Massimiliano Berti, Alberto Maspero, Federico Murgante
We consider the Euler-Korteweg system with space periodic boundary conditions $ x \in \mathbb T^d $. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.
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This short note establishes instability of standing solitary waves in the Euler-Korteweg system.
In this article we study the principle of energy conservation for the Euler-Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler-Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.
We establish a relative energy framework for the Euler-Korteweg system with non-convex energy. This allows us to prove weak-strong uniqueness and to show convergence to a Cahn-Hilliard system in the large friction limit. We also use relative energy to show that solutions of Euler-Korteweg with convex energy converge to solutions of the Euler system in the vanishing capillarity limit, as long as the latter admits sufficiently regular strong solutions.
We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler-Langevin-Korteweg system, which corresponds to the Euler-Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier-Stokes-Langevin-Korteweg system. Introducing a relative entropy f…
The aim of this paper is to investigate the singular relaxation limits for the Euler-Korteweg and the Navier-Stokes-Korteweg system in the high friction regime. We shall prove that the viscosity term is present only in higher orders in the proposed scaling and therefore it does not affect the limiting dynamics, and the two models share the same equilibrium equation. The analysis of the limit is carried out using the relative entropy techniques in the framework of weak, finite energy solutions of the relaxation models converging toward smooth solutions of the equilibrium. The results proved her…
Cahn-Hillard and Keller-Segel systems as high-friction limits of Euler-Korteweg and Euler-Poisson equations
本文研究了带摩擦的Euler-Korteweg和Euler-Poisson方程联合系统，证明了耗散测度值解的存在性，并证明在γ≥2时Cahn-Hillard-Keller-Segel系统的强解是该系统的高摩擦极限。
In this paper, the main objective is to generalize to the Navier-Stokes-Korteweg (with density dependent viscosities satisfying the BD relation) and Euler-Korteweg systems a recent relative entropy [proposed by D. Bresch, P. Noble and J.--P. Vila, (2016)] introduced for the compressible Navier-Stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier-Stokes system [recently obtained simultaneously by I. Lacroix-Violet and A. Vasse…
We prove the convergence in the transonic limit of two-dimensional traveling waves of the E-K system, up to rescaling, toward a ground state of the Kadomtsev-Petviashvili Equation. Similarly, in dimension one we prove the convergence in the transonic limit of solitons toward the soliton of the Korteweg de Vries equation.
On periodic traveling wave solutions with or without phase transition to the Navier-Stokes-Korteweg and the Euler-Korteweg equations
研究等温两相流扩散界面模型的周期行波解，证明Navier-Stokes-Korteweg方程无带相变周期行波解但有非相变解，Euler-Korteweg方程在特定参数下存在带相变周期行波解且给出其渐近性，还通过数值验证了小粘度下的稳定性。
The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg system and the two-fluid Euler Maxwell system. We prove that the lower bound of the lifespan of these systems is no less than the inverse of the $H^s$ $(s>5/2)$ norm of the rotational part of the initial velocity. Our approach is based on energy estimates and the fast time decay results of global solutions to these systems with small irrotational initial da…
Global existence of Euler-Korteweg equations with the non-monotone pressure
针对三维空间中具有非单调压力的可压缩Euler-Korteweg方程，在声速为零的情况下将无旋流体扰动问题转化为准线性薛定谔方程，借助色散估计和正规形方法构造出三维情形的整体散射解。
Under a natural stability condition on the pressure, it is known that for small irrotational initial data, the solutions of the Euler-Korteweg system are global in time. When the initial velocity has a small rotational part, we obtain a lower bound on the time of existence that depends only on the rotational part. In the zero vorticity limit we recover the previous global well-posedness result. Independently of this analysis, we also provide (in a special case) a simple example of solution that blows up in finite time.
