44 research outputs found
Stability of Steady Multi-Wave Configurations for the Full Euler Equations of Compressible Fluid Flow
We are concerned with the stability of steady multi-wave configurations for
the full Euler equations of compressible fluid flow. In this paper, we focus on
the stability of steady four-wave configurations that are the solutions of the
Riemann problem in the flow direction, consisting of two shocks, one vortex
sheet, and one entropy wave, which is one of the core multi-wave configurations
for the two-dimensional Euler equations. It is proved that such steady
four-wave configurations in supersonic flow are stable in structure globally,
even under the BV perturbation of the incoming flow in the flow direction. In
order to achieve this, we first formulate the problem as the Cauchy problem
(initial value problem) in the flow direction, and then develop a modified
Glimm difference scheme and identify a Glimm-type functional to obtain the
required BV estimates by tracing the interactions not only between the strong
shocks and weak waves, but also between the strong vortex sheet/entropy wave
and weak waves. The key feature of the Euler equations is that the reflection
coefficient is always less than 1, when a weak wave of different family
interacts with the strong vortex sheet/entropy wave or the shock wave, which is
crucial to guarantee that the Glimm functional is decreasing. Then these
estimates are employed to establish the convergence of the approximate
solutions to a global entropy solution, close to the background solution of
steady four-wave configuration.Comment: 9 figures
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Stability of Transonic Characteristic Discontinuities in Two-Dimensional Steady Compressible Euler Flows
For a two-dimensional steady supersonic Euler flow past a convex cornered
wall with right angle, a characteristic discontinuity (vortex sheet and/or
entropy wave) is generated, which separates the supersonic flow from the gas at
rest (hence subsonic). We proved that such a transonic characteristic
discontinuity is structurally stable under small perturbations of the upstream
supersonic flow in . The existence of a weak entropy solution and Lipschitz
continuous free boundary (i.e. characteristic discontinuity) is established. To
achieve this, the problem is formulated as a free boundary problem for a
nonstrictly hyperbolic system of conservation laws; and the free boundary
problem is then solved by analyzing nonlinear wave interactions and employing
the front tracking method.Comment: 26 pages, 3 figure
Steady Euler Flows with Large Vorticity and Characteristic Discontinuities in Arbitrary Infinitely Long Nozzles
We establish the existence and uniqueness of smooth solutions with large
vorticity and weak solutions with vortex sheets/entropy waves for the steady
Euler equations for both compressible and incompressible fluids in arbitrary
infinitely long nozzles. We first develop a new approach to establish the
existence of smooth solutions without assumptions on the sign of the second
derivatives of the horizontal velocity, or the Bernoulli and entropy functions,
at the inlet for the smooth case. Then the existence for the smooth case can be
applied to construct approximate solutions to establish the existence of weak
solutions with vortex sheets/entropy waves by nonlinear arguments. This is the
first result on the global existence of solutions of the multidimensional
steady compressible full Euler equations with free boundaries, which are not
necessarily small perturbations of piecewise constant background solutions. The
subsonic-sonic limit of the solutions is also shown. Finally, through the
incompressible limit, we establish the existence and uniqueness of
incompressible Euler flows in arbitrary infinitely long nozzles for both the
smooth solutions with large vorticity and the weak solutions with vortex
sheets. The methods and techniques developed here will be useful for solving
other problems involving similar difficulties.Comment: 43 pages; 2 figures; To be published in Advances in Mathematics
(2019
Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system
We study supersonic flow past a convex corner which is surrounded by
quiescent gas. When the pressure of the upstream supersonic flow is larger than
that of the quiescent gas, there appears a strong rarefaction wave to rarefy
the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears
to separate the supersonic flow behind the rarefaction wave from the static
gas. In this paper, we employ a wave front tracking method to establish
structural stability of such a flow pattern under non-smooth perturbations of
the upcoming supersonic flow. It is an initial-value/free-boundary problem for
the two-dimensional steady non-isentropic compressible Euler system. The main
ingredients are careful analysis of wave interactions and construction of
suitable Glimm functional, to overcome the difficulty that the strong
rarefaction wave has a large total variation.Comment: 34 pages, 2 figures. Accepted by "Discrete & Continuous Dynamical
Systems - A" for publicatio