204 research outputs found
Liquid Crystal Equations with Infinite Energy Local Well-posedness and Blow Up Criterion
In this paper, we consider the Cauchy problem of the incompressible liquid
crystal equations in dimensions. We prove the local well-posedness of mild
solutions to the liquid crystal equations with initial data, in
particular, the initial energy may be infinite. We prove that the solutions are
smooth with respect to the space variables away from the initial time. Based on
this regularity estimate, we employ the blow up argument and Liouville type
theorems to establish vorticity direction type blow up criterions for the type
I mild solutions established in the present paper
Global Strong and Weak Solutions to Nematic Liquid Crystal Flow in Two Dimensions
We consider the strong and weak solutions to the Cauchy problem of the
inhomogeneous incompressible nematic liquid crystal equations in two
dimensions. We first establish the local existence and uniqueness of strong
solutions by using the standard domain expanding method, and then extend such
local strong solution to be a global one, provided the initial density is away
from vacuum and the initial direction field satisfies some geometric structure.
The size of the initial data can be large. Based on such global existence
results of strong solutions, by using compactness argument, we obtain the
global existence of weak solutions with nonnegative initial density.Comment: 20 page
Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density
In this paper, we consider the initial-boundary value problem to the
nonhomogeneous incompressible Navier-Stokes equations. Local strong solutions
are established, for any initial data , with , and if , then
the strong solution is unique. The initial density is allowed to be
nonnegative, and in particular, the initial vacuum is allowed. The assumption
on the initial data is weaker than the previous widely used one that , and no
compatibility condition is required
Global well-posedness of the 1D compressible Navier-Stokes equations with constant heat conductivity and nonnegative density
In this paper we consider the initial-boundary value problem to the
one-dimensional compressible Navier-Stokes equations for idea gases. Both the
viscous and heat conductive coefficients are assumed to be positive constants,
and the initial density is allowed to have vacuum. Global existence and
uniqueness of strong solutions is established for any initial data, which
generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A. V.;
Shelukhin, V. V.: \emph{Unique global solution with respect to time of initial
boundary value problems for one-dimensional equations of a viscous gas, J.
Appl. Math. Mech., 41 (1977), 273--282.) to the case that with nonnegative
initial density
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 Existence of Strong Solutions to Incompressible MHD
We establish the global existence and uniqueness of strong solutions to the
initial boundary value problem for incompressible MHD equations in a bounded
smooth domain of three spatial dimensions with initial density being allowed to
have vacuum, in particular, the initial density can vanish in a set of positive
Lebessgue measure. More precisely, under the assumption that the production of
the quantities and
is suitably small,
with the smallness depending only on the bound of the initial density and the
domain, we prove that there is a unique strong solution to the Dirichlet
problem of the incompressible MHD system.Comment: 10 pages. Communications on Pure and Applied Analysis, 201
Entropy-bounded solutions to the compressible Navier-Stokes equations: with far field vacuum
The entropy is one of the fundamental states of a fluid and, in the viscous
case, the equation that it satisfies is highly singular in the region close to
the vacuum. In spite of its importance in the gas dynamics, the mathematical
analyses on the behavior of the entropy near the vacuum region, were rarely
carried out; in particular, in the presence of vacuum, either at the far field
or at some isolated interior points, it was unknown if the entropy remains its
boundedness. The results obtained in this paper indicate that the ideal gases
retain their uniform boundedness of the entropy, locally or globally in time,
if the vacuum occurs at the far field only and the density decays slowly enough
at the far field. Precisely, we consider the Cauchy problem to the
one-dimensional full compressible Navier-Stokes equations without heat
conduction, and establish the local and global existence and uniqueness of
entropy-bounded solutions, in the presence of vacuum at the far field only. It
is also shown that, different from the case that with compactly supported
initial density, the compressible Navier-Stokes equations, with slowly decaying
initial density, can propagate the regularities in the inhomogeneous Sobolev
spaces
The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: rigorous justification of the hydrostatic approximation
An important feature of the planetary oceanic dynamics is that the aspect
ratio (the ratio of the depth to horizontal width) is very small. As a result,
the hydrostatic approximation (balance), derived by performing the formal small
aspect ratio limit to the Navier-Stokes equations, is considered as a
fundamental component in the primitive equations of the large-scale ocean. In
this paper, we justify rigorously the small aspect ratio limit of the
Navier-Stokes equations to the primitive equations. Specifically, we prove that
the Navier-Stokes equations, after being scaled appropriately by the small
aspect ratio parameter of the physical domain, converge strongly to the
primitive equations, globally and uniformly in time, and the convergence rate
is of the same order as the aspect ratio parameter. This result validates the
hydrostatic approximation for the large-scale oceanic dynamics. Notably, only
the weak convergence of this small aspect ratio limit was rigorously justified
before.Comment: arXiv admin note: text overlap with arXiv:1604.0169
Existence and uniqueness of weak solutions to viscous primitive equations for certain class of discontinuous initial data
We establish some conditional uniqueness of weak solutions to the viscous
primitive equations, and as an application, we prove the global existence and
uniqueness of weak solutions, with the initial data taken as small
perturbations of functions in the space ; in particular, the
initial data are allowed to be discontinuous. Our result generalizes in a
uniform way the result on the uniqueness of weak solutions with continuous
initial data and that of the so-called -weak solutions.Comment: 33 page
Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: smallness on scaling invariant quantity
In this paper, we consider the Cauchy problem to the heat conductive
compressible Navier-Stokes equations in the presence of vacuum and with vacuum
far field. Global well-posedness of strong solutions is established under the
assumption, among some other regularity and compatibility conditions, that the
scaling invariant quantity
is sufficiently small,
with the smallness depending only on the parameters
and in the system. The total mass can be either finite or infinite
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