Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

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Strong Uniform Attractors for Nonautonomous Suspension Bridge-Type Equations

Mathematical Problems in Engineering ◽

10.1155/2012/714696 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19

Author(s):

Xuan Wang ◽

Qiaozhen Ma

Keyword(s):

Hilbert Space ◽

Dynamical Behavior ◽

Type Equation ◽

Strong Solutions ◽

Suspension Bridge ◽

Bridge Type ◽

Uniform Attractors ◽

External Forces ◽

Higher Regularity

We discuss long-term dynamical behavior of the solutions for the nonautonomous suspension bridge-type equation in the strong Hilbert spaceD(A)×H2(Ω)∩H01(Ω), where the nonlinearityg(u,t)is translation compact and the time-dependent external forcesh(x,t)only satisfy condition (C*) instead of translation compact. The existence of strong solutions and strong uniform attractors is investigated using a new process scheme. Since the solutions of the nonautonomous suspension bridge-type equation have no higher regularity and the process associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.

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Stationary Navier–Stokes equations with critically singular external forces: Existence and stability results

Advances in Mathematics ◽

10.1016/j.aim.2013.01.016 ◽

2013 ◽

Vol 241 ◽

pp. 137-161 ◽

Cited By ~ 7

Author(s):

Tuoc Van Phan ◽

Nguyen Cong Phuc

Keyword(s):

Stokes Equations ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Existence And Stability ◽

External Forces ◽

Stationary Navier Stokes Equations ◽

Stability Results

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Local solvability of an inverse problem to the Navier–Stokes equation with memory term

Inverse Problems ◽

10.1088/1361-6420/ab7e05 ◽

2020 ◽

Vol 36 (6) ◽

pp. 065007

Author(s):

Yu Jiang ◽

Jishan Fan ◽

Sei Nagayasu ◽

Gen Nakamura

Keyword(s):

Inverse Problem ◽

Stokes Equation ◽

Local Solvability ◽

Navier Stokes ◽

Memory Term ◽

Navier Stokes Equation ◽

With Memory

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L2-compact uniform attractors for a nonautonomous incompressible non-Newtonian fluid with locally uniformly integrable external forces in distribution space

Journal of Mathematical Physics ◽

10.1063/1.2709845 ◽

2007 ◽

Vol 48 (3) ◽

pp. 032702 ◽

Cited By ~ 10

Author(s):

Caidi Zhao ◽

Shengfan Zhou

Keyword(s):

Newtonian Fluid ◽

Distribution Space ◽

Uniform Attractors ◽

External Forces

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A remark on uniform attractors for the 2D non-autonomous Navier-Stokes equation with damping

Nonlinear Analysis and Differential Equations ◽

10.12988/nade.2013.13008 ◽

2013 ◽

Vol 1 ◽

pp. 67-74

Author(s):

Xin-Guang Yang ◽

Jun-Tao Li

Keyword(s):

Stokes Equation ◽

Navier Stokes ◽

Navier Stokes Equation ◽

Uniform Attractors

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STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199705001878 ◽

2005 ◽

Vol 07 (05) ◽

pp. 553-582 ◽

Cited By ~ 9

Author(s):

YURI BAKHTIN ◽

JONATHAN C. MATTINGLY

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Stationary Solution ◽

Landau Equation ◽

Stationary Solutions ◽

Navier Stokes ◽

Partial Differential ◽

Navier Stokes Equation ◽

With Memory

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

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