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## Cinetic Equations Problem

Cinetic Models Application

Consider the following BGK type model

$\displaystyle (1) \ \ \ \ \begin{cases} \partial_tf+v\cdot \nabla_x f+f=\chi_{n(x,t)}(v), & x,v \in \Bbb{R}\\ f(x,v,0)=f_0(x,v) \geq 0 & f_0 \in L^1(\Bbb{R}^2) \end{cases}$

where

$\displaystyle n(x,t)=\int_\Bbb{R} f(x,v,t)dv$

and

$\displaystyle \chi_n(v)= \begin{cases} 1 & \text{if } 0 \leq v\leq n(x,t), \\ 0 & \text{otherwise} \end{cases}$

I – Let ${f}$ be the solution of ${(1)}$ with ${\chi_{n(x,t)}(v) \in L^1_{t,x,v}}$.

1. Prove that ${f\geq 0}$.

2. Prove that ${\|f(x,v,t)\|_{L^1(\Bbb{R}^2)}=\|f_0\|_{L^1(\Bbb{R}^2)}}$.

3. Prove that if ${f_0(x,v) \leq 1}$ then ${f(x,v,t)\leq 1}$.

II – Let ${\mathcal{T}}$ be the operator defined on ${E=\{\rho(x,t) \geq 0,\ \rho \in L^\infty([0,T];L^1(\Bbb{R}))\}}$ which assigns to ${\rho}$ the function ${n(x,t)=\displaystyle \int_\Bbb{R} f(x,v,t)dv}$ where ${f}$ is the solution of

$\displaystyle (2) \ \ \ \ \begin{cases} \partial_tf+v\cdot \nabla_x f+f=\chi_{\rho(x,t)}(v), & \\ f(x,v,0)=f_0(x,v) \geq 0 & f_0 \in L^1(\Bbb{R}^2) \end{cases}$

4. Let ${C=\displaystyle\left\{ \rho \in E : \int_\Bbb{R} \rho(x,t)dx \leq \|f_0\|_{L^1}\right\}}$. Prove that ${\mathcal{T}}$ maps ${C}$ to ${C}$ and that ${C}$ is a convex closed subset of ${E}$.

5. Prove that for every ${\rho_1,\rho_2}$ in ${C}$ we have

$\displaystyle \|\chi_{\rho_1}-\chi_{\rho_2}\|_{L^1(\Bbb{R}^2)}=\|\rho_1-\rho_2\|_{L^1(\Bbb{R})}$

6. Prove that if we denote by ${f_1,f_2}$ the solutions of ${(2)}$ corresponding to ${\rho_1,\rho_2}$ we have for every ${t \in [0,T]}$ that

$\displaystyle \|f(x,v,t)-f_2(x,v,t)\|_{L^1(\Bbb{R}^2)} \leq (1-e^{-t})\sup_{t \in [0,T]} \|\rho_1-\rho_2\|_{L^1(\Bbb{R}_x)}$

7. Prove that ${\mathcal{T}}$ is a contraction from ${C}$ to ${C}$ and that ${(1)}$ has a unique solution.

III Choose ${f_0=\chi_{R_\varepsilon(x)}}$ bounded in ${L^1(\Bbb{R})\cap L^\infty(\Bbb{R})}$, and denote ${f_\varepsilon}$ the solution of ${(1)}$ which corresponds to ${f_0}$.

8. Prove that if ${v \geq \sup_\varepsilon \|R_\varepsilon\|_{L^\infty}=r}$, then ${f_\varepsilon(x,v,t)=0,\ \forall x \in \Bbb{R}, t \geq 0}$. (we can prove that ${f_\varepsilon \leq \chi_R(v)}$).

9. Prove that ${f_\varepsilon}$ and ${\chi_{n_\varepsilon}}$ are bounded in ${L^\infty([0,T];L^2(\Bbb{R}^2))}$.

10. Prove that ${(n_\varepsilon)_{\varepsilon \geq 0}}$ is compact in ${L^2([0,T]\times B_K)}$ for every ball ${B_K}$ of ${\Bbb{R}_x}$.

11. Prove that, up to a subsequence, we ca pass to the limit in the sense of distributions in ${(1)}$ with ${f_0=\chi_{R_\varepsilon(x)}}$ as ${\varepsilon \rightarrow 0}$.