G. Henkin, A. Shananin, A. Tumanov
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We obtain Schauder estimates for a general class of linear integro-differential equations. The estimates are applied to a scalar non-local Burgers equation and complete the global well-posedness results obtained in \cite{ISV}.
We study the multi-dimensional Burgers equation $u_t + u u_{x_1} + \dots + u^d u_{x_d} = 0$. We prove that the $L^\infty$ norm of entropy solutions of this equation decays polynomially as $t \to \infty$ in terms of the $L^1$ norm of the initial data only.
In this paper, we provide two-sided estimates for the source solution of $d$-dimensional critical fractal Burgers equation $u_t-Δ^{α/2}+b\cdot \nabla\left(u|u|^q\right)=0$, $q=(α-1)/d$, $α\in(1,2)$, $b\in\mathbb{R}^d$, by the density function of the isotropic $α$-stable process.
We consider the KdV-Burgers equation and its linear version in presence of a delay feedback. We prove well-posedness of the models and exponential decay estimates under appropriate conditions on the damping coefficients. Our arguments rely on a Lyapunov functional approach combined with a step by step procedure and semigroup theory.
We present here a version of the existence and uniqueness result of time periodic solutions to the viscous Burgers equation with irregular forcing terms (with Sobolev regularity -1 in space). The key result here is an a priori estimate which is simpler than the previously treated case of forcing terms with regularity -1/2 in time.
We compute some asymptotic limits for solutions of Burgers equation with Cauchy data in $L^1(R)$.
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.
The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\ge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{t\ge0}$ extends uniquely as a continuous semi-group over $L^p(\mathbb{R}^n)$ whenever $1\le p<\infty$, and $u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $p\le q\le \infty$ and $t>0$, $S_t$ actually maps $L^p(\mathbb{R}^n)$ into $L^q(\mathbb{R}^n)$. These results are based upon new dispers…
In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers-Poisson equation for initial data in L^1(R). Additional an Oleinik type estimate is established and some criteria on local smoothness and wave breaking for weak entropy solutions are provided.
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…
In this paper, we investigate the symmetry properties of a variable coefficient nonlinear space-time fractional Burgers' equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators some group invariant solutions are deduced. Further, some exact solutions of fractional Burgers' equation are generated by the invariant subspace method.
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.
We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$α$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $α$ $\le$ 1, is locally well-posed in H^s (R) when s > 3 /2 -- 5$α$ /4. As a consequence, we obtain global well-posedness in the energy space H^{$α$/2} (R) as soon as $α$ > 6/7 .
The formation of singularities in finite time in non-local Burgers' equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise estimates on the blow-up time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.
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.
The aim of this work is to consider the controllability problem of the linear system associated to Korteweg-de Vries Burgers equation posed in the whole real line. We obtain a sort of exact controllability for solutions in $L^2_{loc}(\R^2)$ by deriving an internal observability inequality and a Global Carlemann estimate. Following the ideas contained in \cite{rosier2000}, the problem is reduced to prove an approximate theorem.
We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proof does not rely on energy estimates, but on a combination of the maximum principle and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if the forcing term is zero. To g…
We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two particular choices of the forcing we prove the existence of the periodic globally attracting trajectory and give precise bounds on its shape.
We consider global schemes with H2-estimates for the multivariate Burgers equation and the incompressible Navier-Stokes equation in its Leray projection form. We extend the definition of a global scheme and the global L2-theory of the multivariate Burgers equation to the incompressible Navier-Stokes equation, where we consider estimates of the gradient of the pressure in its Leray projection form in H1 at each time step of the scheme. The extended scheme has a simplified control function (simplified compared to [7]), which controls the growth of the Leray projection term. We use the fact that …
Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable. This is one of the main features of wave turbulence.
