Two-velocity periodic diffusive-scaling model

Fix a period $L>0$ metres and let ${\mathbb{T}}_L:=\mathbb{R}\mathrm{/}(L\mathbb{Z})$. A smooth periodic function on ${\mathbb{T}}_L$ means an $L$-periodic $C^{\mathrm{\infty }}$ function on $\mathbb{R}$. For a periodic measurable function $g$ with ${\int }_0^L\mathrm{\mid }g(x){\mathrm{\mid }}^{2}dx<\mathrm{\infty }$, use the normalized spatial measure $d{m}_L=dx\mathrm{/}L$ on a period and write

$\mathrm{\parallel }g{\mathrm{\parallel }}_{2,L}^2:=\frac{1}{L}{\int }_0^L\mathrm{\mid }g(x){\mathrm{\mid }}^{2}dx\mathrm{.}$

Elements of $L^2({\mathbb{T}}_L,d{m}_L;\mathbb{C})$ are equivalence classes modulo equality almost everywhere. For $n\in \mathbb{Z}$, define the wave number ${\kappa }_n:=2\pi n\mathrm{/}L$, character $e_n(x):=\exp (i{\kappa }_{n}x)$, and Fourier coefficient $\widehat{g}(n):=L^{-1}{\int }_0^{L}g(x)e^{-i{\kappa }_{n}x}dx$. These formulas are independent of the chosen representative of $x$ in ${\mathbb{T}}_L$.

Let $c>0$ be a speed in $\mathrm{m}{\mathrm{s}}^{-1}$, let $\lambda >0$ be a switching-rate parameter in ${\mathrm{s}}^{-1}$, and let $ϵ>0$ be dimensionless. For smooth real initial data $f_{+,0},f_{-,0}$ on ${\mathbb{T}}_L$, the two-velocity periodic model is the linear Cauchy problem below; when these functions represent physical densities, also require $f_{\pm ,0}\ge 0$.

${\mathrm{\partial }}_t{f}_{+}^{ϵ}+\frac{c}{ϵ}{\mathrm{\partial }}_x{f}_{+}^{ϵ}=\frac{\lambda }{{ϵ}^2}(f_{-}^{ϵ}-f_{+}^{ϵ}),{\mathrm{\partial }}_t{f}_{-}^{ϵ}-\frac{c}{ϵ}{\mathrm{\partial }}_x{f}_{-}^{ϵ}=\frac{\lambda }{{ϵ}^2}(f_{+}^{ϵ}-f_{-}^{ϵ}),f_{\pm }^{ϵ}(\cdot ,0)=f_{\pm ,0}\mathrm{.}$

Here $x\in {\mathbb{T}}_L$, $t\ge 0$ is measured in seconds, and $f_{\pm }$ have the units of a linear number density, ${\mathrm{m}}^{-1}$. The speed magnitude is $c\mathrm{/}ϵ$, and from either velocity state the rate of switching to the opposite state is $\lambda \mathrm{/}{ϵ}^2$. Define the total density ${\rho }^{ϵ}:=f_{+}^{ϵ}+f_{-}^{ϵ}$, the flux $j^{ϵ}:=(c\mathrm{/}ϵ)(f_{+}^{ϵ}-f_{-}^{ϵ})$, and the total mass $M^{ϵ}(t):={\int }_0^L{\rho }^{ϵ}(x,t)dx$.

The space $L^p(\mu )$ as the quotient by null functions