Stabilisation by noise on the boundary with dynamical boundary conditions

Together with Klemens Fellner, Stefanie Sonner, and Do Duc Thuan, I have uploaded a paper on arXiv, which deals with stabilising effect of noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions.

Certain dissipative PDEs have either stable or unstable steady states depending on the magnitude of parameters. Consider for example the heat equation in a domain with Dirichlet boundary condition

$u_t - \Delta u - \beta u = 0, \quad u|_{\partial\Omega} = 0, \quad u(0) = u_0\in L^2(\Omega)$.

Multiplying the equation by $u$ in $L^2(\Omega)$ and using the Poincare inequality $\|\nabla u\|_{L^2}^2 \geq \lambda_1\|u\|_{L^2}^2$ we get

$\frac{d}{dt}\|u\|_{L^2}^2 + 2(\lambda_1 - \beta)\|u\|_{L^2}^2 \leq 0$.

Here $\lambda_1$ is the first eigenvalue of Laplace operator $-\Delta$ with homogeneous Dirichlet boundary condition.

By Gronwall’s inequality, one has

$\|u(t)\|_{L^2}^2 \leq e^{-2(\lambda_1 - \beta)t}\|u_0\|_{L^2}^2$

which means that if $\beta < \lambda_1$, then the zero steady state is exponentially stable.

When $\beta \geq \lambda_1$, this stability might be lost. (In fact, one can prove that if $\beta > \lambda_1$, then the zero steady state is unstable). The stability can be regained when, surprisingly, some multiplicative noise is added. More precisely, consider the stochastic equation

$du + (-\Delta u - \beta u)dt = \alpha u dW_t, \quad u|_{\partial \Omega} = 0$

where $W_t$ is a real-valued scalar Wiener process defined in an appropriate probability space, and $dW_t$ denotes the Ito differential. It was proved that the zero steady state is exponentially stable provided

$\beta < \lambda_1 + \frac{\alpha^2}{2}.$

• When $\beta < \lambda_1$ this condition is obviously satisfied, therefore the stability of zero steady state is preserved with the addition of noise.
• When $\beta > \lambda_1$, the zero steady state is exponentially stable for all $\alpha^2 \in (2(\beta - \lambda_1), +\infty)$.

Roughly speaking, the noise shifts all eigenvalues to the right with the distance $\alpha^2/2$, thus helps to stabilise the equation as soon as the first eigenvalue (which is now $\lambda_1 + \alpha^2/2$) is larger than $\beta$. Remark that the larger $\alpha^2$ is, the better the zero steady state can be stabilised. In particular, one can stabilise the equation with an infinite range of noise intensities.

This phenomenon of stabilisation by noise has been studied extensively in the last decades. However, it seems not known if one can also stabilise the equation using the noise only on the boundary. Our paper fills in this gap in the context of a nonlinear Chafee-Infante equation with dynamical boundary conditions. More precisely,

$du + (-\Delta u + u^3 - \beta u) = 0, \text{ in } \Omega, \quad du + (\partial_\nu u + \lambda u)dt = \alpha u dW_t, \text{ on }\partial\Omega$

Our analysis uses in a crucial way the Poincare-Trace inequality: for each $\theta>0$ there exists $C_\theta>0$ such that

$\|\nabla u\|_{L^2(\Omega)}^2 + \theta\|u\|_{L^2(\partial \Omega)}^2 \geq C_\theta \|u\|_{L^2(\Omega)}^2$

Here the dependence of $C_\theta$ on $\theta$ is of importance. Note that even in the limit $\theta \to \infty$, the constant $C_\theta$ remains bounded, for instance $C_\theta \leq \lambda_1$ (by choosing $u\in H_0^1(\Omega)$. An optimal expression of $C_\theta$ is, up to our knowledge, unknown.

Using this Poincare-Trace inequality, we show that one can stabilise the equation with a finite range of noise intensities $\alpha$This differs strongly from the case described above. It is an interesting open problem to decide whether this finite range is due to the boundary noise or merely a technical limitation. However, due to the upper bound of $C_\theta$, we conjecture it to be the latter case.

[4.5.2016] Fujita blow up phenomena and hair trigger effect: the role of dispersal tails

Today ArXiv will be this paper of Matthieu Alfaro.

Firstly, a side interesting fact is the term “hair trigger effect”, which sounds cool to me  🙂

It can be translated naively as “adding just a small hair could change everything”. More precisely, there is a steady state that if you add “just a little bit of something” to it, then the trajectory goes differently and eventually ends up at something totally different. Until here it does sound familiar, doesn’t it? [Yes, to me it’s nothing else but instability of the steady state (in some sense)]

Let’s come back to the paper. The author studied a nonlinear nonlocal evolution problem in $\mathbb R^N$ of the following form

$u_t = J * u - u + u^{1+p}$.

This problem was studied previously with the assumption of compact support of the kernel $J$.

