����������� ������ 49 (1997), 273–278

UDK 517.956.3

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GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR IN TIME

OF SMALL SOLUTIONS TO THE ELLIPTIC-HYPERBOLIC DAVEY-STEWARTSON SYSTEM

Nakao Hayashi and Hitoshi Hirata

We study the initial value problem for the Davey-Stewartson systems

\{ \begin{matrix}i∂_{t}u + c_{0}∂_{x_{1}}^{2}u + ∂_{x_{2}}^{2}u = c_{1}|u|^{2}u + c_{2}u∂_{x_{1}}φ, (x, t) ∈ R^{3}, \\ ∂_{x_{1}}^{2}φ + c_{3}∂_{x_{2}}^{2}φ = ∂_{x_{1}}|u|^{2}, \\ u(x, 0) = ϕ(x), \end{matrix}  (1)

where c_{0}, c_{3} ∈ R, c_{1}, c_{2} ∈ C, u is a complex valued function and φ is a real valued function. The initial data ϕ is C-valued function on R^{n}, and usually it belongs to some kind of Sobolev type spaces. The systems (1) for c_{3} > 0 were derived by Davey and Stewartson [4] and model the evolution equation of twodimensional long waves over finite depth liquid. -Dor - devic-Redekopp [5] showed that the parameter c_{3} can become negative when capillary effects are significant. When (c_{0}, c_{1}, c_{2}, c_{3}) = (1, −1, 2, −1), (−1, −2, 1, 1) or (−1, 2, −1, 1) the system (1) is referred as the DSI, DSII defocusing and DSII focusing respectively in the inverse scattering literature. In [7], Ghidaglia and Saut classified (1) as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respective sign of (c_{0}, c_{3}) : (+, +), (+, −), (−, +) and (−, −). For the elliptic-elliptic and hyperbolic-elliptic cases, local and global properties of solutions were studied in [7] in the usual Sobolev spaces L^{2}, H^{1} and H^{2}. In this paper we consider the elliptic-hyperbolic case. In this case after a rotation in the x_{1}x_{2}-plane and rescaling, the system (1) can be written as

\{ \begin{matrix}i∂_{t}u + Δu = d_{1}|u|^{2}u + d_{2}u∂_{x_{1}}φ + d_{3}u∂_{x_{2}}φ, \\ ∂_{x_{1}}∂_{x_{2}}φ = d_{4}∂_{x_{1}}|u|^{2} + d_{5}∂_{x_{2}}|u|^{2}, \end{matrix}  (2)

AMS Subject Classification: 35 J 45, 35 L 45

Communicated at the 4th Symposium on Mathematical Analysis and Its Applications, Aran - delovac 1997

where Δ = ∂_{x_{1}}^{2} + ∂_{x_{2}}^{2}, d_{1}, ⋯, d_{5} are arbitrary constants. In order to solve the system of equations, one has to assume that φ(·) satisfies the radiation condition, namely, we assume that for given functions φ_{1} and φ_{2}

lim_{x_{2}→∞} φ(x_{1}, x_{2}, t) = φ_{1}(x_{1}, t) and lim_{x_{1}→∞} φ(x_{1}, x_{2}, t) = φ_{2}(x_{2}, t).  (3)

Under the radiation condition (3), the system (2) can be written as

i∂_{t}u + Δu = d_{1}|u|^{2}u + d_{2}u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}(x_{1}, x_{2}′, t)dx_{2}′

+ d_{3}u ∫_{x_{1}}^{∞} ∂_{x_{2}}|u|^{2}(x_{1}′, x_{2}, t)dx_{1}′ + d_{4}u∂_{x_{1}}φ_{1} + d_{5}u∂_{x_{2}}φ_{2} (4)

with the initial condition u(x, 0) = ϕ(x). In what follows we consider the equation (4).

In this paper we use the following notations.

