Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.     

Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller–Segel model with logistic source:

$\{\begin{array}{ccc}\hfill & u_t=\mathrm{?}?(\mathrm{?}u?\chi u\mathrm{?}v)+u?\mu u^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ \hfill & v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^2$ with $\chi ,\mu >0$, nonnegative initial data $u_0$, $v_0$, and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any $\chi ,\mu >0$. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on χ and μ. More, precisely, it is shown that there exists $C=C(u_0,v_0,\mathrm{\Omega })>0$ such that

${6u(?,t)6}_{L^{\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\chi K(\chi ,\mu )N(\chi ,\mu )]$

and

${6v(?,t)6}_{W^{1,\infty }(\mathrm{\Omega })}\le C[1+\frac{1}{\mu }+\frac{{\chi }^{\frac{8}{3}}}{\mu }{K}^{\frac{8}{3}}(\chi ,\mu )]=:CN(\chi ,\mu )$

uniformly on $[0,\infty )$, where

$K(\chi ,\mu )=M(\chi ,\mu )E(\chi ,\mu ),M(\chi ,\mu )=1+\frac{1}{\mu }+\sqrt{\chi }(1+\frac{1}{{\mu }^2})$

and

$E(\chi ,\mu )=\exp ?\left[\frac{\chi C_{GN}^2}{2\min ?\{1,\frac{2}{\chi }\}}\right(\frac{4}{\mu }{6u_{0}6}_{L^1(\mathrm{\Omega })}+\frac{13}{2{\mu }^2}|\mathrm{\Omega }|+{6\mathrm{?}{v}_{0}6}_{L^2(\mathrm{\Omega })}^2\left)\right].$

We notice that these upper bounds are increasing in χ, decreasing in μ, and have only one singularity at $\mu =0$, where the corresponding minimal model (removing the term $u?\mu u^2$ in the first equation) is widely known to possess blow-ups for large initial data.