Non completely solvable systems of complex first order PDE's

Solvable structures for evolution PDEs admitting differential constraints

Solvable structures are exploited in order to find families of explicit solutions to evolution PDEs admitting suitable differential constraints. The effectiveness of the method is verified on several explicit examples.

Completely exceptional $2^\textrm{nd}$ order PDEs via conformal geometry and BGG resolution

By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called 'completely exceptional', where such a transition does not occur after a finite time. A straightforward integration of the completely exceptionality conditions allowed Boillat to show that such PDEs are actually of Monge-Ampere type. In this paper, we first recast these conditions in terms of characteristics, and then we show that the completely exceptional PDEs, with 2 or 3 independent variables, can be described in terms of the conformal geometry of the Lagrangian…

Asymptotic average solutions to linear second order semi-elliptic PDEs: a Pizzetti-type Theorem

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$, where $\mathcal{L}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form.

一类强非共振调制色散偏微分方程的噪声正则化

Regularization by noise for some strongly non-resonant modulated dispersive PDEs

本研究拓展了噪声正则化色散偏微分方程的相关工作，证明强非共振且至多含完全共振项的调制色散PDE，在色散前噪声足够不规则时任意正则性下适定，量化了噪声不规则性要求并验证了多类典型方程的适用性。

Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the starting elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel…

非标准双曲型PDE交通流模型的分析与控制

Analysis and Control of a Non-Standard Hyperbolic PDE Traffic Flow Model

本文研究了非标准双曲型一维非线性交通流模型，证明了其全局解的存在性与唯一性，并提出基于边界反馈的全局稳定控制方法。