线性周期波传播速度的几乎有限性

The Freidlin-Gartner formula for general reaction terms

We devise a new geometric approach to study the propagation of disturbance - compactly supported data - in reaction diffusion equations. The method builds a bridge between the propagation of disturbance and of almost planar solutions. It applies to very general reaction-diffusion equations. The main consequences we derive in this paper are: a new proof of the classical Freidlin-Gartner formula for the asymptotic speed of spreading for periodic Fisher-KPP equations, extension of the formula to the monostable, combustion and bistable cases, existence of the asymptotic speed of spreading for equa…

环面上的一类实向量场系统的正则性传播

Propagation of regularity for a class of systems of real vector fields on torus

本文研究环面上一类实向量场系统，通过证明解的正则性传播定理，刻画了系统的全局亚椭圆性、几乎亚椭圆性和可解性及相应平方和算子的性质。

Spectral multipliers and wave propagation for Hamiltonians with a scalar potential

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $-Δ+V$, where $V$ is a scalar real-valued potential. In this paper, we prove resolvent estimates, a dispersive bound for the perturbed wave propagator, Mihlin multiplier and fractional integration bounds, and the full range of wave equation Strichartz estimates, under optimal or almost optimal scaling-invariant conditions on the potential and on the spectral multipliers themselves.

Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation

We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of "propagation of degeneracy" down to arbitrarily small spatial scales: We combine estimates…

一类非自治积分微分方程解的存在性

Existence Results for Some Nonautonomous Integro-differential Equations

本文利用算子理论和Schauder不动点定理，研究了一类具有伪概自守强迫项的非自治积分微分方程的伪概自守解的存在性.为了说明我们的主要结果，伪几乎自守解的存在性抛物型诺依曼边值问题模型人口遗传学和神经脉冲传播将进行讨论。

Transport in a stochastic Goupillaud medium

This paper is part of a project that aims at modelling wave propagation in random media by means of Fourier integral operators. A partial aspect is addressed here, namely explicit models of stochastic, highly irregular transport speeds in one-dimensional transport, which will form the basis for more complex models. Starting from the concept of a Goupillaud medium (a layered medium in which the layer thickness is proportional to the propagation speed), a class of stochastic assumptions and limiting procedures leads to characteristic curves that are Lévy processes. Solutions corresponding to dis…

具有奇异力的Vlasov方程的粒子近似：混沌的传播

Particles approximations of Vlasov equations with singular forces : Propagation of chaos

我们得到了粒子与$1/|x|^α$型奇异相互作用力相互作用系统的平均场极限和混沌的传播，其中$α<1$在$d \geq 3$维中。我们还提供了奇异力的结果高达$α< d-1$，但有足够大的截止。因此，最后一个结果几乎包括了最有趣的库仑相互作用或引力相互作用的情况，但当奇点$α$的强度较大但接近于1时，它也是有趣的，在这种情况下，它允许非常小的截止。

Regularity and lifespan of small solutions to systems of quasi-linear wave equations with multiple speeds,I: almost global existence

In this paper, we show almost global existence of small solutions to the Cauchy problem for symmetric system of wave equations with quadratic (in 3D) or cubic (in 2D) nonlinear terms and multiple propagation speeds. To measure the size of initial data, we employ a weighted Sobolev norm whose regularity index is the smallest among all the admissible Sobolev norms of integer order. We must overcome the difficulty caused by the absence of the $H^1$-$L^p$ Klainerman-Sobolev type inequality, in order to obtain a required a priori bound in the low-order Sobolev norm. The introduction of good substit…

Abnormal acoustic transmission in a waveguide with perforated screens

We consider the propagation of the piston mode in an acoustic waveguide obstructed by two screens with small holes. In general, due to the features of the geometry, almost no energy of the incident wave is transmitted through the structure. The goal of this article is to show that tuning carefully the distance between the two screens, which form a resonator, one can get almost complete transmission. We obtain an explicit criterion, not so obvious to intuit, for this phenomenon to happen. Numerical experiments illustrate the analysis.

Macroscopic Wave Propagation for 2D Lattice with Random Masses

We consider a simple two-dimemsional harmonic lattice with random, independent and identically distributed masses. Using the methods of stochastic homogenization, we show that solutions with long wave initial data converge in an appropriate sense to solutions of an effective wave equation. The convergence is strong and almost sure. In addition, the role of the lattice's dimension in the rate of convergence is discussed. The technique combines energy estimates with powerful classical results about sub-Gaussian random variables.

Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians

We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result

We investigate the effects of the propagation of a non-degenerate Brownian noise through a chain of deterministic differential equations whose coefficients are rough and satisfy a weak like H{ö}rmander structure (i.e. a non-degeneracy condition w.r.t. the components which transmit the noise). In particular we characterize, through suitable counterexamples , almost sharp regularity exponents that ensure that weak well posedness holds for the associated SDE. As a by-product of our approach, we also derive some density estimates of Krylov type for the weak solutions of the considered SDEs.

Hardy's inequality and (almost) the Landau equation

In this manuscript we establish an $L^\infty$ estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent $γ$ in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincaré inequality, which in turn implies the propagation of some $L^p$ norms of solutions. From here, the $L^\infty$ estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.

Random tensors, propagation of randomness, and nonlinear dispersive equations

Abstract. The purpose of this paper is twofold. We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work arXiv:1910.08492, to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we also solve Conjecture 1.7 in arXiv:1910.08492, and establish almost-sure local well-posedness for semilinear Schrödinger equations in spaces that are subcritical in the probabilistic scaling. The solution we find has an explicit expansion in terms of multilinear Gaussians with adapted random tensor co…