A basic, simple energy method for the Boltzmann equation is presented here. It is based on a new macromicro decomposition of the Boltzmann equation as well as the H-theorem. This allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method. As an illustration, we apply the method for the study of the time-asymptotic, nonlinear stability of the global Maxwellian states. Previous energy method, starting with Grad and finishing with Ukai, involves the spectral analysis and regularity of collision operator through sophisticated weighted norms.

. Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\Phi=\Phi(u,v)$, equivalent to the $\L^1$ distance, which is almost decreasing i.e., $$ \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t&gt;s\geq 0,$$ for every pair of <i></i>-approximate solutions <i>u</i>, <i>v</i> with small total variation, generated by a wave front tracking algorithm. The small parameter <i></i> here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in <i>u</i> and in <i>v</i>. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the ${\vec L}^1$ norm. This provides a new proof of the existence of the

In this paper, self-similarity is illustrated and compared in deterministic and stochastic dissipative systems. Examples are (1) deterministic self-similarity in reaction-diffusion system and Navier-Stokes equations, where solutions eventually decay to zero due to balance of diffusion (viscosity) and nonlinearity;(2) statistical self-similarity in randomly advected passive scalar model of Kraichnan where solutions undergo turbulent decay due to roughness of advection;(3) selfsimilarity in blowup of solutions of fourth order nonlinear parabolic equations of the Cahn-Hillard type. Problems for future research are mentioned, especially those where self-similarity is conjectured based on numerical evidence or physical grounds but mathematically open.

A computational method, based on l1-norm minimization, is proposed for the problem traffic flow correction. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corruptions accurately if there are only a few bad sensors which are located at certain links by the proposed method. We mathematically quantify these links that are robust to miscounts and relate them to the geometric structure of the traffic network. In a more realistic setting, besides the unhealthy link sensors, if small measure noise are present at the other sensors, our method guarantees to give an estimated traffic flow fairly close to the ground-truth. Both toy and real-world examples are provided to demonstrate the effectiveness of the proposed method.

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G-equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.