Notations. We define the weighted Sobolev space as follows

H^{m,l} = {f ∈ L^{2}; ∥(1 − ∂_{x_{1}}^{2} − ∂_{x_{2}}^{2})^{m/2}(1 + x_{1}^{2} + x_{2}^{2})^{l/2}f∥ < ∞},

H^{m,l}(R_{x_{j}}) = {f ∈ L^{2}(R_{x_{j}}); ∥(1 − ∂_{x_{j}}^{2})^{m/2}(1 + x_{j}^{2})^{l/2}f∥_{L^{2}(R_{j})} < ∞},

where ∥ · ∥ denotes the usual L^{2} norm. We let ∂ = (∂_{x_{1}}, ∂_{x_{2}}), J = (J_{x_{1}}, J_{x_{2}}), J_{x_{j}} = x_{j} + 2it∂_{x_{j}}. For simplicity we write L_{x_{j}}^{p} = L^{p}(R_{x_{j}}), L_{x_{1}}^{p}L_{x_{2}}^{q} = L^{p}(R_{x_{1}}; L^{q}(R_{x_{2}})), H_{x_{j}}^{m,l} = H^{m,l}(R_{x_{j}}), ∥ · ∥_{X^{m,l}(t)} = ∑_{|α|≤m} ∥∂^{α} · ∥ + ∑_{|α|≤l} ∥J^{α} · ∥, where α = (α_{1}, α_{2}), |α| = α_{1} + α_{2}, α_{1}, α_{2} ∈ N ∪ {0}.

Local existence of small solutions to (4) was shown when the initial function is in H^{m,l} in [12] for H^{12,0} ∩ H^{0,6}, [8] for H^{m,0} ∩ H^{0,l}, (m, l > 1), [1] for H^{m,0}, (m is sufficiently large integer) and [9] for H^{m,0}, (m ≥ 5/2). Furthermore in [10] without smallness condition on the data local existence of solutions was proved in the analytic function space which consists of real analytic functions. Global existence of small solutions to (4) was also given in [10] when the data are real analytic and satisfy the exponential decay condition.

Recently, H.Chihara [1] established the global existence of small solutions to (4) in higher order Sobolev spaces. Our purpose in this paper is to prove the global existence of small solutions to (4) in the usual weighted Sobolev spaces H^{3,0} ∩H^{0,3}, which is considered as lower order Sobolev class compared to one used in [1], by the calculus of commutator of operators. We shall prove the following.

Theorem 1. Let ϕ ∈ H^{3,0} ∩ H^{0,3}, ∂_{x_{1}}^{j+1}φ_{1} ∈ C(R; L_{x_{1}}^{∞}), ∂_{x_{2}}^{j+1}φ_{2} ∈ C(R; L_{x_{2}}^{∞}), (0 ≤ j ≤ 3), ϵ_{3} and δ_{3} be sufficiently small, where

ϵ_{m} = sup_{t∈R} ∑_{0≤j≤m}(1 + t)^{1+a}(∥(t∂_{x_{1}})^{j}∂_{x_{1}}φ_{1}(t)∥_{L_{x_{1}}^{∞}} + ∥∂_{x_{1}}^{j+1}φ_{1}(t)∥_{L_{x_{1}}^{∞}}
+ ∥(t∂_{x_{2}})^{j}∂_{x_{2}}φ_{2}(t)∥_{L_{x_{2}}^{∞}} + ∥∂_{x_{2}}^{j+1}φ_{2}(t)∥_{L_{x_{2}}^{∞}} ), a > 0,
δ_{m} ≥( ∑_{|α|+|β|≤m} ∥∂_{x_{1}}^{α_{1}}∂_{x_{2}}^{α_{2}}x_{1}^{β_{1}}x_{2}^{β_{2}}ϕ∥^{2})^{1/2}.

Then there exists a unique global solution u of (1.4) such that

u ∈ L_{local}^{∞}(R; H^{3,0} ∩ H^{0,3}) ∩ C(R; H^{2,0} ∩ H^{0,2}),  (5)
sup_{t∈R}( ∑_{|α|+|β|≤2} ∥∂^{α}J^{β}u(t)∥ + ∑_{|α|+|β|≤3}(1 + t)^{−Cδ_{3}}∥∂^{α}J^{β}u(t)∥) ≤ 4δ_{3}. (6)

Corollary 2. Let u be the solution constructed in Theorem 1. Then we have

∥u(t)∥_{L^{∞}} ≤ C(1 + |t|)^{−1}(∥ϕ∥_{H^{3,0}} + ∥ϕ∥_{H^{0,3}}).

Moreover, for any ϕ ∈ H^{3,0} ∩ H^{0,3} there exist u^{±} such that

∥u(t) − U(t)u^{±}∥_{H^{2,0}} → 0 as t → ±∞,

where U(t) = e^{it(∂_{x_{1}}^{2} +∂_{x_{2}}^{2} )}.

