Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation
The following initial value problem with unknown function \(f_\varepsilon(t,{\mathbf r},{\mathbf v})\) is considered: \[ \begin{aligned} \partial_t f_\varepsilon+{1\over\varepsilon} {\mathbf v}_\perp\cdot\nabla_{{\mathbf r}_\perp}f_\varepsilon+v_z\partial_zf_\varepsilon+{1\over\varepsilon^2} E_z\partial_{v_z}f_\varepsilon={1\over\varepsilon^2}Q(f_\varepsilon) &\quad \text{in }\mathbb R^+\times \mathbb R^3_{\mathbf r}\times \mathbb R^3_{\mathbf v},\\ f_\varepsilon|_{t=0}=f_{\text{ini}}({\mathbf r},{\mathbf v}) &\quad\text{in }\mathbb R^3_{\mathbf r}\times\mathbb R^3_{\mathbf v},\end{aligned}\tag{1} \] where \(Q(f)({\mathbf v})=\int\sigma({\mathbf v},{\mathbf v}')\{f({\mathbf v}')(1-f({\mathbf v})){\mathcal M}({\mathbf v})-f({\mathbf v})(1-f({\mathbf v}')){\mathcal M}({\mathbf v}')\})d{\mathbf v}'\), \ \({\mathbf r}=(x,y,z)\),\ \({\mathbf v}=(v_x,v_y,v_z)\),\ \({\mathbf r}_\perp=(x,y)\), \({\mathbf v}_\perp=(v_x,v_y)\), \(\varepsilon>0\) is a small parameter, \({\mathcal M}({\mathbf v})=C\exp(-{\mathbf v}^2/2)\), \(C>0\), \(E_z\) are constants, \(0\leq f_{\text{ini}} \leq 1\), \(f_{\text{ini}},{\mathbf v}^2 f_{\text{ini}}\in L^1(\mathbb R^3_{\mathbf r}\times \mathbb R^3_{\mathbf v})\), \(\int f_{\text{ini}} d{\mathbf v}\in L^\infty(\mathbb R^3_{\mathbf r})\). The authors prove that the problem (1) has a unique solution such that \[ f_\varepsilon\in C^0\big(\mathbb R^+,L^1(\mathbb R^3_{\mathbf r}\times \mathbb R^3_{\mathbf v})\big), \quad 0\leq f_\varepsilon \leq 1,\;{\mathbf v}^2 f_\varepsilon \in L^\infty _{\text{loc}}\big(\mathbb R^+,L^1(\mathbb R^3_{\mathbf r}\times \mathbb R^3_{\mathbf v})\big), \] the sequences \(\{ f_\varepsilon\} \) and \(\{n_\varepsilon:=\int f_\varepsilon \,d{\mathbf v}\} \) converge in \[ L^p((0,T),L^1(\mathbb R^3_{{\mathbf r},\text{loc}}\times \mathbb R^3_{\mathbf v})) \quad\text{and}\quad L^p((0,T),L^1_{\text{loc}} (\mathbb R^3_{\mathbf r})), \quad p\in [1,\infty), \] as \(\varepsilon\to 0\) to the functions \(f\) and \(n\) respectively, each of them is a unique solution of the certain problem.
scientific article; zbMATH DE number 5712532