## [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.