The rate of decay obtained in Corollary 2 is the same as that of solutions to linear Schrödinger equations. Time decay of solutions for the Davey-Stewartson systems (1) was obtained in [3], [7] when (c_{0}, c_{3}) = (+, +) and (c_{0}, c_{3}) = (−, +) and in [10] when (c_{0}, c_{3}) = (+, −) and (c_{0}, c_{3}) = (−, −) under exponential decay conditions on the data.

The strategy of the proof of Theorem 1. For simplicity we consider the following equation

i∂_{t}u + Δu = u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}dx_{2}′,

which have only main nonlinear term. We use following operators K_{x_{1}} and K_{x_{2}}, where

K_{x_{1}} = K_{x_{1}}(v) = ∑_{m=0}^{∞} \frac{A^{m}}{m!} (∫_{−∞}^{x_{1}} ∥v(t, x_{1}′)∥_{L_{x_{2}}^{2}}^{2} dx_{1}′ \frac{D_{x_{1}}}{⟨D_{x_{1}}⟩})^{m}

and

K_{x_{2}} = K_{x_{2}}(v) = ∑_{m=0}^{∞} \frac{A^{m}}{m!} (∫_{−∞}^{x_{2}} ∥v(t, x_{2}′)∥_{L_{x_{1}}^{2}}^{2} dx_{2}′ \frac{D_{x_{2}}}{⟨D_{x_{2}}⟩})^{m},

where ⟨D_{x_{j}}⟩ = (1 + D_{x_{j}}^{2})^{1/2}.

We use Coifman and Meyer’s result ([2], p.154):

Theorem A. Let σ ∈ C^{∞}(R^{m} × R^{m} \ (0, 0)) satisfy

|∂_{ξ}^{α}∂_{η}^{β}σ(ξ, η)| ≤ C_{α,β}(|ξ| + |η|)^{−|α|−|β|}

for (ξ, η) ≠ (0, 0) and any α, β ∈ (N)^{m}. If σ(D) denotes the bilinear operator

σ(D)(a, h)(x) = ∫ ∫ e^{i(x,ξ+η)}σ(ξ, η)â(ξ)ĥ(η)dξdη,

then ∥σ(D)(a, h)∥_{L^{2}(R^{m})} ≤ C∥a∥_{L^{∞}(R^{m})}∥h∥_{L^{2}(R^{m})}.

By virtue of this theorem, we obtain the following commutator estimate.

Lemma 3.

∥[⟨D_{x_{1}}⟩^{1/2}, f]g∥_{L_{x_{1}}^{2}} + ∥[⟨D_{x_{1}}⟩, f]g∥_{L_{x_{1}}^{2}} ≤ C∥⟨D_{x_{1}}⟩f∥_{L_{x_{1}}^{∞}} ∥g∥_{L_{x_{1}}^{2}} .

So, taking A^{2} = 1/δ_{3} (for the definition of δ_{3}, see Theorem 1) and using this lemma, we have

\frac{1}{2} \frac{d}{dt} ∑_{|α|+|β|≤3}(∥K_{x_{1}}∂^{α}J^{β}u(t)∥^{2} + ∥K_{x_{2}}∂^{α}J^{β}u(t)∥^{2})

+ \frac{1}{4δ_{3}^{1/2}} ∑_{|α|+|β|≤3}(∥∥u(t)∥_{L_{x_{2}}^{2}} ∥⟨D_{x_{1}}⟩^{1/2}K_{x_{1}}∂^{α}J^{β}u(t)∥_{L_{x_{2}}^{2}} ∥_{L_{x_{1}}^{2}}^{2}

+ ∥∥u(t)∥_{L_{x_{1}}^{2}} ∥⟨D_{x_{2}}⟩^{1/2}K_{x_{2}}∂^{α}J^{β}u(t)∥_{L_{x_{1}}^{2}} ∥_{L_{x_{2}}^{2}}^{2} )

≤ C(1 + A)^{2}(1 + t)^{−1}∥u(t)∥_{X^{2,2}(t)}^{2}(1 + ∥u(t)∥_{X^{2,2}(t)}^{2})∥u(t)∥_{X^{3,3}(t)}^{2}

+ ∑_{|α|+|β|≤3}(|Im(K_{x_{1}}∂^{α}J^{β}u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}dx_{2}′, K_{x_{1}}∂^{α}J^{β}u)|

