Some asymptotic limits for solutions of Burgers equation

一维分形Burgers方程的渐近自相似激波形成

Asymptotically self-similar shock formation for 1d fractal Burgers equation

本文针对$0<α<\frac{1}{3}$的一维分形Burgers方程，构造了从光滑一般初始数据出发在有限时间内形成首次激波的唯一解，该奇点为稳定自相似激波剖面的渐近自相似$H^6$扰动，并精确计算了其时空位置与Hölder正则性，是首个显式构造并精确刻画此类奇异解的例子。

Burgers-Hilbert 方程的最高尖峰波

Highest Cusped Waves for the Burgers-Hilbert equation

针对 Burgers-Hilbert 方程，研究结合渐近分析与计算机辅助方法，证明了周期性最高尖峰行波解的存在性，并刻画其在零点附近的渐近行为。

On the approximation of vorticity fronts by the Burgers-Hilbert equation

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers-Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers-Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

The asymptotic behaviour of solutions of forced Burgers equation on the circle

We describe the asymptotic behaviour of solutions of unviscid Burgers equation on the circle with time-periodic forcing. These solutions converge to periodic states, but the period of these limit states may be greater than the period of the forcing. We obtain as a corollary a new result on the flow of the associated Hamiltonian system.

BBM-Burgers方程Cauchy问题解的大时间行为

Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation

本文研究BBM-Burgers方程Cauchy问题解的大时间行为，证明其趋向Burgers方程的非线性扩散波，构造了依赖初始数据的第二渐近剖面并导出最优渐近率，重点获得了此前未被研究的慢衰减数据情形下的第二渐近剖面。

弱二维模型中KdV-Burgers前沿的全局渐近稳定性

Global asymptotic stability of KdV-Burgers fronts in a weakly two-dimensional model

本文研究弱二维模型中KdV-Burgers方程前沿解的全局渐近稳定性，通过扩展能量方法证明在窄渠道及特定色散参数条件下，一维前沿为全局吸引子且唯一。

Asymptotic properties of entropy solutions to fractal Burgers equation

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{α/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self…

Estimates for solutions of Burgers type equations and some applications

We obtain precise large time asymptotics for the Cauchy problem for Burgers type equations satisfying shock profile condition. The proofs are based on the exact a priori estimates for (local) solutions of these equations and a recent result of the first and second authors.

On the well-posedness and asymptotic behavior of the generalized KdV-Burgers equation

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg-de Vries Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval $[1,5)$. We first prove the global well-posedness in $H^s(R)$, for $0 \leq s \leq 3$ and $1 \leq p < 2$, and in $H^3(R)$, when $p \geq 2$. For $2 \leq p < 5$, we prove the existence of global solutions in the $L^2$-setting. Then, by using multiplier techniques combined with interpolation the…

Global well-posedness and long-time asymptotics of a general nonlinear non-local Burgers Equation

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads $\partial_t u-F(u) (-Δ)^{s/2} u+(-Δ)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. We are interested in solutions stemming from periodic positive bounded initial data. The given function $F \in C^\infty (R^+)$ must satisfy $F' > 0$ a.e. on (0, +$\infty$). For instance, all the functions $F (u) = u^n$ with n $\in$ N * are admissible non-linearities. We construct global classical solutions starting from smooth positive data, and global weak solutions starting…

Radially Symmetric Stationary Wave for Two-dimensional Burgers Equation

We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that such a stationary wave satisfies nice decay estimates and is time-asymptotically nonlinear stable under radially symmetric perturbation.

