13 research outputs found
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
Symmetries and global solvability of the isothermal gas dynamics equations
We study the Cauchy problem associated with the system of two conservation
laws arising in isothermal gas dynamics, in which the pressure and the density
are related by the -law equation with
. Our results complete those obtained earlier for . We
prove the global existence and compactness of entropy solutions generated by
the vanishing viscosity method. The proof relies on compensated compactness
arguments and symmetry group analysis. Interestingly, we make use here of the
fact that the isothermal gas dynamics system is invariant modulo a linear
scaling of the density. This property enables us to reduce our problem to that
with a small initial density. One symmetry group associated with the linear
hyperbolic equations describing all entropies of the Euler equations gives rise
to a fundamental solution with initial data imposed to the line . This
is in contrast to the common approach (when ) which prescribes
initial data on the vacuum line . The entropies we construct here are
weak entropies, i.e. they vanish when the density vanishes. Another feature of
our proof lies in the reduction theorem which makes use of the family of weak
entropies to show that a Young measure must reduce to a Dirac mass. This step
is based on new convergence results for regularized products of measures and
functions of bounded variation.Comment: 29 page
Isothermal limit of entropy solutions of the Euler equations for isentropic gas dynamics
We are concerned with the isothermal limit of entropy solutions in ∞, containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in ∞ of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent →1. This is achieved by combining careful entropy analysis and refined kinetic formulation with a compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent →1. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to >1. The explicit asymptotic analysis of the Riemann solutions containing the vacuum states is also presented
Vanishing dissipation limit for non-isentropic Navier-Stokes equations with shock data
This paper is concerned with the vanishing dissipation limiting problem of
one-dimensional non-isentropic Navier-Stokes equations with shock data. The
limiting problem was solved in 1989 by Hoff-Liu in [13] for isentropic gas with
single shock, but was left open for non-isentropic case. In this paper, we
solve the non-isentropic case, i.e., we first establish the global existence of
solutions to the non-isentropic Navier-Stokes equations with initial
discontinuous shock data, and then show these solutions converge in
norm to a single shock wave of the corresponding Euler equations
away from the shock curve in any finite time interval, as both the viscosity
and heat-conductivity tend to zero.
Different from [13] in which an integrated system was essentially used,
motivated by [21,22], we introduce a time-dependent shift
to the viscous shock so that a weighted
Poincar\'{e} inequality can be applied to overcome the difficulty generated
from the ``bad" sign of the derivative of viscous shock velocity, and the
anti-derivative technique is not needed. We also obtain an intrinsic property
of non-isentropic viscous shock, see Lemma 2.2 below. With the help of Lemma
2.2, we can derive the desired uniform a priori estimates of solutions, which
can be shown to converge in norm to a single inviscid shock in any
given finite time interval away from the shock, as the vanishing dissipation
limit. Moreover, the shift tends to zero in any
finite time as viscosity tends to zero. The proof consists of a scaling
argument, -contraction technique with time-dependent shift to the shock,
and relative entropy method.Comment: All comments are welcome
Global Solutions of the Compressible Euler-Poisson Equations for Plasma with Doping Profile for Large Initial Data of Spherical Symmetry
We establish the global-in-time existence of solutions of finite
relative-energy for the multidimensional compressible Euler-Poisson equations
for plasma with doping profile for large initial data of spherical symmetry.
Both the total initial energy and the initial mass are allowed to be {\it
unbounded}, and the doping profile is allowed to be of large variation. This is
achieved by adapting a class of degenerate density-dependent viscosity terms,
so that a rigorous proof of the inviscid limit of global weak solutions of the
Navier-Stokes-Poisson equations with the density-dependent viscosity terms to
the corresponding global solutions of the Euler-Poisson equations for plasma
with doping profile can be established. New difficulties arise when tackling
the non-zero varied doping profile, which have been overcome by establishing
some novel estimates for the electric field terms so that the neutrality
assumption on the initial data is avoided. In particular, we prove that no
concentration is formed in the inviscid limit for the finite relative-energy
solutions of the compressible Euler-Poisson equations with large doping
profiles in plasma physics.Comment: 42 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