Global Regularity for the Viscous Boussinesq Equations

On the global well-posedness of axisymmetric viscous Boussinesq system in critical Lebesgue spaces

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in \cite{Gallay,Gallay-Sverak}. Roughly speaking, we show essentially that if the initial data $(v_0,ρ_0)$ is axisymmetric and $(ω_0,ρ_0)$ belongs to the critical space $L^1(Ω)\times L^1(\mathbb{R}^3)$, with $ω_0$ is the initial vorticity associated to $v_0$ …

粘性三维Boussinesq温度锋面的正则性结果

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。

Large time decay properties of solutions to a viscous Boussinesq system in a half space

We consider the long time behavior of weak and strong solutions of the n-dimensional viscous Boussinesq system in the half space, with $n\geq3$ . The $L^r(R^n_+)$-asymptotics of strong solutions and their first three derivatives, with $1\leq r\leq\infty$, are derived combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^\infty-$asymptotics of the second order derivatives the unboundedness of the projection operator $P: L^\infty(R^n_+)\rightarrow L^\infty_σ(R^n_+)$ is dealt by an appropriate decomposition of the nonlinear term.

On the regularity of solutions to the 2D Boussinesq equations satisfying Type I conditions

We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations.

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space-time decay properties for higher order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions in scaling invariant function spaces. We further investigate the asymptotic profiles and decay properties of these strong solution…

Global smooth solutions of the damped Boussinesq equations with a class of large initial data

The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $B^{s}_{p,r}$ or $\dot{B}^{s}_{p,r}$ norms can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together w…

On the global regularity of axisymmetric Navier-Stokes-Boussinesq system

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.

A regularity criterion in weak spaces to Boussinesq equations

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.

A new approach for the regularity of weak solutions of the 3D Boussinesