1 X0,b D 1 + θ 4 + D 1 D y θ + u X0,b D 1 + θ 4 + D 1 (1.17) D y θ + u 3 X0,b D 1 + D y 1 + u j X0,b, where 0 < θ 1 and b = θb 0 + (1 θ)b 1. Now symmetrization via (l 1)-linear interpolation among u,..., u l yields u j X0,b D 1 u 1 X0,b j D α1+ D y α+ D 1 D y α3+ u j X0,b, (1.18) with b, b as before, α 1 = 1 + θ (l 1), α = l 3 (l 1), and α 3 = θ l 1. Using η 1 η, we may replace α + by α in (1.18). Now, for given s 1 > 1 and ε > 0 we choose θ close enough to zero, so that α 1 < s 1 and α 3 < ε, and b 0 (respectively b 1 ) close enough to 1 4 (respectively to 1 ), so that b > 1. Then, assuming by symmetry that u 1 has the largest frequency with respect to the -variable (i. e. 1,...,l ), we obtain u j Xs1 +s +ε,0,0;b u 1 Xs1 +s u +ε,0,0;b j. (1.19) Xs1,s,ε;b j l 4. The same upper bound holds for u j Xs1,s, if η (or even,ε;b if η ), where (respectively η ) are the frequencies in (respectively in y) of the whole product. In the case where 1 - assuming ε min (s 1, 1) and b sufficiently close to 1, the estimate u j Xs1,s,ε;b u j Xs1,s (1.0),ε;b is easily derived by a combination of the standard Strichartz type estimate (1.10) and Sobolev embeddings. So we may henceforth assume 1, η 1, and 1 η 1. l One last simple observation concerning the estimation of u j Xs1,s :,ε;b If we assume in addition to 1,...,l that u 1 has a large frequency with respect to y; i.e., η η 1, then from (1.18) we also obtain (1.0).

7 EJDE-009/8 CAUCHY-PROBLEM FOR GKP-II EQUATIONS 7 5. It remains to estimate l u j Xs1,s,ε;b in the case where 1,...,l (symmetry assumption as before) and η 1 η. By symmetry among u,...,l we may assume in addition that η η 1,3,...,l and hence that, because otherwise previous arguments apply with u 1 and u interchanged. For this distribution of frequencies the symbol of the Fourier multiplier M(u 1 u 3 u l, u ) becomes η η 1/ η 1/ η 1/. Now let P (u 1,..., u l ) denote the projection in Fourier space on { η η 1,3,...,l } { 1 }. Then by (1.15) we obtain for b > 1, b 1 < 1 Dy 1/ P (u 1,..., u l ) X0,b1 D 1/ u X0,b u 1 u 3 u l bl L D 1/ u X0,b D 1 + (u 1 u 3 ) L D 0+ u j bl L u 1 X0,b D 1 + u X0,b D 1 + u 3 X0,b D 1 + D y 1 + u j X0,b, where besides Sobolev type inequalities we have used (1.11) in the last step. Interpolation with (1.17) gives for 0 λ 1 D λ D 1 λ y P (u 1,..., u l ) X0,b1 D λ + u 1 X0,b D 1 + u X0,b D 1 + u 3 X0,b D 1 + D y 1 + u j X0,b. Yet another interpolation, now with (1.16), gives for 0 < θ 1, b = θb 0 + (1 θ)b 1, s = 1 (λ(1 θ) + θ) and s y = 1 (1 θ)(1 λ) D s D sy y P (u 1,..., u l ) X0,b D s+ u 1 X0,b D 1 + θ 4 + D 1 D 1 + D y 1 + u j X0,b. D y θ + u X0,b D 1 + θ 4 + D 1 D y θ + u 3 X0,b The net step is to equidistribute the D y s on u,..., u l. Here we must be careful, because the symmetry between u and u 3,..., u l was broen. But since u has the largest y-frequency we may first shift a D y l 3 (l 1) + onto u and then interpolate among u 3,..., u l in order to obtain D s Dy sy P (u 1,..., u l ) X0,b D s+ u 1 X0,b D β1+ D y β+ D 1 D y β3+ u j X0,b, (1.1) j l 3 where β 1 = 1 + θ 4, β = (l 1) and β 3 = θ. Again we may replace β + by β. Now (1.1) is applied to D s1 D y s D 1 D y ε P (u 1,..., u l ), where we can shift the D s1 partly from the product to u 1 and the D y s partly to u. Moreover, since and η η we have 1 η 1 η, so that a D 1 D y ε θ may be

