The isometric immersion of two-dimensional Riemannian
manifold with negative Gauss curvature into the
three-dimensional Euclidean space is considered through the
Gauss-Codazzi equations for the first and second fundamental
forms. The large $L^\infty$ solution is obtained which leads to
a $C^{1,1}$ isometric immersion. The approximate solutions are
constructed by the Lax-Friedrichs finite-difference scheme with
the fractional step. The uniform estimate is established by
studying the equations satisfied by the Riemann invariants and
using the sign of the nonlinear part. The $H^{-1}$ compactness
is also derived. A compensated compactness framework is applied
to obtain the existence of large $L^\infty$ solution to the
Gauss-Codazzi equations for the surfaces more general than those
in literature.
In this talk, the unique existence of subsonic Euler-Poisson
flow in a finitely long nozzle will be discussed. Moreover,
under physically acceptable boundary conditions, we prove that
the flow is stable in sense of small perturbation of data at the
entrance and exit. Also, some related recent work on supersonic
and transonic shock problem will be introduced.
10:20-10:40 Tea break
This talk is devoted to the well poshness issue for the
self-gravitating Hookean elastodynamics in dimension three. The
solution is constructed near a constant equilibrium. In the
linear level, the system contains wave equations and
Klein-Gordon equations with different speeds.
We establish a unified blow-up criterion for strong
solutions of various compressible models, including baratropic,
fully compressible and heat-conducting magneto hydrodynamic
flows, which is analogous to the Serrin's criterion of
Navier-Stokes equations. It gives an affirmative answer to a
problem proposed by J. Nash in 1958. As an application, we prove
global-in-time classical solutions of compressible flows
allowing vacuum and large fluctunation of initial data provided
the initial energy is small. In the end, we prove a strong
version of Nash’s problem based on new observations.
12:00-14:00
Lunch
We investigate the stability and instability of Parker
problem of a compressible viscous magnetohydrodynamic (MHD)
fluid of zero resistivity in the presence of an modified
gravitational force in a vertical strip domain, in which the
equilibrium magnetic field is horizontal and vertically
stratified, and the velocity of the fluid is non-slip on the
boundary of the strip domain. We establish a discriminant value
$\mm{C_r}$ for the stability and instability of Parker problem.
More precisely, if $\mm{C_r} <0,$ then the Parker problem is
unstable, that is the Parker instability occurs; if
$\mm{C_r}>0$ and the initial perturbation quantities around a
Parker equilibrium state satisfy some relations, then the Parker
problem is stable, where the solution enjoys the algebraic decay
in time. Our stability result presents that the sufficiently
large horizontal equilibrium magnetic field with vertical
stratification has stabilizing effect so that the heavier part
of gas can persistently float on the lighter one under any small
perturbation, and thus prevents the Parker instability from
occurrence.
In this talk, we consider the existence of weak steady
solution and periodic solution for the compressible flows of
nematic liquid crystals in a 3-D bounded domain with no-slip
boundary condition. By an approximation scheme, we establish the
existence of weak steady solution under the hypothesis
$\gamma>1$ for the adiabatic constant, and the existence of
weak periodic solution under the hypothesis $\gamma>3/2$.
Recently there has been some progress on weak solutions of
the Euler systems, such as the uniqueness question of the
admissible weak solutions. We will review some of the recent
results.
16:00-16:20
Tea Break
In this talk, we consider a class of semilinear
pseudo-parabolic equations. By introducing a family of potential
wells, we prove the invariance of some sets, global existence,
nonexistence and asymptotic behavior of solutions with
sub-critical and critical initial energy respectively. Moreover,
we obtain finite time blow-up with sup-critical initial energy
by comparison principle.
Convexity of shocks is frequently observed in many
experimental results and provides better understanding of
mathematical problems with the nonlinear wave, the uniqueness
for instance. We consider the pseudo-transonic shock governed by
the potential flow equation in the self-similar coordinates, and
give a framework to show the strict and uniform convexity by a
nonlinear and global argument. Finally, several applications are
given.
