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1.
Stochastic 2-D Navier—Stokes Equation 总被引:1,自引:0,他引:1
Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded
and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic
Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability
space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions
to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution. 相似文献
2.
S. S. Sritharan 《Applied Mathematics and Optimization》2000,41(2):255-308
This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment
of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive
relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case
with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic
case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and
Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy
viscosity type turbulence closure models.
Accepted 7 June 1999 相似文献
3.
We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related
to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions
of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in
some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman
equations. 相似文献
4.
We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related
to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions
of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in
some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman
equations. 相似文献
5.
Feireisl 《Applied Mathematics and Optimization》2008,47(1):59-78
Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class
of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary.
The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid. 相似文献
6.
Feireisl 《Applied Mathematics and Optimization》2003,47(1):59-78
Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class
of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary.
The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid. 相似文献
7.
A Wentzell–Freidlin type large deviation principle is established for the two-dimensional Navier–Stokes equations perturbed by a multiplicative noise in both bounded and unbounded domains. The large deviation principle is equivalent to the Laplace principle in our function space setting. Hence, the weak convergence approach is employed to obtain the Laplace principle for solutions of stochastic Navier–Stokes equations. The existence and uniqueness of a strong solution to (a) stochastic Navier–Stokes equations with a small multiplicative noise, and (b) Navier–Stokes equations with an additional Lipschitz continuous drift term are proved for unbounded domains which may be of independent interest. 相似文献
8.
We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for
the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection
principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the
space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise,
we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences
of these facts, together with a new sufficient condition for well-posedness.
相似文献
9.
A. G. Bhatt G. Kallianpur R. L. Karandikar J. Xiong 《Applied Mathematics and Optimization》1998,37(2):151-188
A nonlinear Hilbert-space-valued stochastic differential equation where L
-1
(L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity
of L
-1
, the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L
-1
is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions
that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable.
A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions
of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ
0
of the martingale problem posed by the corresponding McKean—Vlasov equation.
Accepted 4 April 1996 相似文献
10.
Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization
of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global
attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result
is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail,
despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides
additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D
Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum
of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and
3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of
the 3D Navier–Stokes equations, established earlier by Doering and Titi.
相似文献