Acta mathematica scientia,Series B
Acta mathematica scientia,Series B
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REGULARIZATION OF PLANAR VORTICES FOR THE INCOMPRESSIBLE FLOW
Daomin CAO, Shuangjie PENG, Shuangjie PENG
Acta Mathematica Scientia 2018, 38 (
5
): 14431467.
Abstract
（
73
）
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In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations
where 0 <
p
< 1, Ω ? R
^{2}
is a bounded simplyconnected smooth domain,
κ
_{i}
(i=1, …,
k
) is prescribed positive constant. The result we prove is that for any given nondegenerate critical point
x
_{0}
=(
x
_{0,1}
, …,
x
_{0,k}
) of the KirchhoffRouth function defined on Ω
^{k}
corresponding to (
κ
_{1}
, …,
κ
_{k}
), there exists a stationary classical solution approximating stationary
k
points vortex solution. Moreover, as λ → +∞, the vorticity set {
y
:
u
_{λ}
>
κ
_{j}
} ∩
B
_{δ}
(
x
_{0,j}
) shrinks to {
x
_{0,j}
}, and the local vorticity strength near each
x
_{0,j}
approaches
κ
_{j}
,
j
=1, …,
k
. This result makes the study of the above problem with
p
≥ 0 complete since the cases
p
> 1,
p
=1,
p
=0 have already been studied in [11, 12] and [13] respectively.
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YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NONMAXIMAL VOLUME GROWTH
Binglong CHEN, Xiping ZHU
Acta Mathematica Scientia 2018, 38 (
5
): 14681484.
Abstract
（
72
）
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The wellknown Yau's uniformization conjecture states that any complete noncompact Kähler manifold with positive bisectional curvature is biholomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress.
In the second part, we will consider Yau's conjecture for manifolds with nonmaximal volume growth. We will show that the finiteness of the first Chern number
C
_{1}
^{n}
is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions,
C
_{1}
^{n}
is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on Kähler manifolds with minimal volume growth.
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STABILITY OF STEADY MULTIWAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW
GuiQiang G. CHEN, Matthew RIGBY
Acta Mathematica Scientia 2018, 38 (
5
): 14851514.
Abstract
（
22
）
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We are concerned with the stability of steady multiwave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady fourwave 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 multiwave configurations for the twodimensional Euler equations. It is proved that such steady fourwave 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 Glimmtype 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 fourwave configuration.
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ONEDIMENSIONAL VISCOUS RADIATIVE GAS WITH TEMPERATURE DEPENDENT VISCOSITY
Lin HE, Yongkai LIAO, Tao WANG, Huijiang ZHAO
Acta Mathematica Scientia 2018, 38 (
5
): 15151548.
Abstract
（
20
）
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This paper is concerned with the construction of global, large amplitude solutions to the Cauchy problem of the onedimensional compressible NavierStokes system for a viscous radiative gas when the viscosity and heat conductivity coefficients depend on both specific volume and absolute temperature. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same is shown to be hold for the global solution constructed. The proof is based on some detailed analysis on uniform positive lower and upper bounds of the specific volume and absolute temperature.
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MACROSCOPIC REGULARITY FOR THE BOLTZMANN EQUATION
Feimin HUANG, Yong WANG
Acta Mathematica Scientia 2018, 38 (
5
): 15491566.
Abstract
（
22
）
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The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cutoff is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of (
x,t
) in the region R
^{3}
×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible NavierStokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible NavierStokes equations.
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RADIAL SYMMETRY FOR SYSTEMS OF FRACTIONAL LAPLACIAN
Congming LI, Zhigang WU
Acta Mathematica Scientia 2018, 38 (
5
): 15671582.
Abstract
（
17
）
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In this paper, we consider systems of fractional Laplacian equations in R
^{n}
with nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions
u
_{i}
in the critical cases by using a direct method of moving planes introduced in ChenLiLi [11] and some new maximum principles in LiWuXu [27].
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STRONG COMPARISON PRINCIPLES FOR SOME NONLINEAR DEGENERATE ELLIPTIC EQUATIONS
Yanyan LI, Bo WANG
Acta Mathematica Scientia 2018, 38 (
5
): 15831590.
Abstract
（
25
）
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In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ▽
^{2}
ψ
+
L
(
x
, ▽
ψ
), including the conformal hessian operator.
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HÖLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS
Tao TAO, Liqun ZHANG
Acta Mathematica Scientia 2018, 38 (
5
): 15911616.
Abstract
（
18
）
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We show the existence of dissipative Hölder continuous solutions of the Boussinesq equations. More precise, for any
β
∈ (0, 1/5), a time interval[0,
T
] and any given smooth energy profile
e
:[0,
T
] → (0, ∞), there exist a weak solution (
v
,
θ
) of the 3d Boussinesq equations such that (
v
,
θ
) ∈
C
^{β}
(
T
^{3}
×[0,
T
]) with
e
(
t
)=∫
_{T3}

v
(
x, t
)
^{2}
d
x
for all
t
∈[0,
T
]. This extend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].
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REVIEW ON MATHEMATICAL ANALYSIS OF SOME TWOPHASE FLOW MODELS
Huanyao WEN, Lei YAO, Changjiang ZHU
Acta Mathematica Scientia 2018, 38 (
5
): 16171636.
Abstract
（
28
）
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The twophase flow models are commonly used in industrial applications, such as nuclear, power, chemicalprocess, oilandgas, cryogenics, biomedical, microtechnology and so on. This is a survey paper on the study of compressible nonconservative twofluid model, driftflux model and viscous liquidgas twophase flow model. We give the research developments of these three twophase flow models, respectively. In the last part, we give some open problems about the above models.
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