Energy Estimate and Convergence of Weak Solutions of a Class of Nonlinear PDEs with Robin Boundary Conditions

We establish strong convergence in $L^2([0,T]\times [0,1])$ of weak solutions to a class of nonlinear PDEs with Robin boundary conditions to weak solutions of the corresponding Neumann and Dirichlet problems as the boundary parameter $\kappa$ tends to $0$ and $\mathrm{\infty }$, respectively. Our approach relies on a particle system whose hydrodynamic limits yield the Robin problem for each fixed $\kappa >0$. An energy estimate derived from the microscopic stochastic dynamics provides the key bounds for passing to the subsequent limits $\kappa \to 0$ and $\kappa \to \mathrm{\infty }$, thereby obtaining the Neumann and Dirichlet problems.