From nonlocal Euler-Korteweg to local Cahn-Hilliard via the high-friction limit
该研究沿用流体力学高摩擦极限研究思路，利用相对熵方法从非局部Euler-Korteweg方程严格推导出非局部及局部退化Cahn-Hilliard方程，为推导无经典解的方程提供了新方法。
We investigate the existence and properties of traveling waves for the Euler-Korteweg system with general capillarity and pressure. Our main result is the existence in dimension two of waves with arbitrarily small energy. They are obtained as minimizers of a modified energy with fixed momentum. The proof follows various ideas developed for the Gross-Pitaevskii equation (and more generally nonlinear Schrödinger equations with non zero limit at infinity). Even in the Schrödinger case, the fact that we work with the hydrodynamical variables and a general pressure law both brings new difficulties …
Lower bounds for the lifespan of solutions to the Euler--Korteweg equations
针对无旋流中的Euler--Korteweg方程初值问题，建立了生命跨度的定量下界，明确依赖Weber数、Mach数和Sobolev范数，并在低Weber数区域中可任意大。
In this paper we analyze the high friction regime for the Navier Stokes Korteweg equations for multicomponent systems. According to the shape of the mixing and friction terms, we shall perform two limits: the high friction limit toward an equilibrium system for the limit densities and the barycentric velocity, and, after an appropriate time scaling, the diffusive relaxation toward parabolic, gradient flow equations for the limit densities. The rigorous justification of these limits is done by means of relative entropy techniques in the framework of weak, finite energy solutions of the relaxati…
The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\geq 3$ for small irrotatio…
We consider a general Euler-Korteweg-Poisson system in $R^3$, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solut…
For an Euler system, with dynamics generated by a potential energy functional, we propose a functional format for the relative energy and derive a relative energy identity. The latter, when applied to specific energies, yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system…
We show that solitary waves for the 2D Euler-Korteweg model for capillary fluids display nonlinear instability when subjected to transverse perturbations.
The zero capillarity limit for the Euler-Korteweg system with no-flux boundary conditions
本文研究了Euler-Korteweg系统在光滑边界域中的小散射极限，采用相对能量方法分析有限能量弱解向压缩性Euler系统强解的收敛性，通过修正极限粒子密度克服边界层影响，无需额外条件限制毛细能异常集中。
On the zero capillarity limit for the Euler-Korteweg system
本文研究带弱毛细张量的Euler-Korteweg方程，在R^d中严格证明其零毛细极限收敛于Euler方程并得到更精确BKW展开，在半空间分析解的先验估计退化性及边界层，扩展改进经典结果。
The high-friction limit in Euler-Korteweg equations for fluid mixtures is analyzed. The convergence of the solutions towards the zeroth-order limiting system and the first-order correction is shown, assuming suitable uniform bounds. Three results are proved: The first-order correction system is shown to be of Maxwell-Stefan type and its diffusive part is parabolic in the sense of Petrovskii. The high-friction limit towards the first-order Chapman-Enskog approximate system is proved in the weak-strong solution context for general Euler-Korteweg systems. Finally, the limit towards the zeroth-ord…
Extending relative entropy for Korteweg-Type models with non-monotone pressure:large friction limit and weak-strong uniqueness
本文利用相对熵技术，通过漂移速度扩展表述，研究了非单调压力下Euler-Korteweg和Navier-Stokes-Korteweg系统的弱-强唯一性及大摩擦极限，推广了已有结果。
In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a $γ$ law for the pressure ($P(ρ)=aρ^γ$, $γ>1$) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where t…
Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg-de Vries equation with initial data in $H^s$ are established for $s>0$. The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below $L^2$ is proved conditional upon conjectured linear Strichartz estimates.
Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions
本文研究了保持结构的有限体积方案对欧拉-纳维-斯托克斯-科特韦格方程组的收敛性，证明其数值解在网格细化下收敛于能量-变分解，提出了一种新的解概念并扩展到纳维-斯托克斯-科特韦格模型。
This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$, then the corresponding solution $u(\cdot,t)$ belongs to $H^l((β,\infty))$ for any $β\in \mathbb R$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $\,l\in \mathbb R^+$.
