565 research outputs found
Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation
We investigate the time-asymptotic stability of planar rarefaction wave for
the three-dimensional Boltzmann equation, based on the micro-macro
decomposition introduced in [24, 22] and our new observations on the underlying
wave structures of the equation to overcome the difficulties due to the wave
propagation along the transverse directions and its interactions with the
planar rarefaction wave. Note that this is the first stability result of planar
rarefaction wave for 3D Boltzmann equation, while the corresponding results for
the shock and contact discontinuities are still completely open.Comment: 45pages. We correct some typoes. The paper will be published on
Kinetic and Related Model
Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier-Stokes equations with temperature-dependent transport coefficients
In this paper, we construct a family of global-in-time solutions of the 3-D
full compressible Navier-Stokes (N-S) equations with temperature-dependent
transport coefficients (including viscosity and heat-conductivity), and show
that at arbitrary times {and arbitrary strength} this family of solutions
converges to planar rarefaction waves connected to the vacuum as the viscosity
vanishes in the sense of . We consider the Cauchy problem in
with perturbations of the infinite global norm, particularly, periodic
perturbations. To deal with the infinite oscillation, we construct a suitable
ansatz carrying this periodic oscillation such that the difference between the
solution and the ansatz belongs to some Sobolev space and thus the energy
method is feasible. The novelty of this paper is that the viscosity and
heat-conductivity are temperature-dependent and degeneracies caused by vacuum.
Thus the a priori assumptions and two Gagliardo-Nirenberg type inequalities are
essentially used. Next, more careful energy estimates are carried out in this
paper, by studying the zero and non-zero modes of the solutions, we obtain not
only the convergence rate concerning the viscosity and heat conductivity
coefficients but also the exponential time decay rate for the non-zero mode.Comment: The version has been updated and the results have been extende
Time-asymptotic stability of generic Riemann solutions for compressible Navier-Stokes-Fourier equations
We establish the time-asymptotic stability of solutions to the
one-dimensional compressible Navier-Stokes-Fourier equations, with initial data
perturbed from Riemann data that forms a generic Riemann solution. The Riemann
solution under consideration is composed of a viscous shock, a viscous contact
wave, and a rarefaction wave. We prove that the perturbed solution of
Navier-Stokes-Fourier converges, uniformly in space as time goes to infinity,
to a viscous ansatz composed of viscous shock with time-dependent shift, a
viscous contact wave and an inviscid rarefaction wave.
This is a first resolution of the challenging open problem associated with
the generic Riemann solution. Our approach relies on the method of
a-contraction with shifts, specifically applied to both the shock wave and the
contact discontinuity wave. It enables the application of a global energy
method for the generic combination of three waves.Comment: arXiv admin note: text overlap with arXiv:2104.0659
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