Chi Hin Chan, Magdalena Czubak
We consider the fractional Burgers' equation on $\R^N$ with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur \cite{Driftdiffusion} and show existence of smooth solutions given any initial datum in $L^2(\R^N)$.
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The SBV regularity of weak entropy solutions to the Burgers-Poisson equation is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.
The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian $α< 1/2,$ and global existence as well as analyticity of solution for $α\geq 1/2.$ We also prove the existence of solutions with very rough initial data $u_0 \in L^p,$ $1 < p < \infty.$ Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.
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.
Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space
该研究针对带耗散项的二维ZKB方程柯西问题，采用结合耗散效应的Fourier限制范数方法，将其适定性的Sobolev空间指标下限拓展至$s>-1/2$，改进了已有相关方程的适定性结果。
Spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in this paper. It is shown that for any sufficiently large time T, there exists an explicit initial data such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions.
The paper recalls two of the regularity results for Burgers' equation, and discusses what happens in the case of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The first regularity result which is considered is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the x-derivative of u when u is an entropy solution of the Cauchy problem for Burgers' equation with bounded initial data. Its extensions to the case of systems is then mentioned. The second regularity result of debate is Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and gener…
Bounded weak solutions of Burgers' equation $\partial_tu+\partial_x(u^2/2)=0$ that are not entropy solutions need in general not be $BV$. Nevertheless it is known that solutions with finite entropy productions have a $BV$-like structure: a rectifiable jump set of dimension one can be identified, outside which $u$ has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for $BV$ solutions. In the present article we show that the set of non-Lebesgue points of $u$ has Hausdorff dimension at most one. In contra…
In this work, we use the spectral Galerkin method to prove the existence of a pathwise unique mild solution of a fractional stochastic partial differential equation of Burgers type in a Hölder space. We get the temporal regularity and using a combination of Galerkin and exponential-Euler methods, we obtain a fully discretization scheme of the solution. Moreover, we calculate the rates of convergence for both approximations (Galerkin and fully discretization) with respect to time and to space.
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.
Well-posedness and regularity for the Fractional Rough Burgers equation in Sobolev spaces with an application
本文研究了在 $H^s(\mathbb T)$ 空间中由时空噪声驱动的分数粗糙 Burgers 方程的适定性。对于高耗散情况，建立了局部适定性；当耗散系数限制在特定区间时，进一步获得了全局适定性。对于低耗散情况，通过正则性分析推导了 para 控制解。
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 prove the existence and the Besov regularity of the density of the solution to a general parabolic SPDE which includes the stochastic Burgers equation on an unbounded domain. We use an elementary approach based on the fractional integration by parts.
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…
On periodic solutions of the Benjamin-Bona-Mahony-Burgers equation
研究在周期外力作用下于 Sobolev 空间中证明 BBM-Burgers 方程周期解的存在性与稳定性，并将分析推广至伪抛物正则化方程。
The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.
Rough Burger-like SPDEs
本文提出了一种在低正则性区域下，通过改进的粗糙路径方法，建立Burgers型随机偏微分方程路径存在性和唯一性的理论框架。
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。
This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtined for the quasi-geostrophic equation.
We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.
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.…
The paper deals with the analysis of Burgers' equation on acyclic metric graphs. The main goal is to establish the existence of weak solutions in the $TV$ -- class of regularity. A key point is transmission conditions in vertices obeying the Kirchhoff law. First, we consider positive solutions at arbitrary acyclic networks and highlight two kinds of vertices, describing two mechanisms of flow splitting at the vertex. Next we design rules at vertices for solutions of arbitrary sign for any subgraph of hexagonal grid, which leads to a construction of general solutions with $TV$ -- regularity for…
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。
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。
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…
Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation
本文研究Burgers方程的色散扰动（含低色散Benjamin-Ono方程）的Cauchy问题，证明了在特定正则性阈值下的局部适定性、无条件唯一性及先验估计，并基于哈密顿结构推导出全局适定性。
We prove that the KdV-Burgers is globally well-posed in $ H^{-1}(\T) $ with a solution-map that is analytic from $H^{-1}(\T) $ to $C([0,T];H^{-1}(\T))$ whereas it is ill-posed in $ H^s(\T) $, as soon as $ s<-1 $, in the sense that the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\T) $ to even ${\cal D}'(\T) $ at any fixed $ t>0 $ small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dissipation part of the KdV-Burgers equation allows to lower the $ C^\infty $ critical index with respect to the KdV equation, it does not permit to imp…
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…
Remarks on the stabilization of large-scale growth in the 2D Kuramoto-Sivashinsky equation
本文通过观察二维无旋 Burgers 方程中散度场的关键作用，提出基于散度的新正则性条件，并将其与 Ladyzhenskaya-Prodi-Serrin 型准则建立联系，进而用于证明修正 KSE 及二维 Burgers-Sivashinsky 方程的全局正则性。
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 …
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 study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable general…
In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. …
We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first…
We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2d incompressible Euler equations and generalized fractional dissipative 2d Boussinesq equations.
