Fumihiko Hirosawa, Jens Wirth
We investigate the long time behaviour of the $L^2$-energy of solutions to wave equations with variable speed. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.
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This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalised Fourier integral operators …
A Critical Density Estimate for the Vlasov--Poisson System: Energy Conservation and Moment Propagation
本文针对全空间和环面上的三维Vlasov-Poisson系统建立临界时空密度估计，解决能量守恒问题并闭合2<k≤3的速度矩传播区间。
Generalized energy conservation for linear wave equations with time-dependent propagation speed
研究具有时变传播速度的波动方程的广义能量守恒条件，通过分析Fourier分量增长推导能量上界，并用反例证明其最优性。
We consider a generalised Webster's equation for describing wave propagation in curved tubular structures such as variable diameter acoustic wave guides. Webster's equation in generalised form has been rigorously derived in a previous article starting from the wave equation, and it approximates cross-sectional averages of the propagating wave. Here, the approximation error is estimated by an a posteriori technique.
In this paper we present a mathematical theory and a numerical method to study the propagation of a three-dimensional (3-D) weak shock front into a polytropic gas in a uniform state and at rest, though the method can be extended to shocks moving into nonuniform flows. The theory is based on the use of 3-D kinematical conservation laws (KCL), which govern the evolution of a surface in general and a shock front in particular. The 3-D KCL, derived purely on geometrical considerations, form an under-determined system of conservation laws. In the present paper the 3-D KCL system is closed by using …
We obtain a result about propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension $N\geq2$. In particular, we investigate conservation of striated and conormal regularity, which is a natural way of generalizing the 2-D structure of vortex patches. The results we get are only local in time, even in the dimension N=2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N=2 or 3, we investigate also propagation of Hölder regularity in the interior of a bounded domain.
Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations
本文提出弱对数凹性概念，证明其可沿广义热半群传播，并基于 HJB 方程的随机控制与反射耦合分析，推导出非凸势下 Schrödinger 算子基态的对数半凹性及抛物方程基本解的双侧对数 Hessian 估计。
The reaction diffusion equation arises in physical situations in problems from population growth, genetics and physical sciences. We consider the generalised Fisher equation in cylindrical coordinates from Lie theory stand point. An invariance method is performed and the optimal set of nonequivalent symmetries is obtained. Finally, the conservation laws are constructed using 'multiplier method'. We determine multipliers as functions of the dependent and independent variables only. The conservation laws are computed and presented in terms of conserved vector corresponding to each multiplier.
We present a new a-priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global $L^2$ bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generali…
Conservation and Breakdown of Modulus of Continuity in the Propagation of Loglog Vortex for 2D Euler equation
本文研究二维 Euler 方程中 loglog 涡在连续扰动下的奇异结构持续性，证明 $0<α<1$ 时连续模可保持传播，并构造光滑扰动说明 $α>1$ 时范数瞬间爆破，从而确立阈值的尖锐性。
Semi-explicit entropic solution to a generalised Riemann problem in some hydrological context
本文研究了一维标量守恒律在特定初始条件下的解，探讨了初始条件跨越阈值时的熵解特性，提供近闭形式表达式。
The article first studies the propagation of well prepared high frequency waves with small amplitude $\eps$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, (1994), about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a new definition of nonlinear flux is stated and compared to classical definitions. Then it is proved that the smoothness expected in Sobolev spaces cannot be exceeded.
Travelling waves in a dispersion-saturating diffusion equation
本文在带扩散与色散正则化的双曲守恒律框架下研究广义Rosenau-KdV方程的单调行波，并在不同参数区域证明了行波的存在性及线性与非线性确定性结果。
$L^2$ Stability of Simple Shocks for Spatially Heterogeneous Conservation Laws
研究具有空间非均匀通量的标量守恒律，确定激波出现条件，在通量混合偏导数条件下建立简单激波剖面的\(L^2\)稳定性，用广义特征和相对熵方法，证明条件严格。
We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting $N$-particle system, and the $N$ times tensorized product of solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results in the case of Lipschitz coefficients, uniform in time propagation of chaos in the case of strictly convex coefficients. But also some recent results as the case of particle in a double well potential.