Burgers-Hilbert 方程的激波相互作用

Shock interactions for the Burgers-Hilbert Equation

研究 Burgers-Hilbert 方程中两激波相互作用点附近解的渐近结构，通过 $H^2$ 正则部分与 |x|ln|x| 型校正项刻画解，并构造一般初值的单激波分段光滑解。

色散-扩散Burgers方程前沿的代数速率渐近吸引

Asymptotic attraction with algebraic rates toward fronts of dispersive-diffusive Burgers equations

本文扩展了BBHY关于色散-扩散Burgers方程前沿稳定性的结果，通过能量估计为KdV-Burgers和分数阶Burgers问题提供了显式代数收敛速率，并推测速率虽非最优但保持代数性。

Burgers型对流扩散方程解中对称性的非线性畸变

Nonlinear distortion of symmetry in solutions to the convection-diffusion equation of Burgers type

本文研究Burgers型对流扩散方程初值问题，明确刻画了解的对称性因非线性效应产生的畸变，发现畸变程度随空间维数奇偶性显著不同，突破了传统认知，且可作为双线性问题研究的参考指标。

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg-de Vries-Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

二维广义Zakharov-Kuznetsov-Burgers方程解的最优衰减估计与渐近轮廓

Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D

本文研究二维广义Zakharov-Kuznetsov-Burgers方程柯西问题，证明解的L^∞最优衰减速率为t^(-3/4)，并结合抛物与薛定谔方程技巧推导出显式渐近轮廓。

Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation

We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball $\mathbb{B}_{r_0}(0)\subset \mathbb{R}^n$ for $n\geq 3$ and some positive constant $r_0>0$, where the boundary data $v_-$ and the far field state $v_+$ of the initial data are prescribed and correspond to a stationary wave. It is shown in \cite{Hashimoto-Matsumura-JDE-2019} that a sufficient condition to guarantee the existence of such a stationary wave is $v_+<0, v_-\leq |v_+|+μ(n-1)/r_0$. Since the stationary wave is no longer monotonic, its nonline…

Unstable shock formation of the Burgers-Hilbert equation

This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates is asymptotic to the first unstable solution to the self-similar inviscid Burgers equation. The blowup profile is a cusp with Hölder 1/5 continuity with explicit blowup time and location. Unlike the previously established stable shocks, the initial data cannot be taken in an open set; instead, we control the two unstable directions by Newton's iteration.

二阶正修正伯格斯方程中的非局部能量耗散与拓扑障碍

Non-Local Energy Dissipation and Topological Obstructions in the Second-Order Positive Modified Burgers' Equation

该研究分析二阶正修正伯格斯方程的非局部积分微分流，发现其非局部耗散会打破能量平衡，导致孤波不稳定，无法形成稳定对称的行波。

The inviscid limit of viscous Burgers at nondegenerate shock formation

We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including $L^\infty$. Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.

Enstrophy growth in the viscous Burgers equation

We study bounds on the enstrophy growth for solutions of the viscous Burgers equation on the unit circle. Using the variational formulation of Lu and Doering, we prove that the maximizer of the enstrophy's rate of change is sharp in the limit of large enstrophy up to a numerical constant but does not saturate the Poincaré inequality for mean-zero 1-periodic functions. Using the dynamical system methods, we give an asymptotic representation of the maximizer in the limit of large enstrophy as a viscous shock on the background of a linear rarefactive wave. This asymptotic construction is used to …

Inviscid limit to the shock waves for the fractal Burgers equation

We show the vanishing viscosity limit to entropy shocks for the fractal Burgers equation in one space dimension. More precisely, we quantify the rate of convergence of the inviscid limit in $L^2$ for large initial perturbations around the entropy shock on any bounded time interval. This is the first result on the inviscid limit to entropy shock for the fractal Burgers equation with the quantified convergence, for large initial perturbations.

Asymptotic stability of viscous shocks in the modular Burgers equation

Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface positio…

Vanishing viscosity limit for viscous Burgers-Vlasov equations

We establish the vanishing viscosity limit of viscous Burgers-Vlasov equations for one dimensional kinetic model about interactions between a viscous fluid and dispersed particles by using compensated compactness technique and the evolution of level sets arguments. The limit we obtained is exactly a finite-energy weak solution to the inviscid equations.