The Burgers' Equation in the Hermite-Sobolev Spaces
该文证明了d维实空间中，当初值属于合适指标的Hermite-Sobolev空间时粘性Burgers方程局部时间解的存在唯一性，还得到了初值为Schwartz空间时解的正则性结果。
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。
We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.
Higher regularity for entropy solutions of conservation laws with geometrically constrained discontinuous flux
本文针对具有间断通量的守恒律熵解正则性低于光滑通量情形的问题，提出在间断通量上施加最优几何条件，从而恢复与同类非线性光滑通量相同的正则性。
We make use of the method of modulus of continuity \cite{K-N-S} and Fourier localization technique \cite{A-H} to prove the global well-posedness of the critical Burgers equation $\partial_{t}u+u\partial_{x}u+Λu=0$ in critical Besov spaces $\dot{B}^{\frac{1}{p}}_{p,1}(\mathbb{R})$ with $p\in[1,\infty)$, where $Λ=\sqrt{-\triangle}$.
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…
This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$ u_t - u |\nabla| u + |\nabla|(u^2) = 0. $$ We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.
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。
Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality
本文研究亚粘性Burgers方程的自相似解，推导近恒等变换方程，构造一阶近似，并用牛顿-拉弗森迭代验证，进而探讨其线性化及二维SQG方程的相关问题。
Asymptotic self-similar blow-up for the regularized Saint-Venant equations
研究正则圣文森特方程中奇点形成机制，通过非线性bootstrap论证建立自相似爆破轮廓稳定性，揭示哈密顿正则化对奇点形成结构的影响。
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 $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$).
We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. Such equations model the incompressible Navier-Stokes systems, generalized viscous Burgers-Hilbert equation and various active scalars. We derive conditions that guarantee the propagation of Lipschitz regularity by the incompressible NSE in the form of a non-local, one dimensional viscous Burgers-type inequality. We show the analogous inequality is always satisfied for …
Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation
本文研究Camassa-Holm方程的奇异性形成，通过构造自相似爆破轮廓、采用调制理论建立估计，证明其梯度爆破解为$C^{3/5}$尖点，首次确定该精细正则性，且发现其与Hunter-Saxton方程奇异性类型一致。
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 study a pressureless Euler system with a nonlinear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian $(-\partial_{xx})^{α/2}$, $α\in (0, 1)$. The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enh…
We study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the "peakon solutions". In particular, we shall show that the local regularity is similar to that exhibited by the solution of the inviscid Burger's equation with the same initial datum. The second goal is to prove that the decay results obtained in a paper of Himonas, Misiołek, Ponce, and Zhou extend to the c…
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.
Averaging lemmas and hypoellipticity
本文使用交换子和基本解方法，为纯动力学输运方程建立平均化引理和亚椭圆估计，并在一般非齐次通量条件下推导出正则性性质，应用于平流-扩散方程及双曲守恒律的动力学形式。
Refined wave breaking for the generalized Fornberg-Whitham equation
本文研究了广义Fornberg-Whitham方程的波破碎现象，通过构造破裂解，揭示了有限时间破裂的机制，并展示了其在自相似变量下的收敛性及正则性性质。
The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.
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.
Regularity results on the flow maps of periodic dispersive Burgers type equations and the Gravity-Capillary equations
本文证明周期色散Burgers方程和重力-毛细波方程的流映射在特定Sobolev空间中的Lipschitz正则性，并验证了先前结果的最优性。
Fractional quadratic obstructions to the local controllability of the Burgers equation
针对线性化系统不可控的Burgers方程，通过分数负指数Sobolev范数量化二次障碍，首次在自然物理PDE中建立连续分数阶障碍尺度，并揭示障碍指数与源剖面正则性的关系。
Space-time fractional stochastic Burgers-type equation
本文研究含Caputo时间分数阶导数、分数阶Laplacian及白噪声的Burgers型方程，通过Da Prato-Debussche分解与粗糙路径理论及Schauder估计，证明解的存在唯一性。
We prove that in arbitrary Carnot groups $\mathbb G$ of step 2, with a splitting $\mathbb G=\mathbb W\cdot\mathbb L$ with $\mathbb L$ one-dimensional, the graph of a continuous function $φ\colon U\subseteq \mathbb W\to \mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $φ$ satisfies, in the distributional sense, a Burgers' type system $D^φφ=ω$, with a continuous $ω$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $φ$ to a Burgers' type system $D^φφ=ω…
We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty}(\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative surface quasi-geostrophic equation and the criti…
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