关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Stochastic equations in infinite dimensions

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

Stochastic differential equations and diffusion processes

The stochastic convective Brinkman-Forchheimer (SCBF) equations or the tamed Navier-Stokes equations in bounded or periodic domains are considered in this work. We show the existence of a pathwise unique strong solution (in the probabilistic sense) satisfying the energy equality (Ito formula) to the SCBF equations perturbed by multiplicative Gaussian noise. We exploited a monotonicity property of the linear and nonlinear operators as well as a stochastic generalization of the Minty-Browder technique in the proofs. The energy equality is obtained by approximating the solution using approximate functions constituting the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. We further discuss about the global in time regularity results of such strong solutions in periodic domains. The exponential stability results (in mean square and pathwise sense) for the stationary solutions is also established in this work for large effective viscosity. Moreover, a stabilization result of the stochastic convective Brinkman-Forchheimer equations by using a multiplicative noise is obtained. Finally, we prove the existence of a unique ergodic and strongly mixing invariant measure for the SCBF equations subject to multiplicative Gaussian noise, by making use of the exponential stability of strong solutions.

Stochastic convective Brinkman-Forchheimer equations

In this work, we consider the two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows. We investigate the well-posedness of such ...

Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures

: In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to the stochastic Euler equations perturbed by additive and multiplicative Lvy noise in two and three dimensions. The existence of a unique maximal solution is also proved.

Stochastic Euler Equations of Fluid Dynamics with Levy Noise

Magneto-hydrodynamics (MHD) is the branch of fluid mechanics which studies the motion of electrically conducted fluid in the presence of magnetic field. MHD system consists of the Navier-Stokes equations, which describe the motion of a fluid in the electric field coupled with the Maxwell equations, which describe the motion of the fluid in the magnetic field (see Chandrasekhar [6]). The deterministic MHD system has been studied by mathematicians and physicists ( e.g. Cabannes [5], Cowling [8], Ladyzhenskaya [20], Landau and Lifshitz [21], Sermange and Temam [33], and Temam [39]) over the past fifty years. Significant contributions in the mathematical study of stochastic MHD system are due to Sritharan and Sundar [34], Sundar [37], and Barbu and Da Prato [3]. The authors in [34] prove the well posedness of the martingale problem associated to two and three-dimensional MHD system perturbed by white noise. In [37], the author study the existence and uniqueness of solutions to the two-dimensional MHD system perturbed by more general noise, namely multiplicative noise and additive fractional Brownian noise. The authors in [3] establish the existence and uniqueness of an invariant measure via coupling methods developed by Odasso [26]. In this paper we consider the stochastic MHD system in the non-dimensional form. LetG ⊂ R be a bounded, open, simply-connected domain with a sufficiently smooth boundary ∂G, then

/pdf/two-dimensional-magneto-hydrodynamic-system-with-jump-5girky5s7j.pdf

Manil T. Mohan

Papers

Stochastic convective Brinkman-Forchheimer equations

Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures

Stochastic Euler Equations of Fluid Dynamics with Levy Noise

Two-dimensional magneto-hydrodynamic system with jump processes: well posedness and invariant measures

Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces