We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only
linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial zeroes. This new difficulty is solved by the use of "good unknowns " and of normal form techniques.
We study the Cauchy problem for one-dimensional dispersive system of Boussinesq type which models weakly nonlinear long wave surface waves. We establish the local well-posedness and ill-posedness of solutions to the system. We also provide criteria for the formation of singularities.
We consider in this paper the well-posedness for the Cauchy problem associated to two-dimensional dispersive systems of Boussinesq type which model weakly nonlinear long wave surface waves. We emphasize the case of the {\it strongly dispersive} ones with focus on the "KdV-KdV" system which possesses the strongest dispersive properties and which is a vector two-dimensional extension of the classical KdV equation.
We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and Strichartz estimates, and implement them together with a fixed point argument to solve the problem locally. Hamiltonian conservation guarantees global well-posedness for small initial data in the one dimensional settings.
This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion. We will establish two different long time existence results for the solutions of the Cauchy problem. The proofs involve normal form transformations suitably modified away from the zero set of the phases.
The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in [4]. It is numerically shown to be stable and a good approximation to the incompressible Euler equations. Here we prove local in time well-posedness. Our proof relies on an energy method and a compactness argument.
The aim of this article is to extend previous works about the asymptotics of an ill-prepared fast rotating, highly stratified incompressible Navier-Stokes system. Thanks to improved Strichartz estimates, we are able not only to cover a case which was unanswered in our previous work (allowing bigger ill-prepared initial data) but also to reduce some assumptions on the initial data. In passing we also widen the range of some parameters and compare two methods to obtain dispersive estimates.
In this article we prove highly improved and flexible Strichartz-type estimates allowing us to generalize the asymptotics we obtained for a stratified and rotating incompressible Navier-Stokes system: for large (and less regular) initial data, we obtain global well-posedness, asymptotics (as the Rossby number $ε$ goes to zero) and convergence rates as a power of the small parameter $ε$. Our approach is lead by the special structure of the limit system: the 3D quasi-geostrophic system.
We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenome…
In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.
This paper is a continuation of a previous work by two of the Authors on long time existence for Boussinesq systems modeling the propagation of long, weakly nonlinear water waves. We provide proofs on examples not considered previously in particular we prove a long time well-posedness result for a delicate "strongly dispersive" Boussinesq system.
This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for $2\times2$ hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a configuration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the h…
In this work, we numerically study the higher-ordered/extended Boussinesq system describing the propagation of water-waves over flat topography. A reformulation of the same order of precision that avoids the calculation of high order derivatives on the surface deformation is proposed. We show that this formulation enjoys an extended range of applicability while remaining stable. Moreover, a significant improvement in terms of linear dispersive properties in high frequency regime is made due to the suitable adjustment of a dispersion correction parameter. We develop a second order splitting sch…
At the core of this article is an improved, sharp dispersive estimate for the anisotropic linear semigroup $e^{R_1 t}$ arising in both the study of the dispersive SQG equation and the inviscid Boussinesq system. We combine the decay estimate with a blow-up criterion to show how dispersion leads to long-time existence of solutions to the dispersive SQG equation, improving the results obtained using hyperbolic methods. In the setting of the inviscid Boussinesq system it turns out that linearization around a specific stationary solution leads to the same linear semigroup, so that we can make use …
In this paper we prove the validity of a long wave Whitham approximation for a system consisting of a Boussinesq equation coupled with a Klein-Gordon equation. The proof is based on an infinite series of normal form transformations and an energy estimate. We expect that the concepts of this paper will be a part of a general approximation theory for Whitham's equations which are especially used in the description of slow modulations in time and space of periodic wave trains in general dispersive wave systems.
This paper is concerned with a two dimensional Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale $\mathcal O(1/\sqrtε)$, where $ε>0$ is a shallowness parameter measuring the ratio of the amplitude of the wave to the mean depth of the fluid. The key ingredients in the proof are frequency loacalised dispersive and Strichartz estimates that depend on $ε$ as well as bilinear estimates in some Strichartz norms.
We study the stability of special, stratified solutions of a 3d Boussinesq system describing an incompressible, inviscid 3d fluid with variable density (or temperature, depending on the context) under the effect of a uni-directional gravitational force. The behavior is shown to depend on the properties of an anisotropic dispersive operator with weak decay in time. However, the dispersive decay also depends on the strength of the gravity in the system and on the profile of the stratified solution, whose stability we study. We show that as the strength of the dispersion in the system tends to in…
We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accu…
We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. (Agnon et al. 1999, J. Fluid Mech., 399 pp. 319-333) and enhanced by Madsen et al. (Madsen et al. 2003, Proc. R. Soc. Lond. A, 459 pp. 1075-1104) which consists in constructing infinite-series Taylor…
This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable, respectively. The well-known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here. The class of nonlocal nonlinea…
In this note we show that all small solutions in the energy space of the generalized 1D Boussinesq equation must decay to zero as time tends to infinity, strongly on slightly proper subsets of the space-time light cone. Our result does not require any assumption on the power of the nonlinearity, working even for the supercritical range of scattering. No parity assumption on the initial data is needed.
We present a new method for the numerical implementation of generating boundary conditions for a one dimensional Boussinesq system. This method is based on a reformulation of the equations and a resolution of the dispersive boundary layer that is created at the boundary when the boundary conditions are non homogeneous. This method is implemented for a simple first order finite volume scheme and validated by several numerical simulations. Contrary to the other techniques that can be found in the literature, our approach does not cause any increase in computational time with respect to periodic …
In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, $u_{tt}-Lu_{xx}=B(- |u|^{p-1}u)_{xx}, ~(p>1)$, where the nonlocality enters through two pseudo-differential operators $L$ and $B$. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and c…
We consider a system describing the long-time dynamics of an hydrodynamical, density-dependent flow under the effects of gravitational forces. We prove that if the Froude number is sufficiently small such system is globally well posed with respect to a $ H^s, \ s>1/2 $ Sobolev regularity. Moreover if the Froude number converges to zero we prove that the solutions of the aforementioned system converge (strongly) to a stratified three-dimensional Navier-Stokes system. No smallness assumption is assumed on the initial data.
In this paper we propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly non resonant dispersive equations. As an example we obtain unconditional well-posedness of the Cauchy problem below $ H^1 $ for a large class of one-dimensional dispersive equations with a dispersion that is greater or equal to the one of the Benjamin-Ono equation. Since this is done without using a gauge transform, this enables us to prove strong convergence results for solutions of viscous versions of these equations towards the purely dispersive solutions.
We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. …
Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces
School of Mathematics and Computer Sciences, Gannan Normal University（赣南师范大学数学与计算机科学学院）；School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）
We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular our resu…
In this paper, we improve the global existence result in [9] slightly. More precisely, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is obtained under the assumption of initial data $(v_0,T_0) \in H^1$ with $\partial_z v_0 \in L^4$. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, in the cases of $H^1$ initial data, $H^1$ initial data with additional regularity $\partial_z v_0 \in L^4$ and $H^2$ initial data, resp…
Consider the Hamiltonian $abcd$ system in one dimension, with data posed in the energy space $H^1\times H^1$. This model, introduced by Bona, Chen and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where $a,c<0$ and $b=d>0$. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this $2\times 2$ system was given by the two authors and Poblete and Pozo, in a strongly dispersive regime, i.e. under essentially the conditions \[ b=d > \frac29, \quad a,c<-\frac1{18}.…
The Boussinesq $abcd$ system is a 4-parameter set of equations posed in $\mathbb{R}_t \times \mathbb{R}_x$, originally derived by Bona, Chen and Saut as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among many particular regimes, depending each of them in terms of the value of the parameters $(a,b,c,d)$ present in the equations, the "generic" regime is characterized by the setting $b,d>0$ and $a,c<0$. The system is hamiltonian if also $b=d$. Th…
In this paper, we derive asymptotic models for the propagation of two and three-dimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities, over an uneven bottom. We assume the thickness of the upper and lower fluids to be of comparable size, and small compared to the characteristic wavelength of the system (shallow water regimes). Following a method introduced by Bona, Lannes and Saut based on the expansion of the involved Dirichlet-to-Neumann operators, we are able to give a rigorous justification of classical models for wea…
A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain $\mathbb{T}$ are studied. We used two distributed controls, supported in $ω\subset\mathbb{T}$ and assumed to be generated by a linear feedback law conserving the "mass" (or "volume"), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in $H^s(\mathbb{T})$, for $s\geq0$. After that, by certain properties of Bourgain spaces we show a property of global exponential stability. This property togeth…
