Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers Equations

大谷昌範 （東京理科大学大学院理学研究科数学専攻 Ｄ３）

1. Introduction

We are concerned with the initial value problem (IVP) for the generalized Benjamin-Ono-Burgers (gBOB) equations

\begin{cases}∂_{t}u + u∂_{x}u − ∂_{x}D_{x}^{1+a}u − ∂_{x}^{2}u = 0, & x, t ∈ ℝ, \\ u(x, 0) = u_{0}(x) ∈ H^{s}(ℝ), \end{cases}  (1)

where 0 ≤ a < 1 and D_{x}^{1+a} is the Fourier multiplier operator with symbol |ξ|^{1+a}. Note that when a = 0, (1) is the ordinary Benjamin-Ono-Burgers equation, and that when a = 1, (1) is the KdV-Burgers equation.

In [8], L. Molinet and F. Ribaud have proved that the KdV-Burgers equation is globally well-posed for s > −1, provided u is real-valued. On the other hand, it is known that the Burgers’ equation ∂_{t}u + u∂_{x}u − ∂_{x}^{2}u = 0, x, t ∈ ℝ is locally well-posed in H^{−1/2} for small initial data [1]. The exponent s = −1/2 is optimal since the uniqueness fails when s < −1/2 [3].

Here, we shall deal with the well-posedness issue of the gBOB equations (1) when 0 ≤ a < 1. Definition 1.1. For s, b ∈ ℝ, X^{s,b} denotes the completion of the Schwartz space S(ℝ^{2}) with respect to the norm

∥F ∥_{X^{s,b}} = (∫_{−∞}^{∞} ∫_{−∞}^{∞}⟨ i(τ − ξ|ξ|^{1+a}) + ξ^{2} ⟩^{2b}⟨ξ⟩^{2s}|F̂(ξ, τ)|^{2}dξdτ)^{1/2} ,  (2)

where ⟨·⟩ = (1 + | · |^{2})^{1/2}. Theorem 1.1. Let s > −(1 + a)/2 with 0 ≤ a < 1 and let b ∈ (1/2, 1). Then for any u_{0} ∈ H^{s}(ℝ) there exist T = T (∥u_{0}∥_{H^{s}}) > 0 and a unique solution u(t) of the IVP (1) satisfying

u(t) ∈ C([0, T], H^{s}(ℝ)),  (3)
u ∈ X^{s−(2b−1),b},  (4)
u∂_{x}u ∈ X^{s−(2b−1),b−1}, ∂_{t}u, ∂_{x}^{2}u ∈ X^{s−(2b+1),b−1}.  (5)

Moreover, the flow map u_{0} ↦ u(t) is locally Lipschitz from H^{s}(ℝ) to Z_{T} ≡ C([0, T], H^{s}(ℝ)) ∩ X^{s−(2b−1),b}. If the solution u is real-valued, u ∈ C((0, +∞), H^{∞}(ℝ)).

We solve the IVP by applying the contraction mapping principle to the following integral equation:

u(t) = ψ(t) [W (t)u_{0}(x) − \frac{χ_{ℝ_{+}(t)}}{2} ∫_{0}^{t} W (t − t′)∂_{x}(ψ_{T}^{2} (t′)u^{2}(t′))dt′] , t ∈ ℝ, (6)

where ψ is a cut-off function satisfying

ψ ∈ C_{0}^{∞}(ℝ), supp ψ ⊂ [−2, 2], ψ ≡ 1 on [−1, 1],  (7)

and ψ_{δ}(t) = ψ(t/δ), and χ_{ℝ_{+}}(t) is the characteristic function of [0, ∞).

W (t) is a linear operator defined on ℝ by setting

F_{x} (W (t)ϕ) (ξ) = exp[−ξ^{2}|t| + iξ|ξ|^{1+a}t ]F_{x}(ϕ)(ξ), t ∈ ℝ, ϕ ∈ S,  (8)

which is an extention of the evolution operator of the linear gBOB equation.

2. Bilinear estimates

The following is a key estimate for the proof of the theorem 1.1: Proposition 2.1. For s > −(1 + a)/2, there exists b > 1/2, C, μ, δ > 0 such that for any u, v ∈ X^{s,b} with compact support in [−T, T], we have

∥∂_{x}(uv)∥_{X^{s,b−1+δ}} ≤ CT^{μ}∥u∥_{X^{s,b}}∥v∥_{X^{s,b}}.  (9)

Idea of the proof of Proposition 2.1. By duality argument, it is equivalent to show that for any w ∈ X^{−s,1−b−δ} with ∥w∥_{X^{−s,1−b−δ}} ≤ 1,

|I| = |⟨∂_{x}(uv), w⟩| ≤ CT^{μ}∥u∥_{X^{s,b}}∥v∥_{X^{s,b}}∥w∥_{X^{−s,1−b−δ}}.  (10)

Putting

f̂(ξ, τ) = ⟨i(τ − ξ|ξ|^{1+a}) + ξ^{2}⟩^{b}⟨ξ⟩^{s}û(ξ, τ),
ĝ(ξ, τ) = ⟨i(τ − ξ|ξ|^{1+a}) + ξ^{2}⟩^{b}⟨ξ⟩^{s}v̂(ξ, τ)
and ĥ(ξ, τ) = ⟨i(τ − ξ|ξ|^{1+a}) + ξ^{2}⟩^{1−b−δ}⟨ξ⟩^{−s}ŵ(ξ, τ),

we see that (10) is equivalent to

|I| ≤ CT^{μ}∥f∥_{L_{x}^{2}L_{t}^{2}}∥g∥_{L_{x}^{2}L_{t}^{2}}∥h∥_{L_{x}^{2}L_{t}^{2}}.  (11)