We study the existence of monotone heteroclinic traveling waves for a general Fisher-Burgers equation with nonlinear and possibly density-dependent diffusion. Such a model arises, for instance, in physical phenomena where a saturation effect appears for large values of the gradient. We give an estimate for the critical speed (namely, the first speed for which a monotone heteroclinic traveling wave exists) for some different shapes of the reaction term, and we analyze its dependence on a small real parameter when this brakes the diffusion, complementing our study with some numerical simulations…
We give a substantially simplified proof of near-optimal estimate on the Kuramoto-Sivashinsky equation from [F. Otto, "Optimal bounds on the Kuramoto-Sivashinsky equation", JFA 2009], at the same time slightly improving the result. The result in the above cited paper relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving…
We consider a generalized Burger's equation (dtu = dxxu - udxu + up - λu)in a subdomain of R, under various boundary conditions. First, using some phase plane arguments, we study the existence of stationary solutions under Dirichlet or Neumann boundary conditions and prove a bifurcation depending on the parameters. Then, we compare positive solutions of the parabolic equation with appropriate stationary solutions to prove that global existence can occur for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions σdtu+dνu = 0. Finally, for many boundary conditions, global ex…
Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev--Petviashvili--Burgers equation in 2D
本文研究二维广义Kadomtsev–Petviashvili–Burgers方程Cauchy问题，构造解在大时间下的近似公式，并通过$L^{\infty}$范数下界证明已知的$t^{-\frac{7}{4}}$衰减速率是最优的。
We show that the homogeneous viscous Burgers equation $(\partial_t-ηΔ) u(t,x)+(u\cdot\nabla)u(t,x)=0,\ (t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^d$ $(d\ge 1, η>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $η>0$ using Schauder estimates. The result exten…
We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle argument to prove local well-posedness for data with Sobolev regularity below $L^2$. We also prove ill-posedness for this type of models and show that the local well-posedness results are sharp in some particular cases viz., when the orders of dissipation $p$, and nonlinearity $k+1$, satisfy…
A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of propagation. The relative orientation of the directions is critical. Within the directional constraints, the Burgers solutions are arbitrary. The several velocities add, and the pressures combine nonlinearly. These solutions cannot exist beyond the time when shocks develop in any of the Burgers solutions.
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。
We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +νΛ^α u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$ Here $f$ is strongly convex and satisfies an additional growth condition, $Λ=\sqrt{-Δ}$, $ν$ is small and positive, while $α\in (1,\ 2)$ is a constant in the subcritical range. For solutions $u$ of this equation, we generalise the results obtained for the case $α=2$ (i.e. when $-Λ^α$ is the Laplacian) in [10]. We obtain sharp estimates for the time-averaged Sobolev norms of $u$ as a f…
Weak error estimates of Galerkin approximations for the stochastic Burgers equation driven by additive trace-class noise
本文研究了由可加迹类噪声驱动的随机粘性Burgers方程的谱Galerkin逼近，证明了其弱收敛速率为2，是已知强收敛速率的两倍，并利用相关Kolmogorov方程的新正则性结果完成证明。
In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encou…
A Priori Estimates for Singular Fractional Stochastic Burgers Equations
针对奇异分数阶随机伯格斯方程，在特定参数条件下结合响应分解与保守 Zvonkin 变换建立余项估计，得到补偿型非局部扩散算子，为该类方程的先验分析提供了关键方法。
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.
In this article we deal with one-dimensional inverse problems concerning the Burgers equation and some related nonlinear systems (involving heat effects and/or variable density). In these problems, the goal is to find the size of the spatial interval from some appropriate boundary observations of the solution. Depending on the properties of the initial and boundary data, we prove uniqueness and non-uniqueness results. In addition, we also solve some of these inverse problems numerically and compute approximations of the interval sizes.
We consider a non-homogeneous generalised Burgers equation: $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η^ω,\quad t \in \R,\ x \in S^1. $$ Here, νis small and positive, f is strongly convex and satisfies a growth assumption, while η^ω is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to t>=2. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. These esti…
$L^2$-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。
Exact solution of the (2+1)-dimensional damping forcing coupled Burgers equation by using Darboux transformation
本文利用达布变换求解(2+1)维阻尼强迫耦合Burgers方程的精确解，推导了N重达布变换，并展示了孤立波和周期波解。
A Short Survey of the Well-posedness of the Two-dimensional Burgers' Equation
本文基于无穷维动力系统框架，利用Galerkin方法、Sobolev空间及能量估计，证明了二维Burgers方程在适当初始与边界条件下全局解的存在性与唯一性。
We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space-time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.
Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2α}u=0,\quad t\in \mathbb{R}^{+}, x\in \mathbb{R}, $$ with $0\leq α\le 1$. We prove a sharp results on the associated Cauchy problem in the Sobolev space $ {H}^s(\mathbb{R})$. For $s>-\min\{\frac {3+2α}4, 1\}$ we give the well-posedness of solutions of the Cauchy problem, while for $\frac 12\leα\le 1$ and for $s<-\min\{\frac {3+2α}4, 1\}$ we show some ill-posedness issues.
