Random Attractors for the Stochastic Discrete Long Wave-Short Wave Resonance Equations

Definition 2.1. (Ω, ℱ, ℙ, (θ_t) _t∈ℝ) are called metric dynamical systems; if θ : ℝ × Ω → Ω is (𝔹(ℝ) × ℱ, ℱ)-measurable, θ₀ = 𝕀, θ_t+s = θ_t∘θ_s for all t, s ∈ ℝ, and θ_tℙ = ℙ for all t ∈ ℝ.

Definition 2.2. A stochastic process ϕ(t, ω) is called a continuous random dynamical system (RDS) over (Ω, ℱ, ℙ, (θ_t) _t∈ℝ) if ϕ is (𝔹(ℝ⁺) × ℱ × 𝔹(H), 𝔹(H))-measurable, and for all ω ∈ Ω

• (i)

  the mapping ϕ : ℝ⁺ × Ω × H → H is continuous;

• (ii)

  ϕ(0, ω) = 𝕀  on   H;

• (iii)

  ϕ(t + s, ω, χ) = ϕ(t, θ_sω, ϕ(s, ω, χ)) for all t, s ≥ 0 and χ ∈ H (cocycle property).

Definition 2.3. A random bounded set B(ω) ⊂ H is called tempered with respect to (θ_t) _t∈ℝ if for a.e. ω ∈ Ω and all ϵ > 0

where d(B) = sup _χ∈B∥χ∥_H.

Definition 2.4. A random set 𝕂(ω) is called an absorbing set in 𝔻 if for all B ∈ 𝔻 and a.e. ω ∈ Ω there exist t_B(ω) > 0 such that

Definition 2.5. A random set 𝔸(ω) is a random 𝔻-attractor for RDS ϕ if

• (i)

  𝔸(ω) is a random compact set, that is, ω → d(χ, 𝔸(ω)) is measurable for every χ ∈ H and 𝔸(ω) is compact for a.e. ω ∈ Ω;

• (ii)

  𝔸(ω) is strictly invariant, that is, ϕ(t, ω, 𝔸(ω)) = 𝔸(θ_tω),   for  all  t ≥ 0 and for a.e. ω ∈ Ω;

• (iii)

  𝔸(ω) attracts all sets in 𝔻, that is, for all B ∈ 𝔻 and a.e. ω ∈ Ω we have

  $$ ()
  where d(X, Y) = sup _χ∈Xinf _y∈Y∥χ−y∥_H,   X, Y ⊂ H.

Definition 2.6. Let ϕ be an RDS on Hilbert space H. ϕ is called asymptotically compact if, for any bounded sequence {χ_n} ⊂ H and t_n → ∞, the set $$ is precompact in H, for any ω ∈ Ω.

Proposition 2.7. Let 𝕂 ∈ 𝔻 be an absorbing set for an asymptotically compact continuous RDS ϕ. Then ϕ has a unique global random 𝔻-attractor

which is compact in H.

Remark 2.8. The special form of multiplicative noise in (2.5) is more suitable than the white noise “adW” and the additive noise “$$”, because it is more approximative to the perturbations of the short wave for this model.

Lemma 3.1. Suppose that f(t) = (f_n(t)) _n∈ℤ ∈ C_B(ℝ, ℓ²). Then, the solution of (1.4)–(1.6) satisfies

for all ω ∈ Ω and ∥f∥ = sup _t∈ℝ | f(t)|².

Proof. We write (1.4) in the form of vector as

Taking the imaginary part of the inner product of (3.2) with u, we obtain So we have By the Gronwall inequality, we get Thus, we derive (3.1).

Remark 3.2. For the general multiplicative noise, we can also choose a suitable process and a change of variable to convert the stochastic equations into deterministic equations.

Theorem 3.3. For any T > 0, (2.5)-(2.6) are well posed and admit a unique solution (u(t), v(t)) ∈ 𝕃²(Ω; C([0, T]; E)). Moreover, the solution of (2.5)-(2.6) depends continuously on the initial data (u₀, v₀).

