7,250 research outputs found
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems
When an upstream steady uniform supersonic flow impinges onto a symmetric
straight-sided wedge, governed by the Euler equations, there are two possible
steady oblique shock configurations if the wedge angle is less than the
detachment angle -- the steady weak shock with supersonic or subsonic
downstream flow (determined by the wedge angle that is less or larger than the
sonic angle) and the steady strong shock with subsonic downstream flow, both of
which satisfy the entropy condition. The fundamental issue -- whether one or
both of the steady weak and strong shocks are physically admissible solutions
-- has been vigorously debated over the past eight decades. In this paper, we
survey some recent developments on the stability analysis of the steady shock
solutions in both the steady and dynamic regimes. For the static stability, we
first show how the stability problem can be formulated as an initial-boundary
value type problem and then reformulate it into a free boundary problem when
the perturbation of both the upstream steady supersonic flow and the wedge
boundary are suitably regular and small, and we finally present some recent
results on the static stability of the steady supersonic and transonic shocks.
For the dynamic stability for potential flow, we first show how the stability
problem can be formulated as an initial-boundary value problem and then use the
self-similarity of the problem to reduce it into a boundary value problem and
further reformulate it into a free boundary problem, and we finally survey some
recent developments in solving this free boundary problem for the existence of
the Prandtl-Meyer configurations that tend to the steady weak supersonic or
transonic oblique shock solutions as time goes to infinity. Some further
developments and mathematical challenges in this direction are also discussed.Comment: 19 pages; 8 figures; accepted by Science China Mathematics on
February 22, 2017 (invited survey paper). doi: 10.1007/s11425-016-9045-
Weak Continuity and Compactness for Nonlinear Partial Differential Equations
We present several examples of fundamental problems involving weak continuity
and compactness for nonlinear partial differential equations, in which
compensated compactness and related ideas have played a significant role. We
first focus on the compactness and convergence of vanishing viscosity solutions
for nonlinear hyperbolic conservation laws, including the inviscid limit from
the Navier-Stokes equations to the Euler equations for homentropy flow, the
vanishing viscosity method to construct the global spherically symmetric
solutions to the multidimensional compressible Euler equations, and the
sonic-subsonic limit of solutions of the full Euler equations for
multidimensional steady compressible fluids. We then analyze the weak
continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding
isometric embeddings in differential geometry. Further references are also
provided for some recent developments on the weak continuity and compactness
for nonlinear partial differential equations.Comment: 29 page
Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity
We are concerned with the global weak continuity of the Cartan structural
system -- or equivalently, the Gauss--Codazzi--Ricci system -- on
semi-Riemannian manifolds with lower regularity. For this purpose, we first
formulate and prove a geometric compensated compactness theorem on vector
bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2),
extending the classical quadratic theorem of compensated compactness. We then
deduce the weak continuity of the Cartan structural system for : For
a family of connection -forms on a
semi-Riemannian manifold , if is uniformly
bounded in and satisfies the Cartan structural system, then any weak
limit of is also a solution of the Cartan
structural system. Moreover, it is proved that isometric immersions of
semi-Riemannian manifolds into semi-Euclidean spaces can be constructed from
the weak solutions of the Cartan structural system or the Gauss--Codazzi--Ricci
system (Theorem 5.1), which leads to the weak continuity of the
Gauss--Codazzi--Ricci system on semi-Riemannian manifolds. As further
applications, the weak continuity of Einstein's constraint equations, general
immersed hypersurfaces, and the quasilinear wave equations is also established.Comment: 64 page
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
Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data
We are concerned with spherically symmetric solutions of the Euler equations
for multidimensional compressible fluids, which are motivated by many important
physical situations. Various evidences indicate that spherically symmetric
solutions of the compressible Euler equations may blow up near the origin at
certain time under some circumstance. The central feature is the strengthening
of waves as they move radially inward. A longstanding open, fundamental
question is whether concentration could form at the origin. In this paper, we
develop a method of vanishing viscosity and related estimate techniques for
viscosity approximate solutions, and establish the convergence of the
approximate solutions to a global finite-energy entropy solution of the
compressible Euler equations with spherical symmetry and large initial data.
This indicates that concentration does not form in the vanishing viscosity
limit, even though the density may blow up at certain time. To achieve this, we
first construct global smooth solutions of appropriate initial-boundary value
problems for the Euler equations with designed viscosity terms, an approximate
pressure function, and boundary conditions, and then we establish the strong
convergence of the viscosity approximate solutions to a finite-energy entropy
solutions of the Euler equations.Comment: 29 page
Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in
We are concerned with the inviscid limit of the Navier-Stokes equations to
the Euler equations for compressible fluids in . Motivated by the
Kolmogorov hypothesis (1941) for incompressible flow, we introduce a
Kolmogorov-type hypothesis for barotropic flows, in which the density and the
sonic speed normally vary significantly. We then observe that the compressible
Kolmogorov-type hypothesis implies the uniform boundedness of some fractional
derivatives of the weighted velocity and sonic speed in the space variables in
, which is independent of the viscosity coefficient . It is shown
that this key observation yields the equicontinuity in both space and time of
the density in and the momentum in , as well as the uniform
bound of the density in and the velocity in independent of
, for some fixed and , where is the
adiabatic exponent. These results lead to the strong convergence of solutions
of the Navier-Stokes equations to a solution of the Euler equations for
barotropic fluids in . Not only do we offer a framework for
mathematical existence theories, but also we offer a framework for the
interpretation of numerical solutions through the identification of a function
space in which convergence should take place, with the bounds that are
independent of , that is in the high Reynolds number limit.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1008.154
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