[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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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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