Ravi P. Agarwal, S. Gala, Maria Alessandra Ragusa
In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces.
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We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations.
In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of regularity for the two variables u and $θ$.
We address the well-posedness for the two-dimensional Boussinesq equations with zero diffusivity in bounded domains. We prove global in time regularity for rough initial data: both the initial velocity and temperature have $ε$ fractional derivatives in $L^{q}$ for some $q>2$ and $ε>0$ arbitrarily small.
On the regularity of weak solution for the Oberbeck-Boussinesq equations
本文在有界区域中为粘性活性化学流体建立了空间与时间均具弱Lebesgue可积性的Prodi-Serrin型正则性准则，推广了Oberbeck-Boussinesq方程弱解的正则性理论。
We prove that a solution to the three-dimensional Boussinesq equations does not blow-up at time T if $\| u_{\le Q}\|_{B^1_{\infty, \infty}}$ is integrable on $(0, T)$, where $u_{\le Q }$ represents the low modes of Littlewood-Paley projection of the velocity $u$.
A single-component regularity criterion and Inviscid limit of axially symmetric MHD-Boussinesq system
该研究针对无粘轴对称MHD-Boussinesq系统给出仅含速度水平旋度分量的临界BKM型爆破准则，同时分析粘性系统的无粘极限并得到粘性系数趋于零时的收敛速率。
In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as $1-α<β< \min\Big\{\fracα{2},\,\, \frac{3α-2}{2α^{2}-6α+5}, \,\,\frac{2-2α}{4α-3}\Big\}$ with $0.77963\thickapproxα_{0}<α<1$. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.
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…
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 $r>1$.
We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. This is the first Prodi-Serrin-type criterion for a hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-…
We study the temperature front problem for the 3D viscous Boussinesq equation. We prove that the $C^{k,γ}$ ($k\geq 1$, $0<γ< 1$) and $W^{2,\infty}$ regularity of a temperature front is locally preserved along the evolution as well as globally preserved under a smallness condition in a critical space. In particular, beside giving another proof of the main result in \cite{GGJ20}, we also extend it to a more general class of regular patch.
The paper deals with the regularity criterion for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution $(u,θ)$ becomes regular provided that $(\nabla_{h}u,\nabla_{h}θ)\in L^{\frac{8}{3}}(0,T;\dot{B}_{\infty ,\infty}^{-1}(\mathbb{R}^{3}))$ Our results improve and extend the well-known results by Fang-Qian for the Navier-Stokes equations.
Regularity results for viscous 3D Boussinesq temperature fronts
本文针对三维不可压缩粘性Boussinesq系统中的非扩散温度锋面，证明了任意大小初始数据的局部存在性，以及小初始数据在临界空间中的全局正则性传播，并处理了分段Hölder温度的一般锋面。
In this paper, we will show that solutions of the three-dimensional non-resistive and non-diffusive MHD-Boussinesq system are globally regular if the initial data is axisymmetric and the swirl components of the velocity and the magnetic vorticity are zero. Our main result extends previous ones on the three-dimensional non-resistive MHD system and non-diffusive Boussinesq system, and the method used here can also be applied to the magnetic Rayleigh-Bénard convection system.
In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed.
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…
This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as $\al>2/3$. This significantly improves upon some recent works. The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by new set of commutator estimates, which may…
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.
Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effects
本文研究涵盖临界正则性初始数据的二维无粘Boussinesq方程的长时间可解性，通过在Besov空间中证明关于流体分层参数的一致局部可解性，并结合爆破准则与Strichartz型估计，对大参数情形获得了与初始数据大小无关的长时间可解性。
This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $Λ^αu$ in the velocity equation and by $Λ^βθ$ in the temperature equation, where $Λ=\sqrt{-Δ}$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when $(α,β)$ is in the critical range: $α>\frac{\sqrt{1777}-23}{24} =0.798103..$, $β>0$ and $α+ β=1$. This result improves the previous work of Jiu, Miao, Wu and Zhang \cite{JMWZ} which obtained the global regularity for $α> \frac{23-\sqrt{145}}{12} \appro…
New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $(u, θ) \in D(A)\times H^1$ improves considerably the main result of the recent article [7]. Our second result on global regularity of the solution $(u, θ) \in V \times H^1$ for both bounded domain and the whole space ${\mathbb R}^2$ is a new one. It has been open and also seems much more challenging than the fi…
This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space $\mathbb{R}^2$. We prove that for initial patches with $W^{2,\infty}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound $\nabla^2u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+γ}$ regularity result in [23], $0<γ<1$, using the scale of Sobolev spaces for the velocity. Furth…
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.