On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below $H^{-\frac{1}{2}}(\mathbb{T})$
本文利用调制限制正规形方法，在不依赖完全可积性理论的前提下，将周期KdV方程的局部适定性从$s\geq -\frac12$推广到$s>-\frac23$，构造了与mKdV已有结果相平行的结论。
This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a certain symmetry (for instance, plane symmetry); these solutions may have unbounded amplitude and contain cavitation regions in which the mass density vanishes. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the ex…
In this paper we investigate the question of the local existence of strong solution for the Korteweg system in critical spaces in dimension $N\geq 1$ provided that the initial data are small. More precisely the initial momentum $ρ_0 u_0$ belongs to $\mbox{bmo}_{T}^{-1}(\mathbb{R}^N)$ for $T>0$ and the initial density $ρ_0$ is in $L^\infty(\mathbb{R}^N)$ and far away from the vacuum. This result extends the so called Koch-Tataru Theorem for the incompressible Navier-Stokes equations to the case of the Korteweg system. It is also interesting to observe that any initial shock on the density is in…
A-priori estimates for generalized Korteweg-de Vries equations in $H^{-1}(\mathbb{R})$
本文研究了广义Korteweg-de Vries方程在$H^{-1}(\mathbb{R})$中的先验估计，提出了一种结合bootstrap论证和局部平滑范数的方法，适用于浅水渠道中长波模型的非积分扰动分析。
The aim of this short paper is threefold. First, we develop an implicit generalization of a constitutive relation introduced by Korteweg [10] that can describe the phenomenon of capillarity. Second, using a sub-class of the constitutive relations (implicit Euler equations), we show that even in that simple situation more than one of the members of the sub-class may be able to describe one or a set of experiments one is interested in describing, and we must determine which amongst these constitutive relations is the best by culling the class by systematically comparing against an increasing set…
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 study special regularity and decay properties of solutions to the IVP associated to the $k$-generalized KdV equations. In particular, for datum $u_0\in H^{3/4^+}(\mathbb R)$ whose restriction belongs to $H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$ we prove that the restriction of the corresponding solution $u(\cdot,t)$ belongs to $H^l((β,\infty))$ for any $β\in \mathbb R$ and any $t\in (0,T)$. Thus, this type of regularity propagates with infinite speed to its left as time evolves.
Consider the scaling $\varepsilon^{1/2}(x-Vt)\to x,\ \varepsilon^{3/2}t\to t$ in the Euler-Poisson system for ion-acoustic waves \eqref{equ1}. We establish that as $\varepsilon\to 0$, the solutions to such Euler-Poisson system converge globally in time to the solutions of the Korteweg-de Vries equation.
On Phase Boundaries in Relativistic Korteweg Fluids
作为系列论文首篇，本文将经典流体相锋研究成果拓展至相对论领域，通过拓展Korteweg毛细理论的洛伦兹不变表述，构造出对应欧拉-Korteweg方程下具有正则异宿轮廓的相锋族。
Sharp local well-posedness for the Schrödinger-Korteweg-de Vries system
本文证明了Schrödinger-Korteweg-de Vries系统在特定初始数据下的尖锐局部存在性，并基于分部积分强解概念与频率限制估计，将全局适定性结果推广至更广的正则性范围。
Improved global well-posedness for the quartic Korteweg-de Vries equation
本文证明了四次Korteweg-de Vries方程在实值初值空间$H^s(\mathbb{R})$（$s>-1/24$）中的整体适定性，改进了已有结果。
We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in $H^s(\mathbb{T})$, $s>2$. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.
This paper is concerned with the existence, uniqueness and time-asymptotic stability of time periodic solutions to the compressible Navier-Stokes-Korteweg system effected by a time periodic external force in $\mathbb{R}^n$. Our analysis is based on a combination of the energy method and the time decay estimates of solutions to the linearized system.
We prove that the modified Korteweg- de Vries equation (mKdV) equation is unconditionally well-posed in $H^s(\mathbb R)$ for $s> \frac 13$. Our method of proof combines the improvement of the energy method introduced recently by the first and third authors with the construction of a modified energy. Our approach also yields \textit{a priori} estimates for the solutions of mKdV in $H^s(\mathbb R)$, for $s>0$, and enables us to construct weak solutions at this level of regularity.
In this paper, we consider the long wavelength limit for the quantum Euler-Poisson equation. Under the Gardner-Morikawa transform, we derive the quantum Korteweg-de Vries (KdV) equation by a singular perturbation method. We show that the KdV dynamics can be seen at time interval of order $O(ε^{-3/2})$. When the nondimensional quantum parameter $H=2$, it reduces to the inviscid Burgers equation.
In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small initial data in $\mathbb R^N$, $N \geq 3$. In this study, the main tools are the maximal $L_p$-$L_q$ regularity and $L_p$-$L_q$ decay properties of solutions to the linearized equations.