+ |Im(K_{x_{2}}∂^{α}J^{β}u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}dx_{2}′, K_{x_{2}}∂^{α}J^{β}u)|).  (7)

The second term of the left hand side of (7) means smoothing properties of solutions to the equation. So we have to estimate the term

∑_{|α|+|β|≤3}(|Im(K_{x_{1}}∂^{α}J^{β}u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}dx_{2}′, K_{x_{1}}∂^{α}J^{β}u)|

+ |Im(K_{x_{2}}∂^{α}J^{β}u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}dx_{2}′, K_{x_{2}}∂^{α}J^{β}u)|).

For this purpose, we pay attention to the special structure of the nonlinear term

u ∫_{x_{2}}^{∞} ∂_{x_{1}}|u|^{2}dx_{2}′ = u\frac{1}{2it} ∫_{x_{2}}^{∞} ūJ_{x_{1}}u − u\overline{J_{x_{1}}u}dx_{2}′.  (8)

This deformation shows this nonlinear term has own time decay in some sence. Using this structure, we can estimate as following,

\frac{1}{2} \frac{d}{dt} ∑_{|α|+|β|≤3}(∥K_{x_{1}}∂^{α}J^{β}u(t)∥^{2} + ∥K_{x_{2}}∂^{α}J^{β}u(t)∥^{2})

+ (\frac{1}{4δ_{3}^{1/2}} − Ce^{Cδ_{3}}) ∑_{|α|+|β|≤3}(∥∥u(t)∥_{L_{x_{2}}^{2}} ∥⟨D_{x_{1}}⟩^{1/2}K_{x_{1}}∂^{α}J^{β}u(t)∥_{L_{x_{2}}^{2}} ∥_{L_{x_{1}}^{2}}^{2}

+ ∥∥u(t)∥_{L_{x_{1}}^{2}} ∥⟨D_{x_{2}}⟩^{1/2}K_{x_{2}}∂^{α}J^{β}u(t)∥_{L_{x_{1}}^{2}} ∥_{L_{x_{2}}^{2}}^{2} ) ≤ C(1 + t)^{−1}δ_{3}∥u(t)∥_{X^{3,3}(t)}^{2} (9) provided that δ_{3} is sufficiently small and

sup_{−T ≤t≤T} ∥u(t)∥_{X^{2,2}(t)}^{2} ≤ 4δ_{3}^{2},  (10)
sup_{−T ≤t≤T}(1 + |t|)^{−Cδ_{3}}∥u(t)∥_{X^{3,3}(t)}^{2} ≤ 4δ_{3}^{2}  (11)

for some time T > 0. We choose δ_{3} satisfying

\frac{1}{4δ_{3}^{1/2}} − Ce^{Cδ_{3}} ≥ 0.

Then we have

∥u(t)∥_{X^{3,3}(t)}^{2} ≤ e^{Cδ_{3}}δ_{3}^{2} + Cδ_{3} ∫_{0}^{t}(1 + s)^{−1}∥u(s)∥_{X^{3,3}(t)}^{2}ds.  (12)

Thus (9) shows that the nonliear term is controlled by the second term of the left hand side of (7) and the right hand side of (9). Global existence theorem is obtained by showing that (10) and (11) hold for any T. In order to prove (10) and (11) for any T > 0 we need (12) and the following inequality

∥u(t)∥_{X^{2,2}(t)}^{2} ≤ e^{Cδ_{3}}δ_{3}^{2} + Cδ_{3} ∫_{0}^{t}(1 + s)^{−1−2Cδ_{3}}∥u(s)∥_{X^{3,3}(t)}^{2}ds.  (13)

The inequality (13) is obtained by the structure of nonlinear term (8) again.

Theorem 1 is obtained by applying the Gronwall inequality to (12) and (13). It seems to be difficult to get the inequality (12) through the methods used in [8], [9] because nonlinear terms are not taken into account to derive smoothing properties of solutions in [8], [9]. On the other hand the operators K_{x_{1}} and K_{x_{2}} are made based on the nonlocal nonlinear terms (the second and the third terms on the right hand side of (4)). The similar operators as those of K_{x_{1}} and K_{x_{2}} have been used in [1] to obtain Theorem 0.1 and the local existence theorem of small solutions to (4) in the usual order Sobolev space.