A regularity criterion for the 3D Boussinesq equations in homogeneous Besov spaces with negative indices
本文针对三维 Boussinesq 方程，利用速度的第三个偏导数在负指标齐次 Besov 空间中的时空积分条件，给出并证明了一个保证解在给定时间区间上保持正则的新准则。
In this paper, we provide conditions, \emph{based solely on the observed velocity data}, for the global well-posedness, regularity and convergence of the Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a Leray-Hopf weak solutions of the three dimensional Boussinesq system. This condition also guarantees the construction of the {\it determining map}. The aforementioned conditions on the (finite-dimensional) velocity observations, which in this case comprise either of a finite-dimensional \emph{modal} projection or finitely many \emph{volume element observations}, are automatic…
Norm inflation with infinite loss of regularity for the generalized improved Boussinesq equation
本文研究了广义改进Boussinesq方程的不适定性，证明了在特定空间中存在正则性无限损失的范数膨胀，并将该结果推广至多维情形及Fourier-Lebesgue等空间。
In this paper we address the temperature patch problem of the 2D viscous Boussinesq system without heat diffusion term. The temperature satisfies the transport equation and the initial data of temperature is given in the form of non-constant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that our partially viscous Boussinesq system admits a unique global regular solution and the initial $C^{k,γ}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ …
The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with $v_0\in H^{\frac12}(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$ and density $ρ_0\in L^2(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$. This respectively enhances the two results recently accomplished in \cite{Danchin-Paicu1, Hmidi-Rousset}. Our formalism is inspired, in particular for the first part from \cite{Abidi} concerning the axisymmetric Navier-Stokes equations once $v_…
Well-posedness and regularity of the Darcy-Boussinesq system in layered porous media
本文研究了层状多孔介质中 Darcy-Boussinesq 系统的适定性与正则性，证明了 2D 和 3D 中整体弱解的存在性及 2D 中的唯一性，并在均匀孔隙率假设下获得了新型分段 $H^2$ 空间中正则解的存在与唯一性。
The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space $\dot{B}_{\infty ,\infty }^{-1}(\mathbb{R}% ^{3})$, that, if the solution of the Boussinesq equation (\ref% {eq1.1}) below (starting with an initial data in $H^{2}$) is such that $% (\nabla u,\nabla θ)\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{-1}(% \mathbb{R}^{3})\right)$, then the solution remains smooth forever after $T$. In this contribution, we prove the same result for weak solutions just by assuming the condition on the velocity $u$ and not o…
Mild solutions to the 3D-Boussinesq system with weakened initial temperature
本研究针对三维粘性Boussinesq系统，在初始温度具有负Sobolev正则性时构造局部温和解，利用系统耦合结构显著降低温度项正则性要求。
In this paper, the integral initial value problems for Boussinesq type equations are studied. The equation include the general differential operators. The existence, uniqueness and regularity properties of solution of these problems are obtained. By choosing differential operators including in the equation, the regularity properties of the Cauchy problem for different type of Boussinesg equations are studied.
Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains
该研究针对二维Boussinesq系统的长时间行为，构造了光滑有界域上的全局持续解，推导了速度和密度的渐近性质并给出收敛到稳态的充要条件，证明采用抛物型极大正则性和初始相容性条件。
Global regularity of the 2D fractional Boussinesq equations with subcritical dissipation
研究二维不可压缩Boussinesq方程在分数阶耗散$(-\Delta)^{\frac\alpha2}u$和$(-\Delta)^{\frac\beta2}\theta$下的整体正则性问题，聚焦于次临界区域$\alpha+\beta>1$，通过推导分数阶Laplacian算子的非线性下界并实施迭代过程，解决了$\alpha\leq\frac23$的剩余情形，得到了最锐利的正则性结果。
The global regularity for the viscous Boussinesq equations is proved.
This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $α+β=1$ when $Λ^α\equiv (-Δ)^{\fracα{2}}$ and $Λ^β$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $α+β=1$ and $0.9132\approx α_0<α<1$. The cases when $α=1$ and when $α=0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by expl…
Long-time validity of the Korteweg-de Vries approximation for the Boussinesq equation
本文证明good Boussinesq方程的KdV近似在对数长时间内有效，通过正则性保持论证和重新标度守恒律迭代局部近似，扩展了先前低正则性结果。
We establish the existence and the uniqueness for the Boussinesq system in the whole 3D space in the critical space of continuous in time with values in the power 3 integrable in space functions for the velocity and square integrable in time with values in the power 3/2 integrable in space.