Proof. By standard existence theorem for ODEs, it follows that (3.8) possess a local solution $$, where [0, T_max] is the maximal interval of existence of the solution of (3.8). Now, we prove that this local solution is a global solution. Let ω ∈ Ω; from (3.8) it follows that

By the Young inequality and (1.8), direct computation shows that

Combining the above inequalities with Lemma 3.1, we obtain

where C₀, C₁ are constants depending on α, β, γ, and ɛ is a sufficiently small positive number. By the Gaussian property of W¹ and W², (3.11) implied that (3.8) admit a global solution $$. The proof of the lemma is completed.

Theorem 3.4. System (2.5)-(2.6) generates a continuous random dynamical system (ϕ(t, θ_−tω)) _t≥0 over (Ω, ℱ, ℙ, (θ_t) _t∈ℝ).

Lemma 3.5. There exists a θ_t invariant set Ω′ ⊂ Ω of full ℙ measure and an absorbing random set 𝕂(ω), ω ∈ Ω′, for the random dynamical system (ϕ(t, θ_−tω)) _t≥0.

Proof. We use the estimates in Theorem 3.3. By (3.11), we have

where $$.

By the Gronwall inequality, we have

Replace ω by θ_−tω in the above inequality to construct the radius of the absorbing set and define Define Then, 𝕂(ω)≜𝕂(0, R(ω)) is a tempered ball by the property of W¹, W², and, for any B ∈ 𝔻,   ω ∈ Ω. Here, 𝔻 denotes the collection of all tempered random sets of Hilbert space H. The proof of the lemma is completed.

Lemma 3.6. Let (u₀, v₀) ∈ 𝕂(ω), the absorbing set given in Lemma 3.5. Then, for every ɛ > 0 and ℙ-a.e. ω ∈ Ω, there exist T(ɛ, ω) > 0 and N(ɛ, ω) > 0 such that the solution (u, v) of system (2.5)-(2.6) satisfies

Proof. Let η(x) ∈ C(ℝ⁺, [0,1]) be a cut-off function satisfying

and |η′(x)| ≤ η₀ (a positive constant).

Taking the inner product of (3.8) with $$ and $$, respectively, we get

We also use the estimates in Theorem 3.3. Similar to (3.11), it follows that

Replace ω by θ_−tω in (3.22). Then, we estimate each term on the right-hand of (3.22); it follows that

Since ∥zⁱ(ω)∥,   i = 1,2, are tempered and zⁱ(θ_tω),   i = 1,2, are continuous in t, there is a tempered function r(ω) > 0 such that Combining (3.23) with (3.24), there is a T₁(ɛ, ω) > T_k such that Next, we estimate Let T^* ≥ (1/C₀)ln (6C₁r(ω)/C₀ɛ) and N₁(ɛ, ω) be fixed positive constants. Then, for t > T^* + T_k and M > N₁(ɛ, ω), by the Lebesgue theorem, we have

Since f(t) ∈ C_B(ℝ, ℓ²) and g(t) ∈ C_B(ℝ, ℓ²), there exists N₂(ɛ, ω) such that for M > N₂(ɛ, ω)

Therefore, let

Then, for $$ and $$, we obtain

Direct computation shows that Therefore, we obtain The proof of the lemma is completed.

Lemma 3.7. The random dynamical systems (ϕ(t, θ_−tω)) _t≥0 are asymptotically compact.

Proof. We use the method of [25]. Let ω ∈ Ω. Consider a sequence (t_n) _n∈ℕ with t_n → ∞ as n → ∞. Since 𝕂(ω) is a bounded absorbing set, for large n, $$, where (u₀, v₀) ∈ 𝕂(ω). Then, there exist (u, v) ∈ E and a sequence (u_n, v_n) (denoted by itself) such that

Next, we show that the above weak convergence is actually strong convergence in E.

From Lemma 3.6, for any ɛ > 0, there exist positive constants N₃(ɛ, ω) and $$ such that, for $$,

Since (u, v) ∈ E, there exists N₄(ɛ, ω) > 0 such that

Let $$, then, from (3.33), there exists $$ such that, for $$,

By (3.34)–(3.36), we obtain that, for $$,

The proof of the lemma is completed.

Theorem 3.8. The random dynamical systems (ϕ(t, θ_−tω)) _t≥0 possess a global random attractor in E.