Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

Sign-changing solutions of the nonlinear heat equation with persistent singularities

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = Δu + |u|^αu$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < α<α_0$, where $α_0=\frac {4} {N-4+2\sqrt{ N-1 } }\in (\frac {2} {N-2}, \frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $μ>0$ that can be arbitrarily large, we prove that for any $u_0 \in \mathrm{L} ^\infty _{\mathrm{loc}} ({\mathbb R}^N \setminus \{ 0 \}) $ which is bounded at infinity and equals $μ|x|^{- \frac {2} {α}}$ in a neighborhood of $0$, there exists a local (in time) solu…

Recurrence of singularities for second order isotropic pseudodifferential operators

Let $H$ be a self-adjoint isotropic elliptic pseudodifferential operator of order $2$. Denote by $u(t)$ the solution of the Schrödinger equation $(i\partial_t - H)u = 0$ with initial data $u(0) = u_0$. If $u_0$ is compactly supported the solution $u(t)$ is smooth for small $t > 0$, but not for all $t$. We determine the wavefront set of $u(t)$ in terms of the wavefront set of $u_0$ and the principal and subprincipal symbol of $H$.

复二次型 Weyl 量子化的 Shubin-Sobolev 奇异性传播

Propagation of Shubin-Sobolev singularities of Weyl-quantizations of complex quadratic forms

本文在各向同性框架下发展 Hörmander 微局部理论，并借此研究了扩散部分由相空间复二次型 Weyl 量子化给定的演化问题的奇异性传播。

A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants

We analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain $Ω$ we investigate the $\varepsilon \to 0$ asymptotics of the variational problem \[\inf \frac{1}{2}\int_Ω\left( \frac{1}{\varepsilon} W(u)+\varepsilon |\nabla u|^2 + L_\varepsilon(\mathrm{div}\, u)^2 \right) \,dx\] within various parameter regimes for $L_\varepsilon > 0.$ Here $u:Ω\to\mathbb{R}^2$ and $W$ is a potential vanishing on the unit circle and …

Inverse transport with isotropic time-harmonic sources

This paper concerns the reconstruction of the scattering coefficient in a two-dimensional transport equation from angularly averaged measurements when the probing source is isotropic and time-harmonic. This is a practical setting in the medical imaging modality called Optical Tomography. As the modulation frequency of the source increases, we show that the reconstruction of the scattering coefficient improves. More precisely, as the frequency $ω$ increases, we show that all frequencies of the scattering coefficient lower than $b$ are reconstructed stably with an accuracy that improves as $ω$ i…

Estimates of solutions of linear Boltzmann equation at large time and spectral singularities

The spectral analysis of the dissipative linear transport (Boltzmann) operator with polynomial collision integral by the Szokefalvi-Nagy - Foias functional model is given. An exact estimate for the reminder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case. In the general case, it is shown that the operator has finitely many eigenvalues and spectral singularities and an absolutely continuous essential spectrum, and an upper estimate for the reminder is established.