In this paper, we study the localization and propagation properties of the edge states associated to a class of magnetic laplacians in $\mathbb{R}^2$. We assume that the intensity of the magnetic field has a fast transition along a regular and compact curve $Γ$. Our main results extend to a general regular curve the study of the localised eigenfunction obtained when $Γ$ is a straight line (i.e. Iwatsuka models). Furthermore, we include in our analysis the case of magnetic fields that slowly change along the curve $Γ$ and we obtain a rigorous and explicit characterization of the asymptotic mass…
We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear wave (NLW), nonlinear Schrödinger (NLS), wave maps (WM), Schrödinger maps (SM), generalised Korteweg-de Vries (gKdV), Maxwell-Klein-Gordon (MKG), and Yang-Mills (YM) equations. The classification of the nonlinearity as \emph{subcritical} (weaker than the linear dispersion at high frequencies), \emph{critical} (comparable to the linear dispersion at all frequencies), or \emph{supercritical} (stronger than the linear dispersion at high frequencies)…
A quasi-linear hyperbolic partial differential equation with a discontinuous flux models geologic carbon dioxide migration and storage. Dual flux curves characterize the model, giving rise to flux discontinuities. One convex flux describes the invasion of the plume into pore space, and the other captures the flow as the plume leaves carbon dioxide bubbles behind, which are then trapped in the pore space. We investigate the method of characteristics, the structure of shock and rarefaction waves, and the result of binary wave interactions. The dual flux property introduces unexpected differences…
In this paper, we are interested in a generalised Vlasov equation, which describes the evolution of the probability density of a particle evolving according to a generalised Vlasov dynamic. The achievement of the paper is twofold. Firstly, we obtain a quantitative rate of convergence to the stationary solution in the Wasserstein metric. Secondly, we provide a many-particle approximation for the equation and show that the approximate system satisfies the propagation of chaos property.
Maximal entropy dissipation numerical scheme for conservation law systems
本文针对守恒律系统，提出一种适配最大耗散原理的数值有限体积法，基于最小化熵泛函构造数值通量，其极限解与Glimm或波前追踪算法的经典弱解进行了对比。
Conservation laws of nonlinear PDEs arising in elasticity and acoustics in Cartesian, cylindrical, and spherical geometries
本文通过尺度齐次性和乘子法，研究了弹性力学与声学中非线性偏微分方程的守恒律，包括剪切波传播模型和参数化本构方程组。
Conservation laws and exact solutions of a nonlinear acoustics equation by classical symmetry reduction
本文研究了广义韦斯特维尔特方程的对称性和保量律，通过约简发现点对称性和局部保量律，推导出潜在系统和非局部保量律，并研究了行波解导致的激波现象。
The main purpose of this review is to summarize the recent advances of the Conservation-Dissipation Formalism (CDF), a new way for constructing both thermodynamically compatible and mathematically stable and well-posed models for irreversible processes. The contents include but are not restricted to the CDF's physical motivations, mathematical foundations, formulations of several classical models in mathematical physics from master equations and Fokker-Planck equations to Boltzmann equations and quasi-linear Maxwell equations, as well as novel applications in the fields of non-Fourier heat con…
Propagation of Love waves in linear elastic isotropic Cosserat materials
本研究提出基于代数Riccati方程的方法分析Cosserat材料中Love波传播，保证亚声速波速存在唯一性，并确认无需人为表面层。
Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions
本文研究了非凸势下弱相互作用扩散的平均场极限，探讨了LSI常数与相变关系，并证明LSI非退化性对时间一致的混沌传播和平衡涨落有重要影响。推广了无界自旋系统结果。
We introduce integral identities to define delta-shock wave type solutions for the multidimensional zero-pressure gas dynamics Using these integral identities, the Rankine-Hugoniot conditions for delta-shocks are obtained. We derive the balance laws describing mass, momentum, and energy transport from the area outside the delta-shock wave front onto this front. These processes are going on in such a way that the total mass, momentum, and energy are conserved and at the same time mass and energy of the moving delta-shock wave front are increasing quantities. In addition, the total kinetic energ…
Models for information propagation on graphs
本文提出并统一了图上信息传播的三类模型（波前传播、广义旅行时间最小化、程函方程），证明其等价性，并应用于半监督学习和信任网络。
We study the semiclassical limit of the (generalised) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in $H^s$ to the solution of the Hopf equation, provided the initial data belongs to $H^s$, ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.
Finite volume schemes for dispersive wave propagation and runup
本文扩展有限体积方法至Boussinesq型色散水波模型，用于模拟一维双向非线性色散波传播，重点研究孤立波相互作用、色散激波形成以及破碎与非破碎长波爬高等非线性现象。
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