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We will talk about the classical limit of the Schrödinger-Poisson system to the Vlasov-Poisson equations as the Planck constant goes to zero. This limit is also frequently called " semiclassical limit ". The coupled Schrödinger-Poisson system for the wave functions {ψ j (t, x)} are transformed to the Wigner-Poisson equations for a " phase space function " f… (More)

We investigate the equation (u t + (f (u)) x) x = f (u)(u x) 2 /2 where f (u) is a given smooth function. Typically f (u) = u 2 /2 or u 3 /3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u tt − c(u)(c(u)u x) x = 0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also… (More)

We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation u tt − c(u)(c(u)u x) x = 0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values.

- Yuxi Zheng
- 1998

We establish rigorously the existence of a three-parameter family of self-similar, globally bounded, and continuous weak solutions in two space dimensions to the compress-ible Euler equations with axisymmetry for γ-law polytropic gases with 1 ≤ γ < 2. The initial data of these solutions have constant densities and outward-swirling velocities. We use the… (More)

- Yuxi Zheng, Guiqiang Chen
- 2006

We establish the existence of a global solution to a regular reflection of a shock hitting a ramp for the pressure gradient system of equations. The setup of the reflection is the same as that of Mach's experiment for the compressible Euler system, i.e., a straight shock hitting a ramp. We assume that the angle of the ramp is close to 90 degrees. The… (More)

We establish the existence of a smooth solution in its elliptic region in the self-similar plane to the pressure-gradient system arisen from the wave-particle splitting of the two-dimensional compressible Euler system of equations. The pressure-gradient system takes the form ρu t +p x = 0, ρv t + p y = 0, ρE t + (up) x + (vp) y = 0. Here (u, v) is the… (More)

We report a novel mutation (erlong, erl) of the cadherin 23 (Cdh23) gene in a mouse model for DFNB12 characterized by progressive hearing loss beginning from postnatal day 27 (P27). Genetic and sequencing analysis revealed a 208 T >C transition causing an amino-acid substitution (70S-P). Caspase expression was upregulated in mutant inner ears. Hearing was… (More)

- Yuxi Zheng, Shuxing Chen
- 2003

We present a global solution to a Riemann problem for the pressure gradient system of equations. The Riemann problem has initially two shock waves and two contact discontinuities. The angle between the two shock waves is set initially to be close to 180 degrees. The solution has a shock wave that is usually regarded as a free boundary in the self-similar… (More)

- Jiequan Li, Tong Zhang, Yuxi Zheng
- 2005

We present a characteristic decomposition of the potential flow equation in the self-similar plane. The decomposition allows for a proof that any wave adjacent to a constant state is a simple wave for the adiabatic Euler system. This result is a generalization of the well-known result on 2-d steady potential flow and a recent similar result on the pressure… (More)

It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves.… (More)