We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner's semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained and some explicit solutions are given by Wigner's semicircle laws. We also construct a bi-Hamiltonian structure for the system of the real and imaginary components of…
Nonlinear enhanced dissipation in viscous Burgers type equations II
本文利用 Hopf-Cole 变换研究粘性 Burgers 方程中无限质量初始数据解的长时间行为，证明其相较热方程具有一般性的增强耗散，并建立各类的全局吸引子。
A Non-Conservative, Non-Local Approximation of the Burgers Equation
本文提出了一种非保守、非局部近似方法，用于Burgers方程的奇异极限分析，通过精确估计和非收敛性证明，探讨了非局部解在不连续初始数据下的行为。
We consider multidimensional stochastic Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. In both cases we show that for positive viscosity $ν>0$ there exists a unique strong global solution in $L^p$ for $p>d$. In the case of torus we also establish a uniform in $ν$ a priori estimate and consider a limit $ν\todown 0$ for potential solutions. In the case of $\Rd$ uniform with respect to $ν$ a priori estimate established if a Beale-Kato-Majda type condition is satisfied.
The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here con- sidered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as $r_t = r_{xx} + 2r_x r$. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be the hereditary. These results are proved v…
We prove $L^2$ stability estimates for entropic shocks among weak, possibly \emph{non-entropic}, solutions of scalar conservation laws $\partial_t u+\partial_x f(u)=0$ with strictly convex flux function $f$. This generalizes previous results by Leger and Vasseur, who proved $L^2$ stability among entropy solutions. Our main result, the estimate \begin{align*} \int_{\mathbb R} |u(t,\cdot)-u_0^{shock}(\cdot -x(t))|^2\,dx\leq \int_{\mathbb R}|u_0-u_0^{shock}|^2 +Cμ_+([0,t]\times\R), \end{align*} for some Lipschitz shift $x(t)$, includes an error term accounting for the positive part of the entropy…
We consider the non-homogeneous generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η,\ t \geq 0,\ x \in S^1. Here f is strongly convex and satisfies a growth condition, νis small and positive, while ηis a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to t>=T_1, where T_1 depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged in time and in ensembl…
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 …
Convergence Rate toward Rarefaction Wave under Periodic Perturbation for Generalized Korteweg-de Vries-Burgers equation
本文研究广义Korteweg-de Vries-Burgers方程在空间周期扰动下稀疏波的稳定性，通过构造携带相同振荡的拟设并利用能量方法，证明了在初始扰动适当小时解趋于稀疏波。
In this paper we first obtain local contraction results in a Hm-norm with respect to time and space for a local scheme. We show that a global controlled scheme preserves higher order regularity with respect to the spatial variables together with polynomial decay of order m of the data at each time step. Here, we simplify the controlled scheme considered in [1] and [2]. Especially, for the simplified controlled scheme we get an upper bound for the Leray projection term. Furthermore, the schemes discussed in [1] and [2] are simplified in the sense that the estimates are achieved without the use …
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。
Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers' equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.
We consider the generalised Burgers equation $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, $$ where $f$ is strongly convex and $ν$ is small and positive. We obtain sharp estimates for Sobolev norms of $u$ (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for small-scale quantities which characterise the decaying Burgers turbulence, i.e. the dissipation length scale, the structure functions and the energy spectrum. The proof uses a quantitative version of an ar…
$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation
本文研究多维Burgers方程粘性激波的大扰动，证明任意大$L^2$扰动在动态位移下以$t^{-1/4}$速率衰减至零，无需小性条件，推广一维结果至多维。
Polynomial mixing for the white-forced hyperviscous Burgers equation on the whole line
研究全直线上白噪声驱动的超粘性伯格斯方程的遍历性，在相空间多个方向受随机强迫假设下，用耦合方法及准则证明动力学以任意幂次多项式速率吸引到唯一不变概率测度。
We consider a multidimensional Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of $\Rd$ we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.
This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linea…
Extinction of multiple shocks in the modular Burgers equation
本文研究模化Burgers方程中多界面激波演化，通过能量估计与数值模拟证明相邻界面会在有限时间内合并并消亡，给出了精确的标度律。
The Inviscid Limit of Viscous Burgers through Shock Formation and Development
该研究扩展了Chaturvedi与Graham的分析，证明了粘性伯格斯方程在激波形成后的无粘极限逐点误差界，结合前人结果得到了无粘极限从初始时间到年轻激波发展的定量描述。
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。