Showing papers by "Peter Constantin published in 2007"

On the Euler equations of incompressible fluids

Abstract: Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the RayleighTaylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

Regularity of Coupled Two-Dimensional Nonlinear Fokker-Planck and Navier-Stokes Systems

Abstract: We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The particles add stress to the fluid and the fluid carries and deforms the particles. Because the particles perform rapid random motion, we assume that the density of particles is carried by a time average of the fluid velocity. The resulting coupled system is shown to have smooth solutions at all values of parameters, in two spatial dimensions.

Regularity of H\"older continuous solutions of the supercritical quasi-geostrophic equation

Abstract: We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha 1-2\alpha$ on the time interval $[t_0, t]$, then it is actually a classical solution on $(t_0,t]$.

Inviscid Limit for Damped and Driven Incompressible Navier-Stokes Equations in $$\mathbb R^2$$

Abstract: We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in \(\mathbb R^2\) . We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance.

A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations

Abstract: In this paper we consider the role that numerical computations—in particular Galerkin approximations—can play in problems modeled by the three-dimensional (3D) Navier–Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem. We then consider Galerkin approximations, and show that if a strong solution exists the Galerkin approximations will converge to it; thus if one is prepared to assume that the Navier–Stokes equations are regular one can justify this particular numerical method rigorously. Combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time. We thus introduce the possibility of rigorous computations of the solutions of the 3D Navier–Stokes equations (despite the lack of rigorous existence and uniqueness results), and demonstrate that numerical investigation can be used to rule out the occurrence of possible singularities in particular examples.

Regularity of inviscid shell models of turbulence.

Abstract: In this paper we continue the analytical study of the sabra shell model of energy turbulent cascade. We prove the global existence of weak solutions of the inviscid sabra shell model, and show that these solutions are unique for some short interval of time. In addition, we prove that the solutions conserve energy, provided that the components of the solution satisfy $\ensuremath{\mid}{u}_{n}\ensuremath{\mid}\ensuremath{\le}C{k}_{n}^{\ensuremath{-}1∕3}[\sqrt{n}\phantom{\rule{0.2em}{0ex}}\mathrm{log}(n+1){]}^{\ensuremath{-}1}$ for some positive absolute constant $C$, which is the analog of the Onsager's conjecture for the Euler's equations. Moreover, we give a Beal-Kato-Majda type criterion for the blow-up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the ``two-dimensional'' parameters regime.

Smoluchowski Navier-Stokes Systems

Abstract: We discuss equilibria, dynamics and regularity for Smoluchowski equations coupled to Navier-Stokes equations. Introduction We consider mixtures of fluids and microscopic inclusions. The microscopic inclusions are characterized by state variables m ∈ M , where M is a compact smooth Riemannian manifold without boundary. The simplest example is that of microscopic rods with directors m ∈ S. The microscopic inclusions evolve stochastically: they are carried by the ambient fluid, agitated by thermal noise and interact with each other. This behavior is modeled in this paper by a Smoluchowski equation for the probability distribution of particles. The inclusions add stresses to the fluid, and thus the system is coupled. When the coupling is negligible, and the inclusions are in statistical equilibrium, then the system is governed by a single time-independent equation, derived by Onsager for colloidal suspensions of rod-like particles. This equation is variational in nature, nonlinear and nonlocal. The free energy has an entropic part and a microscopic selfinteraction part. The selfinteractions are quadratic but nonlocal and indefinite. In the particular case of a specific microscopic model given by a Maier-Saupe potential, Onsager’s equations reduce to few transcendental implicit equations. These can be analyzed, and the limit of strong microscopic interactions can be shown to have a nematic character, which means in this context that the probability distributions concentrate to singular sets in M . High intensity asymptotics for Onsager’s equations have been studied in ([4]). Qualitative properties of solutions were obtained in ([11], [21], [22], [12]). The transition to nematic states as the intensity of the selfinteractions increases, as well as the fact that the infinite dimensional nonlinear nonlocal Onsager equation reduces to few transcendental equations with a variational structure are not isolated features, due to the fact that the Maier-Saupe interactions are particularly limited. In fact, for generic interactions, Onsager’s equation can be written as a sequence of transcendental equations, and the high interaction limit is generically a delta function on M . 1991 Mathematics Subject Classification. Primary 35Q30, 82C31; Secondary 76A05.

H\"older continuity of solutions of supercritical dissipative hydrodynamic transport equations

Abstract: We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($\alpha 0$), and from H\"{o}lder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it does not appear that their approach can be easily extended to establish the H\"{o}lder continuity of $L^\infty$ solutions. In order for their approach to work, we require the velocity to be in the H\"{o}lder space $C^{1-2\alpha}$. Higher regularity starting from $C^\delta$ with $\delta>1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere \cite{CW6}.

Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence

Abstract: In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log ν −1 for all values of the governing parameter e, except for e =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter e becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity.

Energy conservation and Onsager's conjecture for the Euler equations

Abstract: Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space $B^{1/3}_{3,c(\NN)}$. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to $B^{2/3}_{3,c(\NN)}$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D

Abstract: We prove global existence for a nonlinear Smoluchowski equation (a nonlinear Fokker-Planck equation) coupled with Navier-Stokes equations in two dimensions. The proof uses a deteriorating regularity estimate and the tensorial structure of the main nonlinear terms.