arXiv subjects

Michael S. Jolly

Publications and source records attributed to Michael S. Jolly.

25 records · Page 2Linked to original sources

arXiv2015

Continuous Data Assimilation for the 2D Bénard Convection through Velocity Measurements Alone

An algorithm for continuous data assimilation for the two- dimensional Bénard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier-Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the Bénard convection problem associated with the observed (finite dimensional projection of) velocity.

math.AP↗

arXiv2015

Effect of vorticity coherence on energy-enstrophy bounds for the 3D Navier-Stokes equations

Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is Hölder continuous with exponent $r=1/2$ results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, $r>1/2$ (algebraic in Grashof).

math-ph↗

arXiv2014

A determining form for the damped driven Nonlinear Schrödinger Equation- Fourier modes case

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories $X=C_b^1(\mathbb{R}, P_mH^2)$ where $P_m$ is the $L^2$-projector onto the span of the first $m$ Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

math.AP↗

arXiv2013

Time analyticity with higher norm estimates for the 2D Navier-Stokes equations

This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on $[0,L]^2$, and a force $g\in\calD(A^{\frac{α-1}{2}})$, we show there is a fixed strip about the real time axis on which a uniform bound $|A^αu|< m_ανκ_0^α$ holds for each $α\in \bN$. Here $ν$ is viscosity, $\k0=2π/L$, and $m_α$ is explicitly given in terms of $g$ and $α$. We show that if any element in $\calA$ is in $\D(A^α)$, then all of $\calA$ is in $\D(A^α)$, and likewise with $\D(A^α)$ replaced by $C^\infty(Ω)$. We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class.

math.DS↗

arXiv2013

A unified approach to determining forms for the 2D Navier-Stokes equations - the general interpolants case

In this paper we show that the long time dynamics (the global attractor) of the 2D Navier-Stokes equation is embedded in the long time dynamics of an ordinary differential equation, named {\it determining form}, in a space of trajectories which is isomorphic to $C^1_b(\bR; \bR^N)$, for $N$ large enough depending on the physical parameters of the Navier-Stokes equations. We present a unified approach based on interpolant operators that are induced by any of the determining parameters for the Navier-Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, etc... There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, thus its solutions converge, as time goes to infinity, to the set of steady states of the determining form. The second is that these steady states of the determining form are identified, one-to-one, with the trajectories on the global attractor of the Navier-Stokes equations. It is worth adding that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems.

math.DS↗

arXiv2013

Dissipative length scale estimates for turbulent flows - a Wiener algebra approach

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier-Stokes system on an n-torus. This estimate improves or matches previously known estimates provided that certain bounds on the initial data are satisfied. It is argued that for 2D or 3D turbulent flows, the initial data is guaranteed to satisfy these hypothesized bounds on a significant portion of the 2D global attractor or the 3D weak attractor. In these scenarios, the estimate obtained for 3D generalizes and improves upon that of [Doering-Titi], while in 2D, the estimate matches the best known one found in [Kukavica]. A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms.

math.AP↗

arXiv2012

Navier-Stokes equations, determining forms, determining modes, inertial manifolds, dissipative dynamical systems

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous functions of the variable $s\in\mathbb{R}$ with values in certain finite-dimensional linear space. This new evolution ODE, named {\it determining form}, induces an infinite-dimensional dynamical system in the space $X$ which is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$ into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form $v(t,s)=v_0(t+s)$, correspond exactly to initial data $v_0$ that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter). Finally, a unified approach is outlined for an ODE satisfied by a variety of other determining parameters such as nodal values, finite volumes, and finite elements.

math.AP↗