Shock formation for the Burgers-Hilbert equation

We prove finite time blowup of the Burgers-Hilbert equation. We construct smooth initial data with finite $H^5$-norm such that the $L^\infty$-norm of the spacial derivative of the solution blows up. The blowup is an asymptotic self-similar shock at one single point with an explicitly computable blowup profile. The blowup profile is a cusp with Hölder $1/3$ continuity. The blowup time and location are described in terms of explicit ODEs. Our proof uses a transformation to modulated self-similar variables, the quantitative properties of the stable self-similar solution to the inviscid Burgers eq…

Fractional Burgers equation with singular initial condition

We consider the fractional Burgers equation $ Δ^{α/2} u + b\cdot \nabla (u|u|^{(α-1)/β})$ on ${\mathbf R}^d$, $d\geq2$, with {$α\in (1,2)$ and} $β>1$ and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any $L^p({\mathbf R}^d)$, $1\leq p\leq\infty$. Next, we apply the general results to the initial condition $u_0(x)=M|x|^{-β}$, $1<β<d$, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.

Asymptotic structure of cosmological Burgers flows in one and two space dimensions: a numerical study

We study the cosmological Burgers model, as we call it, which is a nonlinear hyperbolic balance law (in one and two spatial variables) posed on an expanding or contracting background. We design a finite volume scheme that is fourth-order in time and second-order in space, and allows us to compute weak solutions containing shock waves. Our main contribution is the study of the asymptotic structure of the solutions as the time variable approaches infinity (in the expanding case) or zero (in the contracting case). We discover that a saddle competition is taking place which involves, on one hand, …

A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space

In this paper we study the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space $ H^{1/2} (\R) $ or $ H^{1/2}(\T) $. We prove the strong convergence in the energy space of the solution to this equation toward the solution of the Benjamin-Ono equation as the dissipation coefficient converges to $ 0 $.

粘性伯格斯方程由点源驱动：弱极限的公式

Viscous Burgers equation driven by point source: a formula for the weak limit

本文研究了由点源驱动的粘性伯格斯方程在粘性系数趋于零时的弱极限问题，提出了一种基于三种类型函数的变分方法。

Pointwise estimates for solutions of fractal Burgers equation

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}u+b\cdot \nabla\left(u|u|^q\right)=0$, $α\in(1,2)$, $b\in\mathbb R^d$ for $q = (α-1)/d$ and $u_0 \in L^1(\mathbb R^d)$. We consider also $q > (α-1)/d$ under additional condition $u_0 \in L^\infty(\mathbb R^d)$. In both cases we assume $u_0\geq0$, which implies that the solution is non-negative. The estimates are given in the terms of the function $P_t u_0$, where $P_t$ is the stable semigroup operator.

粘性 Burgers 型方程中的非线性增强耗散

Nonlinear enhanced dissipation in viscous Burgers type equations

本文构造无限质量初始数据类，证明粘性 Burgers 方程中非线性输运导致比热方程更快的增强耗散，并刻画带边界层的稳定时变渐近剖面。

Critical Keller-Segel meets Burgers on ${\mathbb S}^1$: large-time smooth solutions

We show that solutions to the parabolic-elliptic Keller-Segel system on ${\mathbb S}^1$ with critical fractional diffusion $(-Δ)^\frac{1}{2}$ remain smooth for any initial data and any positive time. This disproves, at least in the periodic setting, the large-data-blowup conjecture by Bournaveas and Calvez. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the method of moduli of continuity by Kiselev, Nazarov and Shterenberg. over a setting where the considered equation has no scaling. This auxiliary result may be interesting by itself.…

Long time behavior for a nonlocal convection diffusion equation

In this paper we consider a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.

