Solvability of the data assimilation problem in the scale of Hilbert spaces for quasilinear singularly perturbed evolutionary problems
Copyright policy )Solvability of the data assimilation problem in the scale of Hilbert spaces for quasilinear singularly perturbed evolutionary problems
The data assimilation problem for a quasilinear evolution equation in a Hilbert space \(H\) with a singular perturbation of the time derivative is analysed. The linear part of the equation is described by a closed, unbounded, selfadjoint, positive definite operator with dense domain of definition and with a compact inverse. The nonlinearity appears with a small coefficient. Problems of that type arise e.g. in modelling large scale atmospheric motion and climate. The author proves solvability of the linear problem and, via the small parameter method, of the nonlinear one. Furthermore, a priori estimates for the solution are obtained. The results are applied to the equations of the two-dimensional motion of a viscous incompressible barotropic fluid on a sphere (including the application to the Navier-Stokes equations). Finally, an iterative process is established to approximate the solution. For this, convergence is proved, and the rate of convergence is estimated.
large scale atmospheric motion and climate, convergence, Numerical solutions to equations with nonlinear operators, Navier-Stokes equations for incompressible viscous fluids, Hilbert space, Initial value problems for linear higher-order PDEs, Nonlinear differential equations in abstract spaces, Meteorology and atmospheric physics, Higher-order parabolic equations, Error bounds for initial value and initial-boundary value problems involving PDEs, iterative process, Navier-Stokes equations, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, singular perturbation, data assimilation, quasilinear evolution equation
large scale atmospheric motion and climate, convergence, Numerical solutions to equations with nonlinear operators, Navier-Stokes equations for incompressible viscous fluids, Hilbert space, Initial value problems for linear higher-order PDEs, Nonlinear differential equations in abstract spaces, Meteorology and atmospheric physics, Higher-order parabolic equations, Error bounds for initial value and initial-boundary value problems involving PDEs, iterative process, Navier-Stokes equations, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, singular perturbation, data assimilation, quasilinear evolution equation
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