Mathematical Models and Methods in Applied Sciences · 1991 · 179 citations · 0 references
Natural SciencesPhysicsComputational Fluid DynamicsFluid DynamicsTransport PhenomenaConvergence AnalysisIncompressible FlowInitial ConditionsNavier-stokes EquationsNumerical ConvergenceWeak FormulationCauchy ProblemConvergenceConvergence HypothesisIncompressible Navier-stokes EquationsKnudsen NumberLinearized Boltzmann EquationStrong SolutionsUniform Convergence
The convergence hypothesis of Bardos, Golse, and Levermore, 1 which leads to the incompressible Navier-Stokes equation as the limit of the scaled Boltzmann equation, is substantiated for the Cauchy problem with initial data small but independent of the Knudsen number ε. The uniform (in ε) existence of global strong solutions and their strong convergence as ε→0 are proved. A necessary and sufficient condition for the uniform convergence up to t=0, which implies the absence of the initial layer, is also established. The proof relies on sharp estimates of the linearized operators, which are obtained by the spectral analysis and the stationary phase method.