The KdV, the Burgers, and the Whitham limit for a spatially periodic Boussinesq model

We are interested in the Korteweg-de Vries (KdV), the Burgers, and the Whitham limit for a spatially periodic Boussinesq model with non-small contrast. We prove estimates between the KdV, the Burgers, and the Whitham approximation and true solutions of the original system which guarantee that these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over the natural time scales of the amplitude equations. The proof is based on Bloch wave analysis and energy estimates. The result is the first justification result of the KdV, the Burgers, an…

$L^2$-收缩性与KdV-伯格斯方程的激波

$L^2$-contraction of Shock Waves for KdV-Burgers Equation

本文研究了KdV-伯格斯方程中粘性色散激波的$L^2$收缩性，证明了其时间渐进稳定性，并探讨了振荡激波的稳定性及结构性质。

Global well-posedness and inviscid limit for the modified Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+ε|\partial_x|^{2α}u=2(u^{3})_x, u(0)=ϕ$, where $0<ε,α\leq 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s (s\geq1)$ for all $ε\in (0,1]$. Moreover, we prove that for any $s\geq 1$ and $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the MKdV equation if $ε$ tends to 0.

Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation $\partial_t u-ε\partial_x^2 u+\mathcal{H}\partial_x^2u+u u_x=0$, where $\mathcal{H}$ denotes the Hilbert transform. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$), whose low-frequency part is scaling critical and high-frequency part is equal to Sobolev space $H^σ$($σ\geq 0$). Furthermore, we also obtain its inviscid limit behavior in $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$).

Lyapunov Exponents for Burgers' Equation

We establish the existence, uniqueness, and stability of the stationary solution of the one-dimensional viscous Burgers equation with the Dirichlet boundary conditions on a finite interval. We obtain explicit formulas for solutions and analytically determine the Lyapunov exponents characterizing the asymptotic behavior of arbitrary solutions approaching the stationary one.

粘性Burgers方程的交叉渐近性与尖锐限制速率

Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation

研究一维粘性Burgers方程在保守边界条件下的交叉渐近行为，揭示临界扩散尺度下解从半直线自相似剖面过渡到区间平衡态的显式交叉剖面，并确定限制的尖锐速率及最优指数界。

Toward nonlinear stability of sources via a modified Burgers equation

Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the …

KdV-伯格斯方程震荡激波的均匀稳定性

Uniform Stability of Oscillatory Shocks for KdV-Burgers Equation

本文研究了KdV-伯格斯方程中震荡激波的均匀稳定性，通过分析激波结构和L²收缩性质，证明了在粘性和色散系数下的时间渐近稳定性和均匀稳定性。

粘性 Fornberg-Whitham 方程解的高阶渐近轮廓

Higher-order asymptotic profiles of the solutions to the viscous Fornberg-Whitham equation

本文研究粘性 Fornberg-Whitham 方程初值问题，证明解的全局存在性及其向非线性扩散波的收敛性，通过构造高阶渐近轮廓分析最优渐近速率与解的详细结构，并探讨非局部色散项的影响。

粘弹性 Burgers 方程的塑性极限——一种海冰动力学的玩具模型

Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics

本文研究了一维粘性 Burgers 方程的塑性极限，提出了一种海冰动力学的玩具模型，通过构造 BV 解并分析极限行为，揭示了塑性效应与应力张量的关系。

具有临界快扩散和奇性的Burgers方程冲击波剖面的非线性稳定性

Nonlinear stability of shock profiles to Burgers' equation with critical fast diffusion and singularity

本文提出了首个研究具有临界快扩散和奇性的Burgers方程的框架，通过引入权函数和加权能量方法克服了$u=0$时的强奇性，证明了在远场消失的粘性冲击波剖面的非线性稳定性，并通过数值模拟验证了理论结果。

Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+ε|\partial_x|^{2α}u+(u^2)_x=0, \ u(0)=ϕ, \end{eqnarray*} where $0<ε,α\leq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s\ (s>s_α)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $ε\in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $ε$ tends to 0.

On the BBM-Burgers Equation: Well-posedness, Ill-posedness and Long Period Limit

In this work we study a dispersive equation with a dissipative term, the Benjamin-Bona-Mahony-Burgers equation. First we prove that the initial value problem for this equation is well-posed in $H^s(\mathbb{R}),$ for $s\geq 0$ and ill-posed if $s< 0.$ The ill-posedness is in the sense that the flow-map cannot be continuous at the origin from $H^s(\mathbb{R})$ to even $\mathcal{D}'(\mathbb{R}).$ Additionally, we establish an exact theory of convergence of the periodic solutions to the continuous one, in Sobolev spaces, as the period goes to infinity.

