scholarly journals Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations

2010 ◽  
Vol 3 (2) ◽  
pp. 269-289 ◽  
Author(s):  
Andrei Fursikov ◽  
Keyword(s):  
Input Data ◽  
Invariant Manifolds ◽  
Stokes Equations ◽  
Existence Theorems ◽  
Local Existence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stable Invariant Manifolds ◽  
Stable Invariant

scholarly journals Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Sourav Mitra
Keyword(s):  
Stokes Equations ◽  
Local Existence ◽  
Interaction Model ◽  
Strong Solutions ◽  
Coupled System ◽  
Elastic Structure ◽  
Navier Stokes ◽  
Beam Equation ◽  
Navier Stokes Equations ◽  
System Coupling

AbstractWe are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.

GENESIS OF A SOURCE/SINK LINE INTO A SINGULAR UPDRAFT

1998 ◽  
Vol 08 (03) ◽  
pp. 431-444 ◽  
Author(s):  
JOËL CHASKALOVIC
Keyword(s):  
Mathematical Models ◽  
Existence And Uniqueness ◽  
Vortex Line ◽  
Stokes Equations ◽  
Local Existence ◽  
Fluid Motion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Local Existence And Uniqueness ◽  
Source Sink

Mathematical models applied to tornadoes describe these kinds of flows as an axisymmetric fluid motion which is restricted for not developing a source or a sink near the vortex line. Here, we propose the genesis of a family of a source/sink line into a singular updraft which can modeled one of the step of the genesis of a tornado. This model consists of a three-parameter family of fluid motions, satisfying the steady and incompressible Navier–Stokes equations, which vanish at the ground. We establish the local existence and uniqueness for these fields, at the neighborhood of a nonrotating singular updraft.

Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In 𝒟 ⊂ ℝ3

2004 ◽  
Vol 47 (1) ◽  
pp. 30-37
Author(s):  
Xinyu He
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Local Existence ◽  
Similar Solution ◽  
Navier Stokes ◽  
Smooth Bounded Domain ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

AbstractLeray's self-similar solution of the Navier-Stokes equations is defined bywhere . Consider the equation for U(y) in a smooth bounded domain D of with non-zero boundary condition:We prove an existence theorem for the Dirichlet problem in Sobolev space W1,2(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t* with t* < +∞, provided the function is permissible.

APPEARANCE OF A SOURCE/SINK LINE INTO A SWIRLING VORTEX

2003 ◽  
Vol 13 (01) ◽  
pp. 121-142 ◽  
Author(s):  
J. CHASKALOVIC ◽  
A. CHAUVIÈRE
Keyword(s):  
Vortex Line ◽  
Stokes Equations ◽  
Local Existence ◽  
Fluid Motion ◽  
Vertical Shear ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Mature Phase ◽  
Local Existence And Uniqueness ◽  
Source Sink

Several mathematical models applied to tornadoes consist of exact and axisymmetric solutions of the steady and incompressible Navier–Stokes equations. These models studied by Serrin,9 Goldshtik and Shtern8 describe families of fluid motions vanishing at the ground and are restricted not to develop a source nor a sink near the vortex line. Therefore, Serrin showed that the flow patterns of the resulting velocity field may have some realistic characteristics to model the mature phase of the lifetime of a tornado, in comparison with atmospheric observations. On the other hand, no reason has been given to motivate the restriction of the absence of a source/sink vortex line. Therefore, we present here the construction and the analysis of a fluid motion driven by the vertical shear near the ground, the rate of the azimuthal rotation and by the intensity of a central source/sink line. We prove the local existence and uniqueness of a family of fluid motions, leading to the genesis of such source/sink lines inside a non-rotating updraft which does not develop, before perturbation, a source nor a sink.

INVARIANT MANIFOLDS FOR STOCHASTIC MODELS IN FLUID DYNAMICS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 439-459
Author(s):  
SALAH-ELDIN A. MOHAMMED
Keyword(s):  
Fluid Dynamics ◽  
Invariant Manifolds ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Joint Work ◽  
Navier Stokes Equations ◽  
Infinite Dimensional ◽  
Stable Manifold Theorem ◽  
Local Stable Manifold

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

scholarly journals A Data Augmentation-Based Technique for Deep Learning Applied to CFD Simulations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1843
Author(s):  
Alvaro Abucide-Armas ◽  
Koldo Portal-Porras ◽  
Unai Fernandez-Gamiz ◽  
Ekaitz Zulueta ◽  
Adrian Teso-Fz-Betoño
Keyword(s):  
Fluid Dynamics ◽  
Turbulent Flows ◽  
Input Data ◽  
Data Augmentation ◽  
Stokes Equations ◽  
Computational Cost ◽  
Navier Stokes ◽  
Cfd Simulations ◽  
Navier Stokes Equations ◽  
Computational Fluid Dynamics Cfd

The computational cost and memory demand required by computational fluid dynamics (CFD) codes simulations can become very high. Therefore, the application of convolutional neural networks (CNN) in this field has been studied owing to its capacity to learn patterns from sets of input data, which can considerably approximate the results of the CFD simulations with relative low errors. DeepCFD code has been taken as a basis and with some slight variations in the parameters of the CNN, while the net is able to solve the Navier–Stokes equations for steady turbulent flows with variable input velocities to the domain. In order to acquire extensive input data to the CNN, a data augmentation technique, which considers the similarity principle for fluid dynamics, is implemented. As a consequence, DeepCFD is able to learn the velocities and pressure fields quite accurately, speeding up the time-consuming CFD simulations.

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