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. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.
In this paper, we investigate the global existence and uniqueness of strong solutions to 2D Boussinesq system with anisotropic thermal diffusion or anisotropic viscosity and with striated initial data. Using the key idea of Chemin to solve 2-D vortex patch of ideal fluid, namely the striated regularity can help to bound the gradient of the velocity, we can prove the global well-posedness of the Boussinesq system with anisotropic thermal diffusion with initial vorticity being discontinuous across some smooth interface. In the case of an anisotropic horizontal viscosity we can study the propagat…
This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\inf…
On decay of solutions to the anisotropic Boussinesq equations near the hydrostatic balance in half space $\mathbb{R}_+^3$
School of Mathematics and Information Sciences, North Minzu University（北方民族大学数学与信息科学学院）；School of Mathematics and Computer Science, Yichun University（宜春学院数学与计算机学院）
The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is $\mathbb R^2$, the stability problem in a Sobolev setting remains open. When the spatial dom…
We show the existence of Hölder continuous periodic solution with compact support in time of the Boussinesq equations with partial viscosity. The Hölder regularity of the solution we constructed is anisotropic which is compatible with partial viscosity of the equations.
We establish the long time existence of solutions for the "Boussinesq-Full Dispersion" systems modeling the propagation of internal waves in a two-layer system. For the two-dimensional Hamiltonian case we prove the global existence of small solutions of the corresponding systems.
We show the existence of an inertial manifold (i.e. a globally invariant, exponentially attracting, finite-dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.
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.
In this paper we deal with the long time existence for the Cauchy problem associated to BBM-type Boussinesq systems of equations which are asymptotic models for long wave, small amplitude gravity surface water waves. As opposed to previous papers devoted to the long time existence issue, we consider initial data with nontrivial behaviour at innity which may be used to model bore propagation.
We prove in this paper a long time existence result for a modified Boussinesq-Peregrine equation in one dimension, describing the motion of Water Waves in shallow water, in the case of a non flat bottom. We first give a local existence result for the original Boussinesq Peregrine equation as derived by Boussinesq and Peregrine in all dimensions. We then introduce a new model which has formally the same precision as the Boussinesq-Peregrine equation, and give a local existence result in all dimensions. We finally prove a long time existence result in dimension 1 for this new equation, without a…
In this paper, we consider the stochastic Boussinesq equations on $\mathbb T^3$ with transport noise and rough initial data. We first prove the existence and uniqueness of the local pathwise solution with initial data in $L^p(Ω;L^p)$ for $p>5$. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.
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 consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and Székelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions.
We study the global well-posedness of a two-dimensional Boussinesq system which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion of type $|\DD|^α$ for the temperature. We prove that for $α>1,$ there exists a unique global solution for initial data with critical regularities.
In this paper we deal with the long time existence for the Cauchy problem associated to some asymptotic models for long wave, small amplitude gravity surface waves. We generalize some of the results that can be found in the literature devoted to the study of Boussinesq systems by implementing an energy method on spectrally localized equations. In particular, we obtain better results in terms of the regularity level required to solve the initial value problem on large time scales.
In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces $M^{s}_{p^\prime,q}(\mathbb{R}^n),$ $n\geq 1.$ After a decomposition of the Boussinesq equation in a $2\times 2$-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in $ M^s_{p,q}\times D^{-1}JM^s_{p,q}$ spaces for suitable $p,q$ and $s,$ including the special case $p=2,q=1$ and $s=0.$ Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.
We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,θ_0)$ are required to be only in the space $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb R^2)\}$, and thus our result generalizes that in [C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004], where the initial data are assumed to be in $H^2(\mathbb R^2)$. The assumption on the initial data is at the minimal level that is required to…
We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in Ocean dynamics. This work improves the global well-posedness results established recently by R. Danchin and M. Paicu for the Boussinesq system with anisotropic viscosity and zero diffusion. Although we follow some of their ideas, in proving the uniqueness result, we have used an alternative approach by writing the transported temperature (density) as $θ= Δξ$ and adapting the techniques of V. Yudovich for the 2D incompress…
Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…
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 …
This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^{0}{\in}B_{2,1}^{5/2}(\RR^3)$ and$ $ρ^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$ with $p>6.$ This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale-Kato-Majda criterion is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.
This paper is devoted to the study of the Cauchy problem for the Boussinesq system with partial viscosity in dimension $N\geq3.$ First we prove a global existence result for data in Lorentz spaces satisfying a smallness condition which is at the scaling of the equations. Second, we get a uniqueness result in Besov spaces with {\it negative} indices of regularity (despite the fact that there is no smoothing effect on the temperature). The proof relies on a priori estimates with loss of regularity for the nonstationary Stokes system with convection. As a corollary, we obtain a global existence a…
In this paper, we discuss with the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well-posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model.
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.
In this paper, the boundary flex control problem of non stationary equation governing the coupled mass and heat flow of a viscous incompressible fluid in a generalized Boussinesq approximation by assuming that viscosity and heat conductivity are dependent on temperature has been studied. The boundary condition for velocity of fluid is non -standard boundary condition: specifically the case where dynamical pressure is given on some part of the boundary and the boundary condition for temperature of fluid is mixed boundary condition has been considered. First, we have proved the existence of exis…
We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.
