We consider the so-called Naiver-Stokes-Korteweg(NSK) equations for the dynamics of compressible barotropic viscous fluids with internal capillarity. We handle the time-asymptotic stability in 1D of the viscous-dispersive shock wave that is a traveling wave solution to NSK as a viscous-dispersive counterpart of a Riemann shock. More precisely, we prove that when the prescribed far-field states of NSK are connected by a single Hugoniot curve, then solutions of NSK tend to the viscous-dispersive shock wave as time goes to infinity. To obtain the convergence, we extend the theory of $a$-contraction with shifts, used for the Navier-Stokes equations, to the NSK system. The main difficulty in analysis for NSK is due to the third-order derivative terms of the specific volume in the momentum equation. To resolve the problem, we introduce an auxiliary variable that is equivalent to the derivative of the specific volume.
We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.
We study the dispersive blow-up phenomena for the Schrödinger-Korteweg-de Vries (S-KdV) system. Roughly, dispersive blow-up has being called to the development of point singularities due to the focussing of short or long waves. In mathematical terms, we show that the existence of this kind of singularities is provided by the linear dispersive solution by proving that the Duhamel term is smoother. It seems that this result is the first regarding systems of nonlinear dispersive equations. To obtain our results we use, in addition to smoothing properties, persistence properties for solutions of t…
We consider the Rosenau-Korteweg-de Vries-regularized long wave and Rosenau- Korteweg-de Vries equations, which contain nonlinear dispersive effects. We prove that, as the diffusion parameter tends to zero, the solutions of the dispersive equations converge to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L^p setting.
We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense (in the spirit of [18]) and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non-dispersion.
Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg-de Vries equation, w…
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.
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 …
This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type with density- and temperature-dependent viscosity, capillarity, and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. We show that the combination of the viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if the strength of the composite wave and the heat conductivity coefficient satisfy some smallness conditions. The proof…
We consider the Rosenau-Korteweg-de Vries-equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converge to the discontinous weak solutions of the Burgers equation. The proof relies on deriving suitable a priori estimates together with an application of tha compansated compactness method in the L^p setting.
We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the poss…
We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_{x}^α\partial_{x}u+u\partial_{x}u=0, & x,t\in\mathbb{R},\,0<α<1, \\ u(x,0)=u_{0}(x).& \\ \end{array} \right. \end{equation} It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This pro…
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws where the far field states are prescribed. We proved that the solution of the Cauchy problem tends toward the constant state as time goes to infinity.
In this paper, we develop dispersive PDE techniques for the Fermi--Pasta--Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg--de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study [45].
We show that the long-time behavior of solutions to the Korteweg-de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schrödinger operator) depends only on the size of the step of the initial data and on the direction, $\frac{x}{t}=const.$, along which we determine the asymptotic behavior of the solution. In turn, the phase shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the scattering data, and is computed explicitly via the Jacobi…
In several cases of nonlinear dispersive PDEs, the difference between the nonlinear and linear evolutions with the same initial data, i.e. the integral term in Duhamel's formula, exhibits improved regularity. This property is usually called nonlinear smoothing. We employ the method of infinite iterations of normal form reductions to obtain a very general theorem yielding existence of nonlinear smoothing for dispersive PDEs, contingent only on establishing two particular bounds. We then apply this theorem to show that the nonlinear smoothing property holds, depending on the regularity of the in…
This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool o…
We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as predicted by inverse scattering theory.
Motivated by the ongoing study of dispersive shock waves in non integrable systems, we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems -- including the generalized Korteweg de Vries equations and the Euler-Korteweg systems -- that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modula…
In this article, we prove that small localized data yield solutions to Higher order Korteweg-de Vries type equation with scattering-supercritical nonlinearity have linear dispersive decay in only a finite length of time. The proof is done by using space-time resonance method and analyzing the oscillatory integrals on the Fourier side.