This work extends the result to the case that $J$ can have unbounded support, but eventually decays at infinity. There two typical cases are:

1. $J$ decays exponentially: $J(\xi) \approx e^{-k|\xi|}$; or
2. $J$ decays algebraically (fat tails): $J(\xi) \approx |\xi|^{-\alpha}$

Depending on these kinds of decay, the author established the Fujita type blow up for the considered equation. That is, depending on the polynomial $p$ (combining with the decay rate of $J$), the solution can either blow up for any nontrivial initial data or blow up for large data and exists globally (extinct) for small initial data.

[3.5.2016] Global existence and Regularity results for strongly coupled nonregular parabolic systems via Iterative methods

Today post is not a new ArXiv article but instead referred from such an article. While reading the Introduction of this (today) ArXiv paper of Dung Le, I came to another old ArXiv paper of him which I’m going to write about (very shortly) now.

The author studied strongly coupled parabolic system of the form

$u_t = \mathrm{div}(A(u)Du) + f(u)$

with initial data and boundary condition, in which $u: \Omega\times R_+ \rightarrow \mathbb R^m$ where $\Omega$ is a bounded domain in $\mathbb R^n$.

The diffusion matrix $A(u)$ and nonlinearity $f(u)$ satisfy some natural conditions. For example, one can take $A(u)$ as a linear matrix satisfying the elliptic condition and $f(u)$ to be polynomial.

Assuming that $p_0 > n$ and initial data $u_0\in W^{1,p_0}(\Omega)$, it is well-known that the system has a unique local classical solution

$u\in C([0,T_{max}), W^{1,p_0}(\Omega))\cap C^{1,2}((0,T_{max})\times\overline{\Omega})$.

Moreover, the criterion for blow-up is: if $T_{max} <+\infty$ then

$\lim_{t\rightarrow T_{max}^{-}}\|u(t)\|_{W^{1,p_0}(\Omega)} = \infty$.

In other words, if we can prove that

$\lim_{t\rightarrow T_{max}^{-}}\|u(t)\|_{W^{1,p_0}(\Omega)} < \infty$                         (*)

then $T_{max} = \infty$, that is the global solution exists globally.

Note that since $p_0 >n$, thanks to the Morrey embedding $W^{1,p_0}(\Omega)\hookrightarrow C^{0,\gamma}(\Omega)$ the condition (*) implies that the solution is bounded in $L^{\infty}(\Omega)$. This fact is usually very hard to prove in general system due to the lack of comparison principle.

In this paper, the author succeeded in providing a weaker criterion for blow-up, or equivalently global existence of solution, that is the solutions needs only to be bounded in $BMO(\Omega)$-norm. To be more precise, the BMO-norm (Bounded Mean Oscillation) is defined as follow

$\|u\|_{BMO(\Omega)} = \|u\|_{L^1(\Omega)} + \sup_{x_0\in\Omega,R>0}\dfrac{1}{|\Omega(x_0,R)|}\int_{\Omega(x_0,R)}|u - u_{x_0,R}|dx$

where $\Omega(x_0,R) = \Omega \cap B(x_0,R)$ and $u_{x_0,R}$ is the mean value of $u$ over $\Omega(x_0,R)$.

Asumming that

$\|u(t)\|_{BMO(\Omega)} \leq C(t) \quad \forall t\in (0,T_{max})$

where $C$ is a continuous function on $(0,T_{max}]$, and for any $\varepsilon >0$ and $(x,t)\in \Omega\times (0,T_{max})$, there exists $R$ such that

$\|u(t)\|_{BMO(B_R(x))} < \varepsilon \quad \forall t\in (0,T_{max})$.

As a corrollary, if

$\lim_{t\rightarrow T_{max}^{-}}\|u(t)\|_{W^{1,n}(\Omega)} < \infty$

then the classical solution exists globally.

[29.4.2016] Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions

In this post, we talked about a paper concerning the, roughly speaking, blow-up profiles of solutions to the critical heat equation in large dimensions ($d\geq 5$). In the ArXiv paper of C. Collot, F. Merle and P. Raphael today, they study the same critical nonlinear heat equation

$u_t = \Delta u + |u|^{\frac{4}{d-2}}u\qquad x\in \mathbb R^d$

and classify the behaviour of solutions around the ground state solitary wave

$Q(x) = \dfrac{1}{\left(1+\dfrac{|x|^2}{d(d-2)}\right)^{(d-2)/2}}$

in the dimension $d\geq 7$.

Given the initial data $u_0$ close enough to the ground state $Q$, the results show that the solution of the heat equation could fall into one of the three scenarios:

(i) Convergence to the ground state: $\exists (\lambda,z)\in\mathbb R_+\times \mathbb R^d$ such that

$\lim\limits_{t\rightarrow +\infty}\left\|u(t,\cdot) - \dfrac{1}{\lambda^{(d-2)/2}}Q\left(\dfrac{\cdot - z}{\lambda}\right)\right\|_{\dot{H}^1} = 0$.

