245 research outputs found
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system
We address the structural stability of 3-D axisymmetric subsonic flows with
nonzero swirl for the steady compressible Euler-Poisson system in a cylinder
supplemented with non small boundary data. A special Helmholtz decomposition of
the velocity field is introduced for 3-D axisymmetric flow with a nonzero
swirl(=angular momentum density) component.
With the newly introduced decomposition, a quasilinear elliptic system of
second order is derived from the elliptic modes in Euler-Poisson system for
subsonic flows. Due to the nonzero swirl, the main difficulties lie in the
solvability of a singular elliptic equation which concerns the angular
component of the vorticity in its cylindrical representation, and in analysis
of streamlines near the axis
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
Incompressible Limit of Solutions of Multidimensional Steady Compressible Euler Equations
A compactness framework is formulated for the incompressible limit of
approximate solutions with weak uniform bounds with respect to the adiabatic
exponent for the steady Euler equations for compressible fluids in any
dimension. One of our main observations is that the compactness can be achieved
by using only natural weak estimates for the mass conservation and the
vorticity. Another observation is that the incompressibility of the limit for
the homentropic Euler flow is directly from the continuity equation, while the
incompresibility of the limit for the full Euler flow is from a combination of
all the Euler equations. As direct applications of the compactness framework,
we establish two incompressible limit theorems for multidimensional steady
Euler flows through infinitely long nozzles, which lead to two new existence
theorems for the corresponding problems for multidimensional steady
incompressible Euler equations.Comment: 17 pages; 2 figures. arXiv admin note: text overlap with
arXiv:1311.398
Variational Structure and Two Dimensional Jet Flows for Compressible Euler System with Non-zero Vorticity
In this paper, we investigate the well-posedness theory of compressible jet
flows for two dimensional steady Euler system with non-zero vorticity. One of
the key observations is that the stream function formulation for two
dimensional compressible steady Euler system with non-zero vorticity enjoys a
variational structure, so that the jet problem can be reformulated as a domain
variation problem. This allows us to adapt the framework developed by Alt,
Caffarelli and Friedman for the one-phase free boundary problems to obtain the
existence and uniqueness of smooth solutions to the subsonic jet problem with
non-zero vorticity. We also show that there is a critical mass flux, such that
as long as the incoming mass flux does not exceed the critical value, the
well-posedness theory holds true
- …