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Journal Articles
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1.
Myoungjean Bae, Gui-Qiang
G. Chen & Mikhail Feldman. Prandtl-Meyer
Reflection for Supersonic Flow past a Solid Ramp. December 2011. pdf-file.
Abstract: We present our recent
results on the Prandtl-Meyer reflection for
supersonic potential flow past a solid ramp. When a steady supersonic flow
passes a solid ramp, there are two possible configurations: the weak shock
solution and the strong shock solution. Elling-Liu's
theorem (2008) indicates that the steady supersonic weak shock solution can
be regarded as a long-time asymptotics of an
unsteady flow for a class of physical parameters determined by certain
assumptions for potential flow. In this paper we discuss our recent
progress in removing these assumptions and establishing the stability
theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical
parameters for potential flow. We apply new mathematical techniques
developed in our recent work to obtain monotonicity
properties and uniform apriori estimates for weak
solutions, which allow us to employ the Leray-Schauder
degree argument to complete the theory for the general case.
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2.
Gui-Qiang G. Chen & James Glimm. Kolmogorov's
Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $\R^3$, November
2011. pdf-file. Communications in Mathematical
Physics (to appear 2012)
Abstract: We are concerned with
the inviscid limit of the Navier-Stokes
equations to the Euler equations in $\R^3$. We first observe that a pathwise Kolmogorov
hypothesis implies the uniform boundedness of the
$\alpha^{th}$-order
fractional derivatives of the velocity for some $\alpha>0$ in the space
variables in $L^2$, which is independent of the viscosity $\mu>0$.
Then
it is shown that this key observation yields the $L^2$-equicontinuity
in the time variable and the uniform bound in $L^q$,
for some $q>2$, of the velocity independent of $\mu>0$. These results
lead to the strong convergence of solutions of the Navier-Stokes
equations to a solution of the Euler equations in $\R^3$. We also consider
passive scalars coupled to the incompressible Navier-Stokes
equations and, in this case, find the weak-star convergence for the passive
scalars with a limit in the form of a Young measure (pdf
depending on space and time). 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
$\mu>0$, that is in the high Reynolds number limit.
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3.
Gui-Qiang G. Chen, Xuemei Deng &
Wei Xiang.
Global Steady Subsonic Flows
through Infinitely Long Nozzles for the Full Euler Equations.
November 2011. pdf-file
Abstract: We are concerned with global steady subsonic flows through
general infinitely long nozzles for the full Euler equations. The problem
is formulated as a boundary value problem in the unbounded domain for a
nonlinear elliptic equation of second order in terms of the stream
function. It is established that, when the oscillation of the entropy and
Bernoulli functions at the upstream is sufficiently small in $C^{1,1}$ and the mass flux is in a suitable regime, there
exists a unique global subsonic solution in a suitable class of general
nozzles. The assumptions are required to prevent from the occurrence of
supersonic bubbles inside the nozzles. The asymptotic behavior
of subsonic flows at the downstream and upstream, as well as the critical
mass flux, have been clarified.
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4.
Gui-Qiang G. Chen, Qiang
Ding & Kenneth Karlsen. On Nonlinear Stochastic
Balance Laws. November 2011. pdf-file. arXiv:1111.5217 Archive for Rational Mechanics and
Analysis (in press, 2012)
Abstract:
We
are concerned with multidimensional stochastic balance laws. We identify a
class of nonlinear balance laws for which uniform spatial $BV$ bounds for
vanishing viscosity approximations can be achieved. Moreover, we establish
temporal equicontinuity in $L^1$ of the
approximations, uniformly
in the viscosity coefficient. Using these estimates, we supply a multidimensional
existence theory of stochastic entropy solutions. In addition, we establish
an error estimate for the stochastic viscosity method, as well as an
explicit estimate for the continuous dependence of stochastic entropy
solutions on the flux and random source functions. Various further generalizations
of the results are discussed.
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5.
Gui-Qiang G. Chen & Hairong
Yuan. Local Uniqueness of Steady
Spherical Transonic Shock-fronts for the Three—Dimensional Full Euler Equations.
November 2011. pdf-file.
arXiv:1112.1750
Abstract: We establish the
local uniqueness of steady transonic shock solutions with spherical symmetry
for the three-dimensional full Euler equations. These transonic
shock-fronts are important for understanding transonic shock phenomena in
divergent nozzles. From mathematical point of view, we show the uniqueness
of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite
elliptic--hyperbolic type. To this end, we develop a decomposition of the
Euler system which works in a general Riemannian manifold, a method to
study a Venttsel problem of nonclassical
nonlocal elliptic operators, and an iteration mapping which possesses
locally a unique fixed point. The approach reveals an intrinsic structure
of the steady Euler system and subtle interactions of its elliptic and
hyperbolic part.
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