Regularity of random attractors for stochastic reaction diffusion equations (bounded domain)

Recently, the long time behavior of stochastic reaction diffusion equations (RDE) is considered by many mathematicians. The equation writes

\begin{cases}du + (-\Delta u + f(u) + \lambda u)dt = g(x)dt + d\omega,\\u|_{\partial\Omega} = 0,\\ u(0) = u_0 \end{cases}                               (1)

in a bounded domain \Omega\subset \mathbb R^N,  where \omega is a two-sided real-valued Weiner process on a probability space.

By putting u = v + z(\theta_t\omega) in which

z(\theta_t\omega) = -\lambda\int_{-\infty}^0e^{\lambda\tau}(\theta_t\omega)d\tau

we get a new deterministic equation with random terms

v_t - \Delta v + f(v+z(\theta_t\omega)) + \lambda v = g(x) + \Delta z(\theta_t\omega).                            (2)

The existence of a random attractor for RDE in L^2(\Omega) is well known, in L^p(\Omega), p>2 is solved recently by many authors. However, the regularity of the random attractor is not well understood. Compare to deterministic case (that says \omega = 0), where many results show that the attractor in fact lies in H^1(\Omega) or even H^2(\Omega), that should be should be something different in here.

Now, look back to the usual method of obtaining the regularity of attractors in deterministic case. One crucial step is usually to have

\|v_t\|_{L^2(\Omega)} \leq C

or roughly speaking, the time derivative of v is bounded (in some sense). In order to do this, because of high growth of the nonlinearity, one (almost) have to take the derivative in time of the equation [I said almost, since there may be a way around, but up to the best of my knowledge, I don’t know]. This couldn’t be done in stochastic case because of the presence of random noise.

Therefore, we have to adapt another method to deal with stochastic RDE. Fortunately, there is a hope coming from some papers of Chengkui Zhong, in which he showed that

a set B in a Hilbert space H is pre-compact if,

for any \epsilon >0 you can find a m – dimensional sub-space H_m such that

i) P(B) is bounded in H_m; and

ii) \|(Id - P)B\|_{H} \leq \epsilon

where P: H\rightarrow H_m is the canonical projection.

We will apply the above result to the RDE equation to prove that the random attractor \mathcal A is actually compact in H_0^1(\Omega). Of course, following the abstract result we set B = \mathcal A and the Hilbert space H = H_0^1(\Omega).

Motivated by the abstract result, inorder to prove that the random attractor \mathcal A is compact in H_0^1(\Omega), we prove that

for every \epsilon >0 there exists H_m\subset H_0^1(\Omega), dim H_m = m such that \|(I - P_m)u\| \leq \epsilon for all u\in \mathcal A.

Transform to the random dynamical system corresponding to the equation (2), we want to prove that

for every \epsilon >0, every \omega, there exist T>0 and m\in \mathbb N such that

\|(I-P_m)v(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_1 \leq \epsilon for all t\geq T.

(Here \|\cdot\|_1 is a H_0^1(\Omega) norm).

How to do that? We will use eigenvalues of - \Delta in H_0^1(\Omega) that says \lambda_1,\lambda_2,...,\lambda_n \rightarrow +\infty and its eigenfunctions e_1, e_2, \ldots, e_n,\ldots.

Denote by H_m = span\{e_1,e_2,\ldots,e_m\} and P_m is the canonical projector from H_0^1(\Omega) to H_m.

For any v\in H_0^1(\Omega), we decompose v = v_1 + v_2 where v_1 = P_m v and v_2 = (I-P_m)v.

Now we only have to prove that

for every \epsilon >0, every \omega, there exist T>0 and m\in \mathbb N such that

\|v_2(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\| \leq \epsilon for all t\geq T.

To do that, we rewrite here equation (2)

v_t - \Delta v + f(v+z(\theta_t\omega)) + \lambda v = g(x) + \Delta z(\theta_t\omega)                     (2)

Multiply (2) by -\Delta v_2, we arrive

\dfrac 12 \dfrac{d}{dt}\|v_2\|_1^2 + \|\Delta v_2\|^2 + (f(v+z(\theta_t\omega)),-\Delta v_2) = (g+\Delta z(\theta_t\omega),-\Delta v_2)

Using Cauchy’s inequality and \|\Delta v_2\|^2 \geq \lambda_{m+1}\|v_2\|_1^2, after standard calculations, we have

\dfrac{d}{dt}\|v_2\|_1^2 + \lambda_{m+1}\|v_2\|_1^2 \leq C(\|g\|^2 + \|f(v+z(\theta_t\omega))\|^2 + \|\Delta z(\theta_t\omega)\|^2).

Using Gronwall’s lemma and replace \omega by \theta_{-t}\omega, we get

\|v_2(t,\theta_{-t}\omega,v_0(\theta_{-t}\omega))\|_1^ 2 \leq Ce^{-\lambda_{m+1}t}\|v_0(\theta_{-t}\omega)\|_1^2

+ C\int_0^te^{-\lambda_{m+1}(t-s)}(\|g\|^2 + \|\Delta z(\theta_{s-t}\omega)\|^2)ds

+ C\int_0^te^{-\lambda_{m+1}(t-s)}\|f(v+z(\theta_{s-t}\omega))\|^2ds                                   (3)

The first term and the second on the right hand side of (3) are easy to proved that they tend to zero as t, m\rightarrow +\infty.

The only thing (and also the technical thing) that we have to deal with is the last term on the right hand side of (3).

This can be implied from

\int_0^t e^{-\lambda (t-s)}\|f(v+z(\theta_{s-t}\omega))\|^2ds <+\infty                     (4)

But how can we get (4)?

This is done in a work of W. Zhao and Y. Li

(L^2,L^p)-random attractors for stochastic reaction diffusion equation on unbounded domains


About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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