There are many open problems on the stability of nonlinear wave patterns to the Boltzmann equation even though the corresponding stability theory has been comparatively well-established for the gas dynamical systems. In this paper, we study the nonlinear stability of a rarefaction wave profile to the Boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut-off. The analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection, and the decomposition on both the solution and the Boltzmann equation introduced in [24, 26] for energy method.

In this paper, we give a condition on the bicharacteristic which guarantees the global existence of the mild solution to the Boltzmann equation with an external force for the hard-sphere model and potentials with angular cutoff in infinite vacuum. This generalizes the previous results to the case when the force can have arbitrary strength. The constructive condition on the bicharacteristic is used to obtain the pointwise estimates on the collision operator so that the global existence comes from the contraction mapping theorem.

The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. Together with the instability result given in [28], it gives an almost necessary and sufficient structural condition for the stability of the three-dimensional Prandtl equations. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the secondary flow, an unstabilizing factor in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three-dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any smooth three-dimensional perturbation.

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources.

For<i>nn</i> systems of conservation laws in one dimension without source terms, the existence of global weak solutions was proved by Glimm [1]. Glimm constructed approximate solutions using a difference scheme by solving a class of Riemann problems.