139 research outputs found
Analysis of some singular solutions in fluid dynamics
Studies on singular flows in which either the velocity fields or the
vorticity fields change dramatically on small regions are of considerable
interests in both the mathematical theory and applications. Important examples
of such flows include supersonic shock waves, boundary layers, and motions of
vortex sheets, whose studies pose many outstanding challenges in both
theoretical and numerical analysis. The aim of this talk is to discuss some of
the key issues in studying such flows and to present some recent progress.
First we deal with a supersonic flow past a perturbed cone, and prove the
global existence of a shock wave for the stationary supersonic gas flow past an
infinite curved and symmetric cone. For a general perturbed cone, a local
existence theory for both steady and unsteady is also established. We then
present a result on global existence and uniqueness of weak solutions to the
2-D Prandtl's system for unsteady boundary layers. Finally, we will discuss
some new results on the analysis of the vortex sheets motions which include the
existence of 2-D vortex sheets with reflection symmetry; and no energy
concentration for steady 3-D axisymmetric vortex sheets
Boundary Layer Problems in the Viscosity-Diffusion Vanishing Limits for the Incompressible MHD Systems
In this paper, we we study boundary layer problems for the incompressible MHD
systems in the presence of physical boundaries with the standard Dirichlet
oundary conditions with small generic viscosity and diffusion coefficients. We
identify a non-trivial class of initial data for which we can establish the
uniform stability of the Prandtl's type boundary layers and prove rigorously
that the solutions to the viscous and diffusive incompressible MHD systems
converges strongly to the superposition of the solution to the ideal MHD
systems with a Prandtl's type boundary layer corrector. One of the main
difficulties is to deal with the effect of the difference between viscosity and
diffusion coefficients and to control the singular boundary layers resulting
from the Dirichlet boundary conditions for both the viscosity and the magnetic
fields. One key derivation here is that for the class of initial data we
identify here, there exist cancelations between the boundary layers of the
velocity field and that of the magnetic fields so that one can use an elaborate
energy method to take advantage this special structure. In addition, in the
case of fixed positive viscosity, we also establish the stability of diffusive
boundary layer for the magnetic field and convergence of solutions in the limit
of zero magnetic diffusion for general initial data.Comment: This paper is translated by published paper in Chinese in "Sciences
in China:Mathematics, Vol 47(2017), No.10, pp1-2
On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity
It is known that smooth solutions to the non-isentropic Navier-Stokes
equations without heat-conductivity may lose their regularities in finite time
in the presence of vacuum. However, in spite of the recent progress on such
blowup phenomenon, it remain to give a possible blowup mechanism. In this
paper, we present a simple continuation principle for such system, which
asserts that the concentration of the density or the temperature occurs in
finite time for a large class of smooth initial data, which is responsible for
the breakdown of classical solutions. It also give an affirmative answer to a
strong version of conjecture proposed by J.Nash in 1950sComment: 17 pages. arXiv admin note: substantial text overlap with
arXiv:1210.593
Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D
In this paper, we prove the global existence of weak solutions to the
non-isothermal nematic liquid crystal system on , based on a new
approximate system which is different from the classical Ginzburg-Landau
approximation. Local energy inequalities are employed to recover the estimates
on the second order spacial derivatives of the director fields locally in time,
which cannot be derived from the basic energy balance. It is shown that these
weak solutions conserve the total energy and while the kinetic and potential
energies transfer to the heat energy precisely. Furthermore, it is also
established that these weak solutions have at most finite many singular times
at which the energy concentration occurs, and as a result, the temperature must
increase suddenly at each singular time on some part of .Comment: 40 page
Global Smooth Supersonic Flows in Infinite Expanding Nozzles
This paper concerns smooth supersonic flows with Lipschitz continuous speed
in two-dimensional infinite expanding nozzles, which are governed by a
quasilinear hyperbolic equation being singular at the sonic and vacuum state.