This talk is concerned with the large-time behavior of
solutions for the one-dimensional compressible Navier-Stokes
system. We show that the combination of viscous contact wave
with rarefaction waves for the non-isentropic polytropic gas is
stable under large initial perturbation without the condition
that the adiabatic exponent $\gamma$ is close to 1, provided the
strength of the combination waves is suitably small.
18:30
Banquet
May 8, Sunday
Consider the dynamics of two layers of immiscible,
compressible, viscous fluid lying atop one another. We establish
a sharp nonlinear stability criterion of the equilibrium. This
is a joint work with Juhi Jang and Ian Tice
This talk examines the initial-value problem for the
two-dimensional magnetohydrodynamic equation with only magnetic
diffusion (without velocity dissipation). There establishes two
main results. The first result features a regularity criterion
in terms of themagnetic field. This criterion comes naturally
from our approach to obtain a global bound for the vorticity.
Due to the lack of velocity dissipation, it is difficult to
conclude the boundedness of the vorticity from the vorticity
equation itself. Instead we derive and involve a new equation
for the combined quantity of the vorticity and a singular
integral operator on the tensor product of the magnetic field.
This criterion may be verifiable. Our second main result is a
weaker version of the small data global existence result, which
is shown by the bootstrap argument.
Entropy weak solutions with bounded periodic initial data
are considered for the system of weakly nonlinear gas dynamics.
Through a modified Glimm scheme, the approximate solution
sequence is constructed, and then a priori estimates are
provided with the methods of approximate characteristics and
approximate conservation laws, which gives the existence and the
uniform bounds for the entropy solutions.
10:00-10:20
Tea Break
I will report our recent result on the bipolar hydrodynamic
model of semiconductors with insulating boundary conditions.
Different from the unipolar case, the corresponding
time-independent model is an elliptic equation with nonlocal
terms. By the calculus of variations, the unique steady state is
gained for N=1,2, without any additional assumption on the
doping profile and the initial densities. Furthermore, the large
time behavior framework of weak solutions and the asymptotic
behavior of smooth solutions are considered. The solutions are
shown to converge to the unique stationary solution with
exponential decay rate.
In this talk, we consider the global existence and
uniqueness of the solutions to the viscous, non-resistive MHD
system. First, we provide a much simplified proof of the main
result in [Lin, Zhang, Comm. Pure Appl. Math. 2014] concerning
the global existence and uniqueness of smooth solutions to the
Cauchy problem for a 3D incompressible complex fluid model under
the assumption that the initial data are close to some
equilibrium states. Beside the classical energy method, the
interpolating inequalities and the algebraic structure of the
equations coming from the incompressibility of the fluid are
crucial in our arguments. We combine the energy estimates with
the $L^\infty$ estimates for time slices to deduce the key $L^1$
in time estimates. The latter is responsible for the global in
time existence. Second, we can obtain the similar result for 2D
the incompressible viscous, non-resistive MHD system under the
assumption that the initial data are close to some equilibrium
states. When the background magnetic field is sufficiently
large, using the idea from Bardos, Sulem and Sulem [Trans. Amer.
Math. Soc. 1988], we can obtain the global strong solutions for
any initial data. Furthermore, we consider the global existence
and uniqueness of the solution for the system with the
time-varying background magnetic field. At last, we remark some
wellposedness results for the 3D viscous and resistive MHD
system.
This paper is dedicated to the global well-posedness issue
of the compressible Oldroyd-B model in the whole space $\mathbb
R^d$ with $d\ge2$. By exploiting the intrinsic structure of the
system, it is shown that this set of equations admits a unique
global solution in a certain critical Besov space provided the
initial data, but not necessarily the coupling parameter, is
small enough. This result extends the previous work by Fang and
the author [{J. Differential Equations}, {256}(2014),
2559--2602] to the non-small coupling parameter case.