1,975 research outputs found
Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
We are concerned with the nonlinear stability of vortex sheets for the
relativistic Euler equations in three-dimensional Minkowski spacetime. This is
a nonlinear hyperbolic problem with a characteristic free boundary. In this
paper, we introduce a new symmetrization by choosing appropriate functions as
primary unknowns. A necessary and sufficient condition for the weakly linear
stability of relativistic vortex sheets is obtained by analyzing the roots of
the Lopatinski\u{\i} determinant associated to the constant coefficient
linearized problem. Under this stability condition, we show that the variable
coefficient linearized problem obeys an energy estimate with a loss of
derivatives. The construction of certain weight functions plays a crucial role
in absorbing error terms caused by microlocalization. Based on the weakly
linear stability result, we establish the existence and nonlinear stability of
relativistic vortex sheets under small initial perturbations by a Nash--Moser
iteration scheme.Comment: 105 pages; to appear in: Arch. Ration. Mech. Anal. 201
Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding
We establish the weak continuity of the Gauss-Coddazi-Ricci system for
isometric embedding with respect to the uniform -bounded solution sequence
for , which implies that the weak limit of the isometric embeddings of the
manifold is still an isometric embedding. More generally, we establish a
compensated compactness framework for the Gauss-Codazzi-Ricci system in
differential geometry. That is, given any sequence of approximate solutions to
this system which is uniformly bounded in and has reasonable bounds on
the errors made in the approximation (the errors are confined in a compact
subset of ), then the approximating sequence has a weakly
convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci
system. Furthermore, a minimizing problem is proposed as a selection criterion.
For these, no restriction on the Riemann curvature tensor is made
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
Isometric embedding via strongly symmetric positive systems
We give a new proof for the local existence of a smooth isometric embedding
of a smooth -dimensional Riemannian manifold with nonzero Riemannian
curvature tensor into -dimensional Euclidean space. Our proof avoids the
sophisticated arguments via microlocal analysis used in earlier proofs.
In Part 1, we introduce a new type of system of partial differential
equations, which is not one of the standard types (elliptic, hyperbolic,
parabolic) but satisfies a property called strong symmetric positivity. Such a
PDE system is a generalization of and has properties similar to a system of
ordinary differential equations with a regular singular point. A local
existence theorem is then established by using a novel local-to-global-to-local
approach. In Part 2, we apply this theorem to prove the local existence result
for isometric embeddings.Comment: 39 page
Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density
We are concerned with the global existence theory for spherically symmetric
solutions of the multidimensional compressible Euler equations with large
initial data of positive far-field density. The central feature of the
solutions is the strengthening of waves as they move radially inward toward the
origin. Various examples have shown that the spherically symmetric solutions of
the Euler equations blow up near the origin at certain time. A fundamental
unsolved problem is whether the density of the global solution would form
concentration to become a measure near the origin for the case when the total
initial-energy is unbounded. Another longstanding problem is whether a rigorous
proof could be provided for the inviscid limit of the multidimensional
compressible Navier-Stokes to Euler equations with large initial data. In this
paper, we establish a global existence theory for spherically symmetric
solutions of the compressible Euler equations with large initial data of
positive far-field density and relative finite-energy. This is achieved by
developing a new approach via adapting a class of degenerate density-dependent
viscosity terms, so that a rigorous proof of the vanishing viscosity limit of
global weak solutions of the Navier-Stokes equations with the density-dependent
viscosity terms to the corresponding global solution of the Euler equations
with large initial data of spherical symmetry and positive far-field density
can be obtained. One of our main observations is that the adapted class of
degenerate density-dependent viscosity terms not only includes the viscosity
terms for the Navier-Stokes equations for shallow water (Saint Venant) flows
but also, more importantly, is suitable to achieve our key objective of this
paper. These results indicate that concentration is not formed in the vanishing
viscosity limit even when the total initial-energy is unbounded.Comment: 57 page
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
A Fluid Dynamic Formulation of the Isometric Embedding Problem in Differential Geometry
The isometric embedding problem is a fundamental problem in differential
geometry. A longstanding problem is considered in this paper to characterize
intrinsic metrics on a two-dimensional Riemannian manifold which can be
realized as isometric immersions into the three-dimensional Euclidean space. A
remarkable connection between gas dynamics and differential geometry is
discussed. It is shown how the fluid dynamics can be used to formulate a
geometry problem. The equations of gas dynamics are first reviewed. Then the
formulation using the fluid dynamic variables in conservation laws of gas
dynamics is presented for the isometric embedding problem in differential
geometry.Comment: arXiv admin note: substantial text overlap with arXiv:0805.243
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