Asymptotic results for pressureless magneto--hydrodynamics

We are interested in the life span and the asymptotic behaviour of the solutions to a system governing the motion of a pressureless gas, submitted to a strong, inhomogeneous magnetic field $ \e^{-1} B(x)$, of variable amplitude but fixed direction -- this is a first step in the direction of the study of rotating Euler equations. This leads to the study of a multi--dimensional Burgers type system on the velocity field $ u_\e$, penalized by a rotating term $ \e^{-1} u_\e \wedge B(x)$. We prove that the unique, smooth solution of this Burgers system exists on a uniform time interval $ [0,T]$. We …

环面$\n\b\bb{m\na\nt\nh\nb\nb\b{o\nb\b{\n\bT\nb}$上KdV-Burgers与mKdV-Burgers方程的一致适定性及无粘极限

Uniform well-posedness and Inviscid limit for the KdV-Burgers and mKdV-Burgers equations on $\mathbb{T}$

本文研究环面上周期KdV-B与mKdV-B方程的一致适定性及无粘极限，分别在对应Sobolev空间建立无条件一致适定性，证明扩散系数趋于0时解收敛到对应无粘方程的解。

Singular limit of the generalized Burgers equation with absorption

We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $\R\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ in $\R$, as $m\to\infty$ for any $p>1$ and $u_0\in L^1(\R)\cap L^{\infty}(\R)$ or as $p\to\infty$ for any $m>1$ and $u_0\in L^{\infty}(\R)$ . We also show that in general $\underset{p\to\infty}\lim\underset{m\to\infty}\lim u_{m,p}\ne\underset{m\to\infty}\lim\underset{p\to\infty}\lim u_{m,p}$.

多维度Burgers方程平面激波的相对熵与无粘极限

Relative Entropy and the Inviscid Limit for Planar Shocks of the Multi-D Burgers' Equation

本文将近期研究的粘性消失极限分析框架应用于多维粘性Burgers方程，证明大粘性激波在特定扰动下的L²稳定性，并研究其向无粘激波的收敛速率。

Introduction à l'équation de Burgers stochastique et à la burgulence

This paper is an introduction to the theory of 1d stochastic Burgers equation under periodic boundary conditions and with a stochastic force, sufficiently smooth in the space variable. We prove the classical results on the existence and uniqueness of solutions, study their regularity and discuss their properties when the time goes to infinity or the viscosity goes to zero. The latter limit describes the turbulence in the Burgers equation, named by U. Frish "the burgulence". Our paper may be used as an elementary introduction to the modern methods of stochastic PDE.

Mixing in Reaction-Diffusion Systems: Large Phase Offsets

We consider Reaction-Diffusion systems on $\mathbb{R}$, and prove diffusive mixing of asymptotic states $u_0(kx - ϕ_{\pm}, k)$, where $u_0$ is a periodic wave. Our analysis is the first to treat arbitrarily large phase-offsets $ϕ_d = ϕ_{+}- ϕ_{-}$, so long as this offset proceeds in a sufficiently regular manner. The offset $ϕ_d$ completely determines the size of the asymptotic profiles, placing our analysis in the large data setting. In addition, the present result is a global stability result, in the sense that the class of initial data considered are not near the asymptotic profile in any s…

反应-扩散系统中周期波列抵抗$C_{\mathrm{ub}}$-扰动的非线性稳定性与渐近行为

Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against $C_{\mathrm{ub}}$-perturbations

本文提出反应-扩散系统中周期波列的非线性稳定性理论，仅依赖纯$L^\infty$估计，通过引入时空相位调制和Cole-Hopf变换，证明了扩散谱稳定波列抵抗$C_{\mathrm{ub}}$-扰动的稳定性及其渐近收敛行为。

Source-solutions for the multi-dimensional Burgers equation

We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $\rightarrow$ +$\infty$. MSC2010: 35F55, 35L65. Notations. We denote $\times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its …