We study a Boussinesq system in a bounded domain with an outlet boundary portion where fluid can leave or re-enter. On this boundary part, we consider a do-nothing condition for the fluid flow, and a new artificial condition for the heat transfer that couples nonlinearly the fluid velocity and temperature. The latter can be further adjusted if convective or conductive phenomena are dominant. We prove existence and, in some cases, uniqueness of weak solutions to stationary and evolutionary problems by a fixed point strategy under suitable assumptions on the data. A variety of numerical tests sh…
In this paper, a generalized Boussinesq equation that couples the mass and heat flows in a viscous incompressible uid is considered. The kinematic viscosity and the heat conductivity are assumed to be dependent on the temperature. The boundary condition on the velocity of fluid is non-standard where the dynamical pressure is given on some part of the boundary, and the temperature of fluid is represented in a mixed boundary condition. The existence of the weak solution is proved by the Galerkin approximation scheme, and the uniqueness is also obtained under the condition on the weak solution th…
We establish the existence and uniqueness of local strong pathwise solutions to the stochastic Boussinesq equations with partial diffusion term forced by multiplicative noise on the torus in $\mathbb{R}^{d},d=2,3$. The solution is strong in both PDE and probabilistic sense.In the two dimensional case, we prove the global existence of strong solutions to the Boussinesq equations forced by additive noise using a suitable stochastic analogue of a logarithmic Gronwall's lemma. After the global existence and uniqueness of strong solutions are established, the large deviation principle (LDP) is prov…
We show the existence of finite kinetic energy solution with prescribed kinetic energy to the 2d Boussinesq equations with diffusive temperature on torus.
Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…
The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the Bénard system are also discussed.
The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,α}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,α}$ finite time blow-up solution for Boussinesq equations where the temperature has diffusion and finite energy. Generally speaking, the diffusion of temperature smooths the solution of the sy…
We obtain a one-parameter family $$(u_μ(x,t),η_μ(x,t))_{μ\geq μ_0}=(ϕ_μ(x-ω_μ t),ψ_μ(x-ω_μ t))_{μ\geq μ_0}$$ of traveling-wave solutions to the Boussinesq system $$u_t+η_x+uu_x+cη_{xxx}=0,η_t+u_x+(ηu)_x+au_{xxx}=0$$ in the case $a,c<0$, with non-null speeds $ω_μ$ arbitrarily close to $0$ ($ω_μ\xrightarrow[μ\to+\infty]{} 0$). We show that the $L^2$-size of such traveling-waves satisfies the uniform (in $μ$) estimate $\|ϕ_μ\|_2^2+\|ψ_μ\|_2^2\leq C\sqrt{|a|+|c|},$ where $C$ is a positive constant. Furthermore, $ϕ_μ$ and $-ψ_μ$ are smooth, non-negative, radially decreasing functions which decay ex…
In this article we consider the Boussinesq system supplemented with some dissipation terms. These equations model the propagation of a waterwave in shallow water. We prove the existence of a global smooth attractor for the corresponding dynamical system.
In this paper we establish local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}$ to the viscous, zero thermal-diffusive Boussinesq equations in $\mathbb{R}^n , n = 2,3$. Beale-Kato-Majda type blow-up criterion has been established in three-dimensions with respect to the $BMO$-norm of the vorticity. We further prove the local-in-time existence and blow-up criterion for non-viscous and fully ideal Boussinesq systems. Commutator estimates due to Kato and Ponce (1988) \cite {KP} and Fefferman et. al. (2014) \cite {Fe} play important roles in the calculations.
We study the existence, uniqueness as well as regularity issues for the two-dimensional incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces. The optimal regularity exponent ranges are considered.
We study the local and global well-posedness of the periodic boundary value problem for the nonlinear Schrödinger-Boussinesq system. The existence of periodic pulses as well as the stability of such solutions are also considered.
We investigate large time existence of solutions of the Navier-Stokes-Boussinesq equations with spatially almost periodic large data when the density stratification is sufficiently large. In 1996, Kimura and Herring \cite{KH} examined numerical simulations to show a stabilizing effect due to the stratification. They observed scattered two-dimensional pancake-shaped vortex patches lying almost in the horizontal plane. Our result is a mathematical justification of the presence of such two-dimensional pancakes. To show the existence of solutions for large times, we use $\ell^1$-norm of amplitudes…
We address a question concerning the issue of existence to a Boussinesq type system with a heat source. The problem is studied in the whole two dimensional plane and the heat source is a measure transported by the flow. For arbitrary initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructing solutions and make us work in a non-standard framework of inhomogeneous Besov spaces of the $L^\infty(0,T;B^s_{p,\infty})$-type. Application of the Lagrangian coordinates yields uniqueness omitting difficulties with comparison …
We prove the existence of a large family of two-dimensional standing waves, that are triple periodic solutions, for a Boussinesq system which describes two-way propagation of water waves in a channel. Our proof uses the Lyapunov-Schmidt method to find the bifurcation standing waves.
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.
In this paper, we study the problem of analyticity of smooth solutions of the inviscid Boussinesq equations. If the initial datum is real-analytic, the solution remains real-analytic on the existence interval. By an inductive method we can obtain lower bounds on the radius of spatial analyticity of the smooth solution.
We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.
We prove the local in time existence and a blow up criterion of solution in the Hölder spaces for the inviscid Boussinesq system in $R^{N},N\geq2$, under the assumptions that the initial values $θ_{0},u_{0}\in C^{r}$, with $r>1$.
The Boussingesq equations was introduced in understanding the coupling nature of the thermodynamics and the fluid dynamics. We show the existence of continuous periodic weak solutions of the Boussinesq equations which satisfies the prescribed kinetic energy or some other prescribed property. Our results represent the conversions between internal energy and mechanical energy.
