Commuting differential operators with regular singularities

Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities

We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic re…

On the regularity problem of complex Monge-Ampere equations with conical singularities

In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2π)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coordinates. This shows the weak Kähler-Einstein metrics constructed by Guenancia-Paun \cite{GP}, and independently by Yao \cite{GT}, are all actually strong-conical Kähler-Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kähler-Ricci flat metri…

具有可变奇异性和测度数据的半线性混合局部-非局部方程的正则性与存在性

Regularity and existence for semilinear mixed local-nonlocal equations with variable singularities and measure data

本文证明了一类具有可变奇异指数和测度数据的混合局部-非局部奇异问题弱解的存在性与正则性，其中源项可同时为测度，该现象在常数奇异指数情形下也是新的。

Double spiral singularities for a flow of regular planar curves

In this paper we study the singularity formation for the geometric flow of complex curves $$z_t = -z_{xxx} + \frac{3}{2}øz_{x} z_{xx}^2,$$ that was derived [R. E. Goldstein and D. M. Petrich, {\em Phys. Rev. Lett.}, 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlevé II transcendents and furthermore we es…

On global regularity and singularities of Navier-Stokes- and Euler equation solutions

Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the …

Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations

In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary conditions or to the nonlinearity of the equations can effect whether the equations develop finite-time singularities. In particular, we aim to underscore the idea that in analytical and computational investigations of the blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary conditions may need to be taken into greater account. We als…

含可变奇异性与测度数据的混合局部-非局部抛物方程的正则性与存在性

Regularity and existence for a mixed local-nonlocal parabolic equation with variable singularities and measure data

该研究针对含奇异非线性项与测度数据的混合局部-非局部抛物问题，引入可变奇异指数，证明了弱解的存在性、非存在性、正则性及渐近行为，相关结果对纯局部算子也成立。

Uniform Regularity close to Cross Singularities in an Unstable Free Boundary Problem

We introduce a new method for the analysis of singularities in the unstable problem $$Δu = -χ_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neig…

On Type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations

Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.

Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values

In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results.

On partial regularity for the steady Hall magnetohydrodynamics system

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $Ω\subset \Bbb R^3$. In particular we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $Ω=\Bbb R^3$, we show that the set of possible singularities is compact.

Inhomogeneous microlocal propagation of singularities in Fourier Lebesgue spaces

Some results of microlocal continuity for pseudodifferential operators whose non regular symbols belong to weighted Fourier Lebesgue spaces are given. Inhomogeneous local and microlocal propagation of singularities of Fourier Lebesgue type are then studied, with applications to some classes of semilinear equations.

Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal $L^{q}$-regularity space for all times and is instantaneously smooth in space and time, where the maximal $L^{q}$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

Distributional solutions of the stationary nonlinear Schrödinger equation: singularities, regularity and exponential decay

We consider the nonlinear Schrödinger equation $-Δu + V(x) u = Γ(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-Δ+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,Γ$ are bounded on $\R^n\setminus B_1(0)$ and satisfy suitable Hölder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1<p<\frac{n}{n-2}$, however, we show that every distributional solution of the more general equat…

Propagation of singularities for the wave equation on conic manifolds

For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, non-direct, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave.

有限时间出现奇点的正则平面曲线流的数值研究

Numerical study of a flow of regular planar curves that develop singularities at finite time

本文数值研究了Goldstein-Petrich几何流，该流在Euler方程下描述平面涡斑演化，并再现了Perelman-Vega证明的有限时间角状奇点形成。

On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy problem for the inhomogeneous heat equation, when the free term lies in a weighted parabolic Hölder or Sobolev space with discrete asymptotics. This generalizes a result previously obtained by Coriasco, Schrohe, and Seiler (Thm. 7.2 in Math. Z. 244 (2003), 235--269) by different mean…

Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition

We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions, and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularising terms, of parametrices with prescribed b-wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the b-wavefront set.