Nonlinear Landau Equation Research Articles

Regularization estimates and hydrodynamical limit for the Landau equation

In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and we obtain the instantaneous expected anisotropic gain of regularity (see [54] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.

Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if $H^r_x(L^2_v), r >3/2$ norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position $x$ and velocity $v$ variables for any time $t>0$. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight associated with the kinetic transport operator.

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker–Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.

From a colored glass condensate to the gluon plasma: Equilibration in high energy heavy ion collisions

The initial distribution of gluons at the very early times after a high-energy heavy ion collision is described by the bulk scale ${Q}_{s}$ of gluon saturation in the nuclear wave function. The subsequent evolution of the system towards kinetic equilibrium is described by a nonlinear Landau equation for the single particle distributions [A. H. Mueller, Nucl. Phys. B572, 227 (2000); Phys. Lett. B 475, 220 (2000)]. In this paper, we solve this equation numerically for the idealized initial conditions proposed by Mueller, and study the evolution of the system to equilibrium. We discuss the sensitivity of our results on the dynamical screening of collinear divergences. In a particular model of dynamical screening, the convergence to the hydrodynamic limit is seen to be rapid relative to hydrodynamic time scales. The equilibration time, the initial temperature, and the chemical potential are shown to have a strong functional dependence on the initial gluon saturation scale ${Q}_{s}.$

Quasi-linear saturation of coherent backward ionization modes

Quasi-linear saturation of the amplitude of coherent backward ionization waves is studied using the slowly varying parameters or Van der Pol’s method. The result is a nonlinear Landau equation which describes the growth and saturation of the wave amplitude. The analytical expression of the most important nonlinear saturation term predicts results in accordance with earlier experimental observations.