Truyen Nguyen

Pressureless Euler/Euler–Poisson Systems via Adhesion Dynamics and Scalar Conservation Laws

The “sticky particles” model at the discrete level is employed to obtain global solutions for a class of systems of conservation laws among which lie the pressureless Euler and the pressureless attractive/repulsive Euler–Poisson system with zero background charge. We consider the case of finite, nonnegative initial Borel measures with finite second-order moment, along with continuous initial velocities of at most quadratic growth and finite energy. We prove the time regularity of the solution for the pressureless Euler system and obtain that the velocity satisfies the Oleinik entropy condition, which leads to a partial result on uniqueness. Our approach is motivated by earlier work of Brenier and Grenier, who showed that one-dimensional conservation laws with special initial conditions and fluxes are appropriate for studying the pressureless Euler system.

Estimating Shannon and Constrained Capacities of Bernoulli-Gaussian Impulsive Noise Channels in Rayleigh Fading

This paper presents a novel approach to tightly estimate the ergodic Shannon and constrained capacities of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading environments where channel gains are known at the receiver, but not at the transmitter. We first show that the differential entropy of the BG impulsive noise can be established in closed-form using Gaussian hypergeometric function 2F1(1, 1; ·; ·). The Shannon capacity is then calculated via upper and lower bounds. Specifically, we derive in closed-form two upper bounds on the Shannon capacity using the assumption of a Gaussian output and using full knowledge of noise state, respectively. Under the assumption of a Gaussian input, we propose a novel approach to calculate a lower bound by examining the instantaneous output entropy in two regions of channel gains. In the high-gain region, the lower bound is evaluated via the upper bound obtained under the Gaussian output assumption. In the other region, we apply the piecewise-linear curve fitting (PWLCF) method to estimate the lower bound. It is then demonstrated that the lower bound can be calculated with a predetermined accuracy. By establishing the difference between the lower bound and the two upper bounds, we show that the lower bound can be used to effectively estimate the Shannon capacity. Finally, we detail a PWLCF-based method to estimate the constrained capacity for a finite-alphabet constellation. To this end, we first propose a numerical technique to calculate the instantaneous entropy of the output using 2-dimensional (2-D) Gauss-Hermite quadrature formulas. The average output entropy is then obtained using the PWLCF method. Combined with the closed-form expression of the entropy of the BG impulsive noise, the constrained capacity can be effectively estimated.

Gradient Estimates and Global Existence of Smooth Solutions to a Cross-Diffusion System

We investigate the global time existence of smooth solutions for the Shigesada--Kawasaki--Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global

Interior second derivative estimates for solutions to the linearized Monge-Ampère equation

W^{1,p}

Estimating information rates of Bernoulli-Gaussian impulsive noise channels in Rayleigh fading

-estimates of Calderon--Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli--Peral perturbation technique together with a new two-parameter scaling argument.

Geometric properties of boundary sections of solutions to the Monge–Ampère equation and applications

Let Ω ⊂ Rn be a bounded convex domain and φ ∈ C(Ω) be a convex function such thatφ is sufficiently smooth on∂Ω and the Monge–Ampere measure det D2φ is bounded away from zero and infinity in Ω. The corresponding linearized Monge–Ampere equation is trace(ΦD2u) = f , where Φ := det D2φ (D2φ)−1 is the matrix of cofactors of D2φ. We prove a conjecture in [GT] about the relationship between Lp estimates for D2u and the closeness between det D2φ and one. As a consequence, we obtain interior W2,p estimates for solutions to such equation whenever the measure det D2φ is given by a continuous density and the function f belongs to Lq(Ω) for some q > max {p,n}.

Homogenization and convergence of correctors in Carnot groups

This paper presents simple methods to tightly estimate the information rate achieved by a Gaussian input and the constrained capacity of a finite-alphabet input of a Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading. Specifically, under the assumption of a Gaussian input, we propose a novel approach to calculate the achievable rate by examining the instantaneous output entropy in two regions of channel gains. In the high-gain region, the rate is evaluated via an upper bound obtained under the Gaussian output assumption. In the other region, we apply the piecewise-linear curve fitting (PWLCF) method to estimate the rate. It is then demonstrated that the information rate achieved by Gaussian inputs can be effectively calculated with a pre-determined accuracy. For a finite-alphabet input, we detail a PWLCF-based method to estimate the constrained capacity. In particular, we first propose a numerical technique to calculate the instantaneous output entropy using 2-dimensional Gauss-Hermite quadrature formulas. The average output entropy is then obtained using PWLCF. Combined with the closed-form expression the entropy of the BG impulse noise, an accurate estimation of the constrained capacity is finally established.

On the capacity of Bernoulli-Gaussian impulsive noise channels in Rayleigh fading

Abstract In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge–Ampere equation: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of Besicovitch type, a covering theorem and a strong type p – p estimate for the maximal function corresponding to boundary sections. Moreover, we show that the Monge–Ampere setting forms a space of homogeneous type.

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of p -Laplacian type

ABSTRACT We consider homogenization of differential operators of the form where is a family of linearly independent vector fields in ℝ N that by commutation generate the Lie algebra of a Carnot group, a ij (ξ) are periodic functions in the sense of the group, and δ1/ε are the dilations in the group. We establish Meyers type estimates for the horizontal gradients Xu = (X 1 u,…,X m u) of solutions to equations defined with general vector fields satisfying Hörmanders condition, and use them to prove convergence of the horizontal gradients of correctors in L 2+Θ, Θ > 0.

One-Dimensional Pressureless Gas Systems with/without Viscosity

In this paper, we investigate the channel capacity of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading via lower and upper bounds. To this end, we first show that the differential entropy of the BG impulse noise can be established in closed-form using Gaussian hypergeometric function 2F1 (1, 1; .; .). This closed-form expression allows us to derive a lower bound on the capacity limit obtained by a Gaussian input using the Gauss-Hermite quadrature formula. We also derive in closed-form two upper bounds on the channel capacity. The first upper bound is obtained under the assumption of full knowledge of noise state, while the second upper bound is developed using a Gaussian distributed output. At high power regions, the lower bound achieved by Gaussian inputs and the upper bound generated by Gaussian inputs are indistinguishable. These two bounds can therefore be used as an accurate estimation for the channel capacity. When the channel input power is small compared to the power of the impulsive noise component, the lower bound obtained by using a Gaussian input and the upper bound under the perfect knowledge of impulse noise state are almost identical, which are useful to predict the capacity. The establishment of the lower bound and the two upper bounds in closed-form helps us to confirm the near-optimality of the Gaussian input in a wide range of input power levels over BG impulsive noise channels in Rayleigh fading.