[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.

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[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.

 

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[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}.

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[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.

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[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. 

The renewal equation reads as

\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,

the main result reads as

\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).

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[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.

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Why study “pullback attractors”?

Pullback attractors for PDE systems was my first research project. Consider a non-autonomous (partial) differential equation, with t> \tau \in \mathbb R,

u'(t) = A(t)u(t) \qquad u(\tau) = u_{\tau}

with u_\tau \in X for a Banach space X. If the evolution equation has a unique solution in X for each u_\tau \in X, the it generates an evolution process U(t,\tau): X \rightarrow X where U(t,\tau)u_{\tau} is the solution of the equation at the time t with initial data u_\tau at the time \tau.

The pullback attractor for \{U(t,\tau)\} is a set of compact sets \{A(t)\}_{t\in\mathbb R} which is invariant, i.e. U(t,\tau)A(\tau) = A(t) and is pullback attracting all bounded sets, that is for any fixed time t\in \mathbb R,

\lim_{\tau \rightarrow -\infty}d_{H}(U(t,\tau)B, A(t)) = 0

for any bounded set B.

The main difference between the pullback attractor and the usual global attractor is that the pullback attractor attracts bounded set in a “pullback” sense, that is instead of sending the final time t to infinity, we send the initial time \tau to minus infinity. In other words, instead of starting at a initial time and going to the future, what we do with the pullback attractor is to fix the current time and go back to history.

I have been asked many times the question “What does that mean that you go back to the history, while the main idea of attractors to investigate the large time behavior or equivalently the future?“. To be honest, I did not know a convincing answer.

Luckily, after attending a talk of Stefanie Sonner today I now have in mind a good answer for that question.

Consider the simple non-autonomous ODE equation

u' = -u + t and initial data u(\tau) = u_{\tau} \in \mathbb R.

By solving the equation we easily get that

u(t) = e^{-(t-\tau)}(u_{\tau} - \tau + 1) + t - 1.

It is obvious that \lim_{t\rightarrow+\infty}u(t) = +\infty and there is no hope for a compact (or even bounded) attracting set as t\rightarrow+\infty.

However, if we take the differences between two solutions u and v (with different initial data) then we have

(u - v)' = -(u-v) or equivalently (u-v)(t) = e^{-(t-\tau)}(u_\tau - v_{\tau}).

This suggests that there is still an asymptotic behavior of the equation that needs to be studied. And that is where the pullback attractor theory comes in.

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