| Abstract: |
| When the Boltzmann equation is studied on a bounded domain $\Omega$, certain physically motivated boundary conditions have to be used. In our context, we look at the specular reflection boundary condition, which imposes a symmetry on the solution $f(t,x,v)$ in the velocity variable based on the tangent plane at $x \in \partial \Omega$. Microscopically, this condition models the situation where particles bounce against the boundary with perfect restitution and no friction, conserving energy (but not momentum). Building the well-posedness theory for this model with such a boundary condition baked-in can be quite cumbersome (as compared to the well-posedness theory on the full space), since one must start at low-regularity. In this work we investigate an alternate approach. We study a full-space model with a Vlasov-like force field produced by a potential barrier outside of $\Omega$. We show that, under some uniform bounds on the solution, as the potential barrier becomes infinitely strong, the solution $f$ indeed converges to a weak solution (as constructed in Ouyang-Silvestre 2023) of the Boltzmann equation with the boundary condition. The argument mainly relies on determining the precise shape of characteristic curves produced by this potential barrier outside of $\Omega$, and showing that they re-enter $\Omega$ quickly, and with velocity converging to that of specular reflection, in accordance with the microscopic intuition. |
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