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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}


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


\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}.

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