And we can rewrite

I = ∫_{ℝ^{4}} \frac{ξ\overline{ĥ(ξ, τ)}⟨ξ⟩^{s}}{⟨iσ + ξ^{2}⟩^{1−b−δ}} \frac{ĝ(ξ_{1}, τ_{1})⟨ξ_{1}⟩^{−s}}{⟨iσ_{1} + ξ_{1}^{2}⟩^{b}} \frac{f̂(ξ − ξ_{1}, τ − τ_{1})⟨ξ − ξ_{1}⟩^{−s}}{⟨iσ_{2} + (ξ − ξ_{1})^{2}⟩^{b}} dξdτdξ_{1}dτ_{1}, (12)

where

σ = τ − ξ|ξ|^{1+a}, σ_{1} = τ_{1} − ξ_{1}|ξ_{1}|^{1+a}, σ_{2} = τ − τ_{1} − (ξ − ξ_{1})|ξ − ξ_{1}|^{1+a}. (13)

We divide the integral region ℝ^{4} into three parts; D_{1} = { (ξ, ξ_{1}, τ, τ_{1}) ∈ ℝ^{4} : |ξ| ≤ 2|ξ_{1}|, |ξ_{1}| ≥ 1, |ξ − ξ_{1}| ≥ 1 }, D_{2} = { (ξ, ξ_{1}, τ, τ_{1}) ∈ ℝ^{4} : |ξ| ≥ 2|ξ_{1}|, |ξ_{1}| ≥ 1, |ξ − ξ_{1}| ≥ 1 }. D_{3} = { (ξ, ξ_{1}, τ, τ_{1}) ∈ ℝ^{4} : |ξ_{1}| ≤ 1 or |ξ − ξ_{1}| ≤ 1 }. Making use of Schwarz inequality and the following algebraic relation, we derive the estimate (11).

The algebraic smoothing relation holds among σ, σ_{1} and σ_{2}:

max{|σ|, |σ_{1}|, |σ_{2}|} ≥ C|ξ_{1}|^{1+a}|ξ − ξ_{1}|,  (14)
or max{|σ|, |σ_{1}|, |σ_{2}|} ≥ C|ξ||ξ_{1}|^{a}|ξ − ξ_{1}|,  (15)
or max{|σ|, |σ_{1}|, |σ_{2}|} ≥ C|ξ_{1}||ξ − ξ_{1}|^{1+a},  (16)
or max{|σ|, |σ_{1}|, |σ_{2}|} ≥ C|ξ||ξ_{1}||ξ − ξ_{1}|^{a}.  (17)

Remark. When a = 1, it follows from σ_{1} + σ_{2} − σ = 3ξξ_{1}(ξ − ξ_{1}) that max{|σ|, |σ_{1}|, |σ_{2}|} ≥ |ξξ_{1}(ξ − ξ_{1})|. See [2], [5].

3. Linear estimates

Finally, we collect some linear estimates to establish the theorem 1.1. Proposition 3.1. Let s ∈ ℝ and b ∈ [0, 1]. There exists C > 0 such that

∥ψ(t)W (t)ϕ∥_{X^{s,b}} ≤ C∥ϕ∥_{H^{s+2b−1}}  (18)

for any ϕ ∈ H^{s+2b−1}(ℝ). Proposition 3.2. Let s ∈ ℝ and let b > 1/2. For δ ∈ (0, 1], we have

∥ψ_{δ}F ∥_{X^{s,b}} ≤ Cδ^{(1−2b)/2}∥F ∥_{X^{s,b}}.  (19)

Proposition 3.3. Let s ∈ ℝ and let b > 1/2. For 0 < δ < 1/2, there exists C_{δ} > 0 such that, for any v ∈ X^{s,b−1+δ},

∥χ_{ℝ_{+}}(t)ψ(t) ∫_{0}^{t} W (t − t′)v(t′)dt′∥_{X^{s,b}} ≤ C_{δ}∥v∥_{X^{s,b−1+δ}}.  (20)

Proposition 3.4. Let s ∈ ℝ, b ≥ 1/2 and δ > 0. For all f ∈ X^{s,b−1+δ},

t ↦ ∫_{0}^{t} W (t − t′)f(t′)dt′ ∈ C(ℝ_{+}, H^{s+2δ}(ℝ)).  (21)

Moreover, we have

∥χ_{ℝ_{+}}(t)ψ(t) ∫_{0}^{t} W (t − t′)f(t′)dt′∥_{L^{∞}(ℝ_{+},H^{s+2δ})} ≤ C∥f∥_{X^{s,b−1+δ}}.  (22)

Lemma 3.5. Let v be with compact support in time in [−T, T]. For any θ > 0, there exists μ = μ(θ) > 0 such that

∥F_{x,t}^{−1}(\frac{v̂(ξ, τ)}{⟨τ − ξ|ξ|^{1+a}⟩^{θ}} )∥_{L_{x,t}^{2}(ℝ^{2})} ≤ CT^{μ}∥v∥_{L_{x,t}^{2}(ℝ^{2})}.  (23)

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