We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation ()^<i></i>/2, 0 &lt; <i></i> 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then <i></i> is a classical solution in . This result improves our previous result in [18].

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.

We investigate the fractional Schrödinger equation with a periodic PT-symmetric potential. In the inverse space, the problem transfers into a first-order nonlocal frequency-delay partial differential equation. We show that at a critical point, the band structure becomes linear and symmetric in the one-dimensional case, which results in a nondiffracting propagation and conical diffraction of input beams. If only one channel in the periodic potential is excited, adjacent channels become uniformly excited along the propagation direction, which can be used to generate laser beams of high power and narrow width. In the two-dimensional case, there appears conical diffraction that depends on the competition between the fractional Laplacian operator and the PT-symmetric potential. This investigation may find applications in novel on-chip optical devices.

This paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries, which is described by a degenerate and strongly coupled parabolic system. We prove the global (in time) existence of weak solution by constructing an approximate solution with some standard smoothing. The proof is based on the physical nature of gas convection, in which the heat (energy) flux in convection is determined by the mass (vapor) flux in convection.

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half-space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.