In this article, we study an initial-boundary-value problem of the sixth order Boussinesq equation on a half line with nonhomogeneous boundary conditions: \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x>0\mbox{, }t>0,\] \[u(x,0)=φ(x), u_t(x,0)=ψ''(x),\] \[ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t),\] where $β=\pm1$. It is shown that the problem is locally well-posed in $H^s(\mathbb{R}^+)$ for $-\frac12<s\leq 0$ with initial condition $(φ,ψ)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$ and boundary condition $(h_1,h_2,h_3) $ in the product space $H^{\frac{s+1}{3}}(…
On the existence, regularity and uniqueness of $L^p$-solutions to the steady-state 3D Boussinesq system in the whole space
该研究探讨三维全空间稳态Boussinesq系统的$L^p$解性质，证明不同$p$范围下解的存在性与正则性，得到Navier-Stokes方程新正则准则及齐次情形的Liouville型结果。
In this work, we revisit the study by M. E. Schonbek [11] concerning the problem of existence of global entropic weak solutions for the classical Boussinesq system, as well as the study of the regularity of these solutions by C. J. Amick [1]. We propose to regularize by a "fractal" operator (i.e. a differential operator defined by a Fourier multiplier of type $ε|ξ|^λ, \, (ε,λ) \in\,\mathbb{R}_+\times ] 0,2]$). We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type $H^s(\mathbb{R}),\,s > \frac {1}{2},$, uniformly in the regularizing parameters…
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.
In this paper we establish some regularity criteria for the 3D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. We also obtain some uniform estimates for the corresponding 2D case when the fluid viscosity coefficient is a positive constant.
Global regularity and infinite Prandtl number limit of temperature patches for the 2D Boussinesq system
本文证明了二维非扩散Boussinesq系统温度斑块边界$C^{k+γ}$正则性的全局保持，解决了曲率全局控制问题，并通过无穷Prandtl数极限将已有保持结果从$k\in\{0,1,2\}$推广到$k\geq 1$。
We continue to study the initial-boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: \begin{equation*} \begin{cases} u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x,t\in \mathbb{R}^+,\\ u(x,0)=φ(x), u_t(x,0)=ψ''(x), \\ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t), \end{cases} \end{equation*} where $β=\pm1$. We show that the problem is locally analytically well-posed in the space $H^s(\mathbb{R}^+)$ for any $ s> -\frac34 $ with the initial-value data $$(φ,ψ)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)…
Here we investigate the so-called temperature patch problem for the incompressible Boussinesq system with partial viscosity, in the whole space $\mathbb{R}^N$ $(N \geq 2)$, where the initial temperature is the characteristic function of some simply connected domain with $C^{1, \varepsilon}$ H{ö}lder regularity. Although recent results in [1, 15] ensure that an initially $C^1$ patch persists through the evolution, whether higher regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue globally in time, in the 2-D case for large initial…
In this paper we study the regularity properties of the "good" Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [Himonas-Mantzavinos 2015], being the first result that constructs solutions for the initial and boundary value problem of the "good" Boussinesq equation below the $L^2$ space. …
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…
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.
The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the "vortex stretching", we seek unique global small solution…
Global regularity of temperature patches for the 3D non-diffusive Boussinesq system with large Prandtl number
研究大普朗特数下三维非扩散布辛涅斯克系统强解全局适定性问题，通过证明相关强解全局存在唯一，建立非恒定温度斑块初始数据下的边界正则性全局持续性，还证明了普朗特数趋于无穷时的极限及收敛情况。
Inspired by the recently published paper \cite{Hassainia-Hmidi}, the current paper investigates the local well-posedness for the generalized $2d-$Boussinesq system in the setting of regular/singular vortex patch. Under the condition that the initial vorticity $ω_{0}={\bf 1}_{D_0}$, with $\partial D_0$ is a Jordan curve with a Hölder regularity $C^{1+\EE},\;0<\EE<1$ and the density is a smooth function, we show that the velocity vector field is locally well-posed and we also establish local-in-time regularity persistence of the advected patch. Although, in the case of the singular patch, the an…
This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq-Navier-Stokes equations. The velocity $u$ in this system is related to the vorticity $ω$ through the relations $u=\nabla^\perp ψ$ and $Δψ= Λ^σ(\log(I-Δ))^γω$, which reduces to the standard velocity-vorticity relation when $σ=γ=0$. When either $σ>0$ or $γ>0$, the velocity $u$ is more singular. The "quasi-velocity" $v$ determined by $\nabla\times v =ω$ satisfies an equation of very special structure. This paper establishes the global regular…
This paper is concerned with the initial-boundary value problem to 2D magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. First, we establish the global weak solutions under the minimal initial assumption. Then by imposing higher regularity assumption on the initial data, we obtain the global strong solution with uniqueness. Moreover, the exponential decay estimate of the solution is obtained.
We prove the global well-posedness of the two-dimensional Boussinesq equations with zero viscosity and positive diffusivity in bounded domains for rough initial data [ $u_{0}\in L^{2}$, $\text{curl}\,u_{0}\in L^{\infty}$ and $θ_{0}\in B^{2-2/p}_{q,p}$ with $p\in (1,\infty)$, $q\in (2,\infty)$ ]. Our method is based on the maximal regularity for heat equation.
In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schrödinger equation and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for one dimensional perio…
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 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.
As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].
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