(ii) Decaying to zero:

$\lim\limits_{t\rightarrow+\infty}\|u(t,\cdot)\|_{H^1\cap L^{\infty}} = 0$.

(iii) Blow up in Type I:

$\|u(t,\cdot)\|_{L^{\infty}} \approx (T-t)^{-(d-2)/4}$.

[28.4.2016] Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces

Young-Pil Choi studied in this paper the existence and large time behaviour of solution to the following nonlinear Vlasov-Fokker-Planck equation

$\partial_t F + v\cdot \nabla_xF + \nabla_v\cdot((u_F - v)F) = \Delta_vF$

where $F = F(x,v,t), x, v\in \mathbb R^d, t\geq 0$ and

$u_F(x,t) = \dfrac{\int_{\mathbb R^d}vF(x,v,t)dv}{\int_{\mathbb R^d}F(x,v,t)dv}.$

Assuming the initial data $F_0$ close enough to the global Maxwellian

$M = M(v) = \dfrac{1}{(2\pi)^{d/2}}exp\left(-\dfrac{|v|^2}{2}\right)$

the author proved that the solution to the mentioned problem exists globally in the classical sense, and converges to the Maxwellian $M$ with an algebraic rate, i.e.

$\|f(t)\|_{H^s} \leq C(\|f_0\|_{H^s} + \|f_0\|_{L^2_v(L^1)})(1+t)^{-d/4}$

where

$F = M + \sqrt{M}f$.

Moreover, if the spatial is periodic, then the convergence is exponential.

[27.4.2016] Generalized entropy method for the renewal equation with measure data

In this paper of P. Gwiazda and E. Wiedemann, they apply the generalized entropy method, which is initated in a series of work of B. Perthame and collaborators, cf. this paper, to show the exponential convergence to equilibrium for the renewal equation with measure initial data.

$\partial_t n(t,x) + \partial_{x}n(t,x) = 0$ on $\mathbb R_+^2$

$n(t,x=0) = \int_0^{\infty}B(y)n(t,y)dy$

$n(t=0,x) = n_0(x)$

This equation has been extensively studied recently by many authors due to its application to biology.

The convergence to equilibrium by using relative entropy method for this problem was known for $L^1$-initial data. By denoting $N(x)$ and $\varphi(x)$ are the solutions to an eigenvalue problem and its dual, and defining

$h(t,x) = n(t,x) - N(x)\int_0^{\infty}n_0(y)\phi(y)dy$

with some suitable function $\phi$, then we have the large time behaviour of $n(t,x)$ as follows

$\int_{0}^{\infty}|h(t,x)|\varphi(x)dx \leq e^{-\mu(t-y_0)}\int_0^{\infty}|h(y_0,x)|\varphi(x)dx$.

This result was based on the so-called entropy method (see this paper for more details).

For measure initial data, the arguments applied to $L^1$-initial data seems not to be directly applicable. However, looking at the convergence result, we would expect similar results for measure data (with some suitable changes).

This is what was done in the paper of Gwiazda and Wiedemann. The main idea is to use recession function $f^{\infty}$ for a function $f$ defined as

$f^{\infty}(z) = \lim\limits_{s\rightarrow \infty}\dfrac{f(sz)}{s},\quad z\in \mathbb R^n-\{0\}$

provided that $f$ grows mostly linearly. By expoloiting these functions, the authors succeeded in choosing a convex function making the entropy method works in the case of measure initial data. Denote by

$g(t,x) = n(t,x) - N(x)\int_0^{\infty}\varphi(x)dn_0(x)dx$,

$\int_0^{\infty}\eta(x)d|g(t,x)| \leq e^{-\sigma(t-y_0)}\int_0^{\infty}\eta(x)d|g(0,x)|$

for some bounded function $\eta$ positive on $supp(\varphi)$.

[26.4.2016] Green’s function and infinite-time bubbling in the critical nonlinear heat equation

In this paper of Carmen Cortazar, Manuel del Pino, Monica Musso they studied the profile of solutions which blow-up in in-finite time of the critical nonlinear heat equation

$u_t - \Delta u = u^{\frac{n+2}{n-2}}$

in a bounded domain $\Omega\subset \mathbb R^n$ with $n\geq 5$.

The solutions are shown to have bubbling type of behaviour, that means, there exist $k$ points in $\Omega$ which are the only blow-up points of such a solution.

Moreover, such a solution can be constructed approximately by: let $q_1, q_2, \ldots, q_k$ be the bubble points, then

$u(x,t) \approx \sum\limits_{j=1}^{k}\alpha_n\left(\dfrac{\mu_j(t)}{\mu_j(t)^2 + |x- \xi_j(t)|^2}\right)^{(n-2)/2}$

where $\xi_j(t) \rightarrow q_j$ and $0<\mu_j(t) \rightarrow 0$ as $t\rightarrow +\infty$.