8 8 A. GRÜNROCK EJDE-009/8 thrown from the product onto u. The result is P (u 1,..., u l ) Xs1,s,ε;b D s1+1 θ s+ j 3 u 1 X0,b D β1+ D y β+s sy+θ D 1 D y ε θ+β3+ u X0,b D β1+ D y β D 1 D y β3+ u j X0,b. Here β s and by choosing θ < min (ε, 3(l 1), s 1 1 ) and λ such that s y = β +θ we can achieve that s θ s < s 1 + s + ε, β + s s y + θ s, ε θ + β 3 < ε, as well as β 1 < s 1, β 3 < ε. This gives as desired. P (u 1,..., u l ) Xs1,s,ε;b u 1 Xs1 +s +ε,0,0;b j u j Xs1,s,ε;b References [1] Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, GAFA 3 (1993), and 09-6 [] Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy-Problem for the Zaharov-System, J. of Functional Analysis 151 (1997), [3] Grünroc, A., Panthee, M., Silva, J.: On KP-II type equations on cylinders. Ann. I. H. Poincare - AN (009), doi: /j.anihpc [4] Grünroc, A.: Bilinear space-time estimates for linearised KP-type equations on the threedimensional torus with applications. J. Math. Anal. Appl. 357 (009) [5] Hadac, M.: On the local well-posedness of the Kadomtsev-Petviashvili II equation. Thesis, Universität Dortmund, 007 [6] Hadac, M.: Well-posedness for the Kadomtsev-Petviashvili equation (KPII) and generalisations, Trans. Amer. Math. Soc., S (08) , 008 [7] Hayashi, N., Naumin, P. I., and Saut, J.-C.: Asymptotocs for large time of global solutions to the generalized Kadomtsev-Petviashvili equation, Commun. Math. Phys. 01, (1999) [8] Iório, R. J., and Nunes, W. V. L.: On equations of KP-type, Proc. Royal Soc. Edin., 18A (1998), [9] Isaza, P., Mejia, J.: Local and global Cauchy problems for the Kadomtsev-Petviashvili (KPII) equation in Sobolev spaces of negative indices, Comm. Partial Differential Equations, 6 (001), [10] Isaza, P., Mejia, J., and Stallbohm, V.: Regularizing effects for the linearized Kadomtsev- Petviashvili (KP) equation, Revista Colombiana de Matemáticas 31, (1997) [11] Kenig, C. E., Ponce, G., Vega, L.: Wellposedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, CPAM 46 (1993), [1] Kenig, C. E., Ponce, G., Vega, L.: Quadratic forms for the 1 - D semilinear Schrödinger equation, Transactions of the AMS 348 (1996), [13] Kenig, C. E., Ziesler, S. N.: Local well posedness for modified Kadomstev-Petviashvili equations. Differential Integral Equations 18 (005), no. 10, [14] Kenig, C. E., Ziesler, S. N.: Maimal function estimates with applications to a modified Kadomstev-Petviashvili equation Commun. Pure Appl. Anal. 4 (005), no. 1, [15] Saut, J.-C.: Remars on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J. 4 (1993), no. 3, [16] Taaoa, H.: Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Differential Equations, 5 (10-1), , 000 [17] Taaoa, H., Tzvetov, N.: On the local regularity of the Kadomtsev-Petviashvili-II equation, IMRN (001), no.,

9 EJDE-009/8 CAUCHY-PROBLEM FOR GKP-II EQUATIONS 9 [18] Tzvetov, N.: Global low regularity solutions for Kadomtsev-Petviashvili equation, Differential Integral Equations, 13 (10-1), , 000 Ael Grünroc Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut Beringstraße 1, Bonn, Germany address: gruenroc@math.uni-bonn.de