A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

This paper aims to mathematically advance the field of quantitative thermo-acoustic imaging. Given several electromagnetic data sets, we establish for the first time an analytical formula for reconstructing the absorption coefficient from thermal energy measurements. Since the formula involves derivatives of the given data up to the third order, it is unstable in the sense that small measurement noises may cause large errors. However, in the presence of measurement noise, the obtained formula, together with a noise regularization technique, provides a good initial guess for the true absorption coefficient. We finally correct the errors by deriving a reconstruction formula based on the least square solution of an optimal control problem and prove that this optimization step reduces the errors occurring and enhances the resolution.

In this paper, we study the existence of stationary solutions to the VlasovPoissonBoltzmann system when the background density function tends to a positive constant with a very mild decay rate as| x|. In fact, the stationary VlasovPoissonBoltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln (e+| x|)) for some > 0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the VlasovPoissonFokkerPlanck system, J. Math. Anal. Appl. 202 (1996) 10581075] where the decay rate (1+| x|) 1 2 is assumed.

We prove that the strong detonation travelling waves for a viscous combustion model are nonlinearly stable, that is, for a given perturbation of a travelling wave, the solution converges to a shifted travelling wave asymptotically as<i>t</i> . The rate of convergence is obtained. This work is a continuation of a previous work by done T.P. Liu and the first author in [7].

The multi-scale zero relaxation singular limit for gas dynamics in thermal non-equilibrium with multiple non-equilibrium modes in multi-dimensions with physical boundaries from non-equilibrium to thermal equilibrium of compressible Euler flow is proved in this paper for analytical data by establishing the uniform local-in-time estimates. A cancellation mechanism is utilized to deal with the nonlinear singular terms that cause the increase in both time and space derivatives in energy estimates. The rates of the relaxations corresponding to different non-equilibrium modes tending to zero discussed in this paper can be arbitrarily different.