We study persistence properties of solutions to some canonical dispersive models, namely the semi-linear Schrödinger equation, the $k$-generalized Korteweg-de Vries equation and the Benjamin-Ono equation, in weighted Sobolev spaces $H^s(\R^n)\cap L^2(|x|^ldx),\;s,\,l>0$
We generalize the Abstract Interpolation Lemma proved by the authors in [2]. Using this extension, we show in a more general context, the persistence property for the generalized Korteweg-de Vries equation, see (1.2), in the weighted Sobolev space with low regularity in the weight. The method used can be applied for other nonlinear dispersive models, for instance the multidimensional nonlinear Schrodinger equation.
We are concerned with the convergence of a numerical scheme for the initial-boundary value problem associated to the Korteweg-de Vries- Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several uid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in uid dynamics. We prove here the convergence of both semi-discrete as well as fully-discrete finite difference schemes for the Kawahara equation. Finally, the convergence is illustratred b…
We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related variational problem is nonlinearly stable to period preserving perturbations. We also discuss when the associated linearized equation admits exponentially growing solutions. The proof utilizes the variational structure of the equation.
We are concerned with the convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier- Galerkin spectral method and in time by the explicit leap-frog scheme. For the re…
In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. In contrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy.
In this article, we derive a viscous Boussinesq system for surface water waves from Navier-Stokes equations. We use neither the irrotationality assumption, nor the Zakharov-Craig-Sulem formulation. During the derivation, we find the bottom shear stress, and also the decay rate for shallow water. In order to justify our derivation, we derive the viscous Korteweg-de Vries equation from our viscous Boussinesq system and compare it to the ones found in the bibliography. We also extend the system to the 3-D case.
We provide a detailed study of the dynamics obtained by linearizing the Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability, we prove global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. Furthermore, we provide both a leading-order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters. This requires a global-in-time study of th…
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…
A compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The dispersive term is due to quantum effects described through the Bohm potential and the viscosity term is of linear type. It is shown that small-amplitude viscous-dispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio et al. (Phys. D 402, 2020, p. 132222; Appl. Math. Comput. 385, 2020, p. 125450). The proof is based on spectral energy estimates which profit from the mono…
In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove global existence of finite energy weak solutions for large initial data. Contrary to previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The convergence of the approximating solutions is obtained by introducing suitable truncations in the momentum equations of the velocity field and the mass density at different scales and use only the a prio…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg--de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the $x/t$ plane splits into $g+1$ soliton regions which are interlaced by $g+1$ oscillatory regions, where $g+1$ is the number of spectral gaps. In the soliton regions the solution is asymptotically given by a number of solitons travelling on top of finite-gap solutions which are in the same isospectral class as the backg…
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases…
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…
Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force
School of Mathematics, Nanjing University（南京大学数学系）；The Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）；School of Mathematical Sciences, Ocean University of China（中国海洋大学数学科学学院）
We show the existence of ground state solutions to the following stationary system coming from some coupled fractional dispersive equations such as: nonlinear fractional Schrödinger (NLFS) equations (for dimension $n=1,\, 2,\, 3$) or NLFS and fractional Korteweg-de Vries equations (for $n=1$), $$ \left \{ \begin{array}{ll} (-Δ)^{s} u+ λ_1 u &= u_1^{3}+βuv,\quad u\in W^{s,2}(\mathbb{R}^n), (-Δ)^{s} v + λ_2 v &= \frac 12 v^{2}+\frac 12 βu^2,\quad v\in W^{s,2}(\mathbb{R}^n), \end{array} \right. $$ where $λ_j>0$, $j=1,2$, $β\in \mathbb{R}$, $n=1,\, 2,\, 3$, and $\frac n4< s<1$. Precisely, we prove…
The Korteweg-de Vries equation has a central place in a model for waves on shallow water and it is an example of the propagation of weakly dispersive and weakly nonlinear waves. Its history spans a period of about sixty years, starting with experiments of Scott Russell in 1834, followed by theoretical investigations of, among others, Lord Rayleigh and Boussinesq in 1871 and, finally, Korteweg and De Vries in 1895. In this essay we compare the work of Boussinesq and Korteweg-de Vries, stressing essential differences and some interesting connections. Although there exist a number of articles, re…
