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., $\mu=\bar{\mu} \theta^{\alpha}, ~ \lambda=\bar{\lambda} \theta^{\alpha},~ \kappa=\bar{\kappa} \theta^{\alpha}$, for constants $0\leq \alpha\leq 1/(\gamma-1)$, $\bar{\mu}>0,~2\bar{\mu}+n\bar{\lambda}\geq0,~\bar{\kappa}\geq 0$, and adiabatic exponent $\gamma>1$). 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 $1/(\gamma-1)$ 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, $u_t=\Delta e^{-\Delta u},$ 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