Dispersive decay for the mass-critical generalized Korteweg-de Vries equation and generalized Zakharov-Kuznetsov equations
本文研究质量临界 gKdV 方程及作为其多维推广的广义 Zakharov-Kuznetsov 方程，证明非线性解可达到线性衰减率，并给出小初值情形下的逐点衰减估计。
On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data
本文研究多维可压缩Navier-Stokes-Korteweg系统在任意大初始数据下的全局强解存在性，采用改进的截断分析和修改的Nash-Moser迭代方案解决密度不可积性问题，将大数据理论从有限域扩展到无限域。
We consider the isentropic Navier-Stokes-Korteweg equations with a non-decreasing pressure on the whole space $\mathbb{R}^n$ $(n \ge 2)$, where the system describes the motion of compressible fluids such as liquid-vapor mixtures with phase transitions including a variable internal capillarity effect. We prove the existence of a unique global strong solution to the system in the $\mathrm{L}^p$-in-time and $\mathrm{L}^q$-in-space framework, especially in the maximal regularity class, by assuming $(p, q) \in (1, 2) \times (1, \infty)$ or $(p, q) \in \{2\} \times (1, 2]$. We show that the system i…
In this paper, we consider the stability for line solitary waves of the two dimensional Zakharov-Kuznetsov equation on $\mathbb{R}\times\mathbb{T}_L$ which is one of a high dimensional generalization of Korteweg-de Vries equation , where $\mathbb{T}_L$ is the torus with the period $2πL$. The orbital and asymptotic stability of the one soliton of Korteweg-de Vries equation on the energy space has been proved by Benjamin, Pego and Weinstein and Martel and Merle. We regard the one soliton of Korteweg-de Vries equation as a line solitary wave of Zakharov-Kuznetsov equation on $\mathbb{R}\times\mat…
We consider the fractional Korteweg-de Vries equation $u_t + u u_x - |D|^αu_x = 0$ in the range of $-1<α<1$ , $α\neq0$. Using basic Fourier techniques in combination with the modified energy method we extend the existence time of classical solutions with initial data of size $\varepsilon$ from $\frac{1}{\varepsilon}$ to a time scale of $\frac{1}{\varepsilon^2}$. This analysis, which is carried out in Sobolev space $H^N(\mathbb{R})$, $N \geq 3$, answers positively a question posed by Linares, Pilod and Saut (SIAM J. Math. Anal. 46 (2014), no. 2, 1505-1537).
A unified duality framework for barotropic, quantum and Korteweg fluids
提出一个统一的对偶变分框架，涵盖正压欧拉、量子欧拉和欧拉-科特韦格系统，证明了大时间区间上对偶格式的相容性，并建立了连续无真空初值下变分对偶解的存在性及Dafermos原理。
In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(\mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumpt…
We consider the orbital stability of solitons of the Kadomtsev--Petviashvili-I equation in $\mathbb{R} \times (\mathbb{R}/2π\mathbb{Z})$ which is one of a high dimensional generalization of the Korteweg--de Vries equation. Benjamin showed that the Korteweg--de Vries equation possesses the stable one soliton. We regard the one soliton of the Korteweg--de Vries equation as a line soliton of the Kadomtsev--Petviashvili-I equation. Zakharov and Rousset--Tzvetkov proved the orbital instability of the line solitons of the Kadomtsev--Petviashvili-I equation on $\mathbb{R}^2$. In the case of the Kadom…
We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existen…
We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $\FF L^{α,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.
Self-similar solutions for the generalized fractional Korteweg-de Vries equation
本文研究了广义分数阶 Korteweg-de Vries 方程的 Cauchy 问题，证明了在初始数据满足小性假设且参数组合广泛取值时，可构造出该问题的自相似解。
In this paper, we consider global weak solutions to com-pressible Navier-Stokes-Korteweg equations with density dependent viscosities , in a periodic domain $Ω= \mathbb T^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the densit…
This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time …
Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipativ…
We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+ηLu + uu_x=0;\; x\in \mathbb{T},\; t\geq0, $$ where $η>0$, $(Lu)^{\wedge}(k)=-Φ(k)\hat{u}(k)$, $k\in \mathbb{Z}$ and $Φ\in \mathbb{R}$ is bounded above. Particular cases of this problem are the Korteweg-de Vries-Burgers equation for $Φ(k)=-k^2$, the derivative Korteweg-de Vries-Kuramoto-Sivashinsky equation for $Φ(k)=k^2-k^4$, and the Ostrovsky-Stepanyams-Tsimring equation for $Φ(k)=|k|-|k|^3$.
Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients
本文基于热方程、Stokes方程和Lamé方程的极大正则性，证明了具有变毛细率和变黏性系数的不可压Navier-Stokes-Korteweg系统在临界空间中解的全局存在性与唯一性。
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