Nonlocal Hamilton-Jacobi Equations BGK equation Averaging Diffusion limit Convergence Partial differential equations Timoshenko beam Existence Modal analysis Diffusion Conduction Porous media Reaction-diffusion Micro-macro decomposition Regularity Data assimilation Kinetic equations Finite element method Navier-Stokes Equations FOS Mathematics Highly oscillatory problems Conservation laws Dislocation dynamics DATA ASSIMILATION Schrödinger equation Invariant measure Density matrix Two-scale convergence Formal series Entropy Ensemble forecasting Scattering $L^1-$dependence in time Population dynamics Boundary layer Adjoint model Uncertainty quantification Error estimate Numerical simulations Fluid-structure interaction Numerical range Subdivision Asymptotic Analysis Asymptotic preserving schemes Elasticity Global existence Ensemble forecasts MPCC Homogenization Robust optimization Hamilton-Jacobi equations Nonlocal front propagation Stability Turbulence models Turbulence White noise Fine-scale substructure Lower-bound gradient estimate Navier-Stokes equations Reaction-diffusion system PIV Viscosity solutions Spectral set Concrete Large eddy simulation Maximum principle Long-time behavior 65L20 Chapman-Enskog expansion Finite elements Markov chain Controllability Vlasov-Poisson equation Numerical analysis Level-set approach Eikonal equation B-series Oceanography Deconvolution models Backward error analysis Interaction fluide-structure Fractured porous media Dynamical system Binary alloys Optimal control Numerical simulation Cardiac electrophysiology Asymptotic analysis Vlasov equation Stochastic partial differential equations Degenerate parabolic equations Shape optimization Optimization Finite Element Method Nonlinear Schrödinger equation Stroboscopic averaging Fluid mechanics Asymptotic behavior Robust control Uniform accuracy