Showing papers by "Peter Constantin published in 1999"
TL;DR: In this article, the authors studied solutions to the 2D quasi-geostrophic (QGS) equation and proved global existence and uniqueness of smooth solutions for strong and weak solutions.
Abstract: We study solutions to the 2D quasi-geostrophic (QGS) equation $$ \frac{\partial \theta}{\partial t}+u\cdot
abla\theta + \kappa (-\Delta)^{\alpha}\theta=f $$ and prove global existence and uniqueness of smooth solutions if $\alpha\in (\frac{1}{2},1]$; weak solutions also exist globally but are proven to be unique only in the class of strong solutions. Detailed aspects of large time approximation by the linear QGS equation are obtained.
421 citations
TL;DR: In this article, an inequality of the type N≤CR1/3(1+log+R)2/3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing three-dimensional Rayleigh-Benard convection in the limit of infinite Prandtl number was proved.
Abstract: We prove an inequality of the type N≤CR
1/3(1+log+
R)2/3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing three-dimensional Rayleigh–Benard convection in the limit of infinite Prandtl number
180 citations
TL;DR: In this paper, the authors study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria.
Abstract: We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2
147 citations
TL;DR: In this paper, the authors derived upper bounds for the Nusselt number in infinite Prandtl number rotating convection, and showed that when the rotation is fast enough the purely conductive solution is the globally and nonlinearly attractive fixed point; the critical rotation rate also depends on boundary conditions.
31 citations
TL;DR: A systematic exploration of many different initial conditions reveals no evidence of singular solutions for finite-time blowup in an active scalar equation similar to the Euler equation.
Abstract: We study the formation of thermal fronts in an active scalar equation that is similar to the Euler equation. For a particular initial condition, an earlier candidate for finite-time blowup, the front forms in a generalized self-similar way with constant hyperbolicity at the center. The behavior belongs to a class of scenarios for which finite-time blowup is impossible. A systematic exploration of many different initial conditions reveals no evidence of singular solutions. [copyright] [ital 1999] [ital The American Physical Society]
23 citations
TL;DR: In this article, the authors consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction.
Abstract: We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physical sense in general situations and extends the often ill-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of flows. These "percolating" flows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that these can produce only sub-linear enhancement of the bulk burning rate.