Remark 1. We cannot apply above method to hyperbolic-hyperbolic Davey-Stewartson system. In fact, if we estimate similarly as above, we have

\frac{1}{2} \frac{d}{dt} ∑_{|α|+|β|≤3}(∥K_{x_{1}}∂^{α}J^{β}u(t)∥^{2} + ∥K_{x_{2}}∂^{α}J^{β}u(t)∥^{2})

+ \frac{1}{4δ_{3}^{1/2}} ∑_{|α|+|β|≤3}(∥∥u(t)∥_{L_{x_{2}}^{2}} ∥⟨D_{x_{1}}⟩^{1/2}K̃_{x_{1}}∂^{α}J^{β}u(t)∥_{L_{x_{2}}^{2}} ∥_{L_{x_{1}}^{2}}^{2}

+ ∥∥u(t)∥_{L_{x_{1}}^{2}} ∥⟨D_{x_{2}}⟩^{1/2}K̃_{x_{2}}∂^{α}J^{β}u(t)∥_{L_{x_{1}}^{2}} ∥_{L_{x_{2}}^{2}}^{2} )

≤ C(1 + A)^{2}(1 + t)^{−1}∥u(t)∥_{X^{2,2}(t)}^{2}(1 + ∥u(t)∥_{X^{2,2}(t)}^{2})∥u(t)∥_{X^{3,3}(t)}^{2}

+ Ce^{Cδ_{3}} ∑_{|α|+|β|≤3}∥∥u(t)∥_{L_{x_{2}}^{2}} ∥⟨D_{x_{1}}⟩^{1/2}K̃_{x_{1}}∂^{α}J^{β}u(t)∥_{L_{x_{2}}^{2}} ∥_{L_{x_{1}}^{2}}^{2}  (14)

under the condition (10) and (11), where

K̃_{x_{1}} = ∑_{m=0}^{∞} \frac{A^{m}}{m!} (∫_{−∞}^{x_{2}} ∥v(t, x_{2}′)∥_{L_{x_{1}}^{2}}^{2} dx_{2}′ \frac{D_{x_{1}}}{⟨D_{x_{1}}⟩})^{m} = e^{A ∫_{−∞}^{x_{2}} ∥v(t,x_{2}′)∥_{L{2 \atop x_{1}}}^{2} dx_{2}′ \frac{Dx_{1}}{⟨Dx_{1} ⟩}}

and

K̃_{x_{2}} = ∑_{m=0}^{∞} \frac{A^{m}}{m!} (∫_{−∞}^{x_{1}} ∥v(t, x′_{1})∥_{L_{x_{2}}^{2}}^{2} dx_{1}′ \frac{D_{x_{2}}}{⟨D_{x_{2}}⟩})^{m} = e^{A ∫_{−∞}^{x_{1}} ∥v(t,x′_{1})∥_{L{2 \atop x_{2}}}^{2} dx_{1}′ \frac{Dx_{2}}{⟨Dx_{2} ⟩}} ,

but we can easy to see that the last term of the right-hand side of (14) cannot be controlled by the second term of the left-hand side of (14).

Remark 2. For Davey-Stewartson systems, we can define formally the energy similar as the usual nonlinear Schrödinger equation if c_{1}, c_{2} ∈ R. But unfortunately, this energy is not conserved in elliptic-hyperbolic and hyperbolic-hyperbolic cases. So, we cannot use usual H^{1} a-priori estimate by energy. This is one reason that the global existence theorem of this system is difficult.

REFERENCES

[1] H. Chihara, The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation, preprint (1995).

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[8] N. Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems, Ann. Inst. Henri Poincaré, Physique théorique (1996), to appear.

[9] N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev spaces, Proc. Royal Soc. of Edinburgh Sec. A (1997), to appear.

[10] N. Hayashi and J. C. Saut, Global existence of small solutions to the Davey-Stewartson and the Ishimori systems, Diff. Integral Eqs. 8 (1995), 1657–1675.

[11] H. Kumano-go, Pseudo-Differential Operators, The MIT press, 1974.

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(received 22.09.1997., in revised form 19.02.1998.)

Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan E-mail: nhayashi@rs.kagu.sut.ac.jp Department of Mathematics, Sophia University, 7-1, Kioicho, Chiyoda-ku, Tokyo 102, Japan E-mail: h-hirata@mm.sophia.ac.jp