scholarly journals Global potential, topology, and pattern selection in a noisy stabilized Kuramoto–Sivashinsky equation

2020 ◽  
Vol 117 (38) ◽  
pp. 23227-23234
Author(s):  
Yong-Cong Chen ◽  
Chunxiao Shi ◽  
J. M. Kosterlitz ◽  
Xiaomei Zhu ◽  
Ping Ao
Keyword(s):  
Stochastic Dynamics ◽  
Nonlinear Partial Differential Equations ◽  
Saddle Points ◽  
Stationary Solutions ◽  
Stochastic Simulations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Topological Network ◽  
Sivashinsky Equation ◽  
General Method

We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen.37, L25–L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto–Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.

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

2005 ◽  
Vol 07 (05) ◽  
pp. 553-582 ◽  
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.

Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny–Lin equation

2017 ◽  
Vol 14 (12) ◽  
pp. 1750170 ◽  
Author(s):  
Saeede Rashidi ◽  
S. Reza Hejazi
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Group Analysis ◽  
Navier Stokes ◽  
Kawahara Equation ◽  
Navier Stokes Equation ◽  
Invariance Properties ◽  
Fractional Differential ◽  
Fifth Order ◽  
Sivashinsky Equation

This paper investigates the invariance properties of the time fractional Benny–Lin equation with Riemann–Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto–Sivashinsky equation and Navier–Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny–Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

The exponential behavior and stabilizability of the stochastic 3D Navier–Stokes equations with damping

2019 ◽  
Vol 31 (07) ◽  
pp. 1950023 ◽  
Author(s):  
Hui Liu ◽  
Lin Lin ◽  
Chengfeng Sun ◽  
Qingkun Xiao
Keyword(s):  
Weak Solutions ◽  
Multiplicative Noise ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Exponential Behavior ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
The Mean ◽  
The Stability

The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The stability of weak solutions to the stochastic 3D Navier–Stokes equations with damping is proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions are proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The stabilization of these equations is obtained for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text].

scholarly journals A Hopf Bifurcation in the Planar Navier–Stokes Equations

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Gianni Arioli ◽  
Hans Koch
Keyword(s):  
Hopf Bifurcation ◽  
Viscous Fluid ◽  
Kinematic Viscosity ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Computer Assisted ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Navier Boundary Conditions

AbstractWe consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

Divergent expansion, Borel summability and three-dimensional Navier–Stokes equation

2008 ◽  
Vol 366 (1876) ◽  
pp. 2775-2788 ◽  
Author(s):  
Ovidiu Costin ◽  
Guo Luo ◽  
Saleh Tanveer
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Local Existence ◽  
Three Dimensional ◽  
Nonlinear Pdes ◽  
Navier Stokes ◽  
Borel Summability ◽  
Numerical Computations ◽  
Navier Stokes Equation ◽  
Existence Proofs ◽  
General Data

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier–Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/ t or some positive power of 1/ t ). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally. Moreover, computation of the solution of the IE over an interval [0, p 0 ] provides sharper control of its p →∞ behaviour. Preliminary numerical computations give encouraging results.

Sign in / Sign up

Export Citation Format

Share Document