7,942 research outputs found
Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations
This paper considers the immediate blowup of classical solutions to the
vacuum free boundary problem of non-isentropic compressible Navier--Stokes
equations, where the viscosities and the heat conductivity could be constants,
or more physically, the degenerate, temperature-dependent functions which
vanish on the vacuum boundary (i.e., ,
for constants ,
, and
adiabatic exponent ).
In our previous study (Liu and Yuan, Math. Models Methods Appl. Sci. (9) 12,
2019), with three-dimensional spherical symmetry and constant shear viscosity,
vanishing bulk viscosity and heat conductivity, we established a class of
global-in-time large solutions, with bounded entropy and entropy derivatives,
under the condition the decaying rate of the initial density to the vacuum
boundary is of any positive power of the distance function to the boundary. In
this paper we prove that such classical solutions do not exist for any small
time for non-vanishing bulk viscosity, provided the initial velocity is
expanding near the boundary.
When the heat conductivity does not vanish, it is automatically satisfied
that the normal derivative of the temperature of the classical solution across
the free boundary does not degenerate; meanwhile, the entropy of the classical
solution immediately blowups if the decaying rate of the initial density is not
of power of the distance function to the boundary.
The proofs are obtained by the analyzing the boundary behaviors of the
velocity, entropy and temperature, and investigating the maximum principles for
parabolic equations with degenerate coefficients
Temperature Dependence of the Effective Bag Constant and the Radius of a Nucleon in the Global Color Symmetry Model of QCD
We study the temperature dependence of the effective bag constant, the mass,
and the radius of a nucleon in the formalism of the simple global color
symmetry model in the Dyson-Schwinger equation approach of QCD with a
Gaussian-type effective gluon propagator. We obtain that, as the temperature is
lower than a critical value, the effective bag constant and the mass decrease
and the radius increases with the temperature increasing. As the critical
temperature is reached, the effective bag constant and the mass vanish and the
radius tends to infinity. At the same time, the chiral quark condensate
disappears. These phenomena indicate that the deconfinement and the chiral
symmetry restoration phase transitions can take place at high temperature. The
dependence of the critical temperature on the interaction strength parameter in
the effective gluon propagator of the approach is given.Comment: 10 pages, 9 figure
Gradient flow approach to an exponential thin film equation: global existence and latent singularity
In this work, we study a fourth order exponential equation, derived from thin film growth on crystal surface in multiple
space dimensions. We use the gradient flow method in metric space to
characterize the latent singularity in global strong solution, which is
intrinsic due to high degeneration. We define a suitable functional, which
reveals where the singularity happens, and then prove the variational
inequality solution under very weak assumptions for initial data. Moreover, the
existence of global strong solution is established with regular initial data.Comment: latent singularity, curve of maximal slope. arXiv admin note: text
overlap with arXiv:1711.07405 by other author
On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory
It is well known that the self-consistent field (SCF) iteration for solving
the Kohn-Sham (KS) equation often fails to converge, yet there is no clear
explanation. In this paper, we investigate the SCF iteration from the
perspective of minimizing the corresponding KS total energy functional. By
analyzing the second-order Taylor expansion of the KS total energy functional
and estimating the relationship between the Hamiltonian and the part of the
Hessian which is not used in the SCF iteration, we are able to prove global
convergence from an arbitrary initial point and local linear convergence from
an initial point sufficiently close to the solution of the KS equation under
assumptions that the gap between the occupied states and unoccupied states is
sufficiently large and the second-order derivatives of the exchange correlation
functional are uniformly bounded from above. Although these conditions are very
stringent and are almost never satisfied in reality, our analysis is
interesting in the sense that it provides a qualitative prediction of the
behavior of the SCF iteration
- …