The flow satisfies the slip condition on the walls and the flow velocity is
prescribed at the inlet. First, it is proved that if the incoming flow is away
from the sonic and vacuum state and its streamlines are rarefactive at the
inlet, then a flow in a straight nozzle never approaches the sonic and vacuum
state in any bounded region. Furthermore, a sufficient and necessary condition
of the incoming flow at the inlet is derived for the existence of a global
smooth supersonic flow in a straight nozzle. Then, it is shown that for each
incoming flow satisfying this condition, there exists uniquely a global smooth
supersonic flow in a symmetric nozzle with convex upper wall. It is noted that
such a flow may contain a vacuum. If there is a vacuum for a global smooth
transonic flow in a symmetric nozzle with convex upper wall, it is proved that
for the symmetric upper part of the flow, the first vacuum point along the
symmetric axis must be located at the upper wall and the set of vacuum points
is the closed domain bounded by the tangent half-line of the upper wall at this
point to downstream and the upper wall after this point. Moreover, the flow
speed is globally Lipschitz continuous in the nozzle, and on the boundary
between the gas and the vacuum, the flow velocity is along this boundary and
the normal derivatives of the flow speed and the square of the sound speed both
are zero. As an immediate consequence, the local smooth transonic flow obtained
in [10] can be extended into a global smooth transonic flow in a symmetric
nozzle whose upper wall after the local flow is convex
On an Elliptic Free Boundary Problem and Subsonic Jet Flows for a Given Surrounding Pressure
This paper concerns compressible subsonic jet flows for a given surrounding
pressure from a two-dimensional finitely long convergent nozzle with straight
solid wall, which are governed by a free boundary problem for a quasilinear
elliptic equation. For a given surrounding pressure and a given incoming mass
flux, we seek a subsonic jet flow with the given incoming mass flux such that
the flow velocity at the inlet is along the normal direction, the flow
satisfies the slip condition at the wall, and the pressure of the flow at the
free boundary coincides with the given surrounding pressure. In general, the
free boundary contains two parts: one is the particle path connected with the
wall and the other is a level set of the velocity potential. We identify a
suitable space of flows in terms of the minimal speed and the maximal velocity
potential difference for the well-posedness of the problem. It is shown that
there is an optimal interval such that there exists a unique subsonic jet flow
in the space iff the length of the nozzle belongs to this interval.
Furthermore, the optimal regularity and other properties of the flows are
shown.Comment: accepted on SIAM J. Math. Ana
Existence of Global Steady Subsonic Euler Flows through Infinitely Long Nozzles
In this paper, we study the global existence of steady subsonic Euler flows
through infinitely long nozzles without the assumption of irrotationality. It
is shown that when the variation of Bernoulli's function in the upstream is
sufficiently small and mass flux is in a suitable regime with an upper critical
value, then there exists a unique global subsonic solution in a suitable class
for a general variable nozzle. One of the main difficulties for the general
steady Euler flows, the governing equations are a mixed elliptic-hyperbolic
system even for uniformly subsonic flows. A key point in our theory is to use a
stream function formulation for compressible Euler equations. By this
formulation, Euler equations are equivalent to a quasilinear second order
equation for a stream function so that the hyperbolicity of the particle path
is already involved. The existence of solution to the boundary value problem
for stream function is obtained with the help of the estimate for elliptic
equation of two variables. The asymptotic behavior for the stream function is
obtained via a blow up argument and energy estimates. This asymptotic behavior,
together with some refined estimates on the stream function, yields the
consistency of the stream function formulation and thus the original Euler
equations.Comment: 1 figur
Incompressible inviscid resistive MHD surface waves in 2D
We consider the dynamics of a layer of an incompressible electrically
conducting fluid interacting with the magnetic field in a two-dimensional
horizontally periodic setting. The upper boundary is in contact with the
atmosphere, and the lower boundary is a rigid flat bottom. We prove the global
well-posedness of the inviscid and resistive problem with surface tension
around a non-horizontal uniform magnetic field; moreover, the solution decays
to the equilibrium almost exponentially. One of the key observations here is an
induced damping structure for the fluid vorticity due to the resistivity and
transversal magnetic field.Comment: 36p
Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities
This paper concerns the existence of global weak solutions to the barotropic
compressible Navier-Stokes equations with degenerate viscosity coefficients. We
construct suitable approximate system which has smooth solutions satisfying the
energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate.
Then, after adapting the compactness results due to Mellet-Vasseur [Comm.
Partial Differential Equations 32 (2007)], we obtain the global existence of
weak solutions to the barotropic compressible Navier-Stokes equations with
degenerate viscosity coefficients in two or three dimensional periodic domains
or whole space for large initial data. This, in particular, solved an open
problem in [P. L. Lions. Mathematical topics in fluid mechanics. Vol. 2.
Compressible models. Oxford University Press, 1998]
Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum
This paper concerns the global well-posedness and large time asymptotic
behavior of strong and classical solutions to the Cauchy problem of the
Navier-Stokes equations for viscous compressible barotropic flows in two or
three spatial dimensions with vacuum as far field density. For strong and
classical solutions, some a priori decay with rates (in large time) for both
the pressure and the spatial gradient of the velocity field are obtained
provided that the initial total energy is suitably {small.} Moreover, by using
these key decay rates and some analysis on the expansion rates of the essential
support of the density, we establish the global existence and uniqueness of
classical solutions (which may be of possibly large oscillations) in two
spatial dimensions, provided the smooth initial data are of small total energy.
In addition, the initial density can even have compact support. This, in
particular, yields the global regularity and uniqueness of the re-normalized
weak solutions of Lions-Feireisl to the two-dimensional compressible barotropic
flows for all adiabatic number provided that the initial total
energy is small.Comment: arXiv admin note: substantial text overlap with arXiv:1004.4749,
arXiv:1207.374
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