Stochastic Navier-stokes Equations with Fractional Brownian Motions
نویسندگان
چکیده
The aim of this dissertation is to study stochastic Navier-Stokes equations with a fractional Brownian motion noise. The second chapter will introduce the background results on fractional Brownian motions and some of their properties. The third chapter will focus on the Stokes operator and the semigroup generated by this operator. The Navier-Stokes equations and the evolution equation setup will be described in the next chapter. The main goal is to prove the existence and uniqueness of solutions for the stochastic Navier-Stokes equations with a fractional Brownian motion noise under suitable conditions. The proof is given with full details for two separate cases based on the value of the Hurst parameter H: 1/2 < H < 1 and 1/8 < H < 1/2.
منابع مشابه
Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations*
In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique max...
متن کاملOn Parabolic Volterra Equations Disturbed by Fractional Brownian Motions
Aim of this paper is to study the parabolic Volterra equation u(t) + (b ∗Au)(t) = (QB)(t), t ≥ 0, on a separable Hilbert space. Throughout this work the operator −A is assumed to be a differential operator like the Laplacian, the elasticity operator, or the Stokes operator. The random disturbance Q1/2BH is modeled to be a system independent vector valued fractional Brownian motion with Hurst pa...
متن کاملTwo-dimensional Stochastic Navier-stokes Equations with Fractional Brownian Noise
We study the perturbation of the two-dimensional stochastic Navier-Stokes equation by a Hilbert-space-valued fractional Brownian noise. Each Hilbert component is a scalar fractional Brownian noise in time, with a common Hurst parameter H and a specific intensity. Because the noise is additive, simple Wiener-type integrals are suffi cient for properly defining the problem. It is resolved by sepa...
متن کاملThe Internal Stabilization by Noise of the Linearized Navier-stokes Equation
i=1 Vi(t)ψi(ξ)β̇i(t), ξ ∈ O, where {βi}i=1 are independent Brownian motions in a probability space and {ψi}i=1 is a system of functions on O with support in an arbitrary open subset O0 ⊂ O. The stochastic control input {Vi}i=1 is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution. Mathematics S...
متن کاملInternal Stabilization by Noise of the Navier--Stokes Equation
One shows that the Navier-Stokes equation in O⊂Rd, d = 2, 3, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller V (t, ξ) = ∑N i=1 Vi(t)ψi(ξ)β̇i(t), ξ ∈ O, where {βi}i=1 are independent Brownian motions and {ψi}i=1 is a system of functions on O with support in an arbitrary open subset O0 ⊂ O. The stochastic control input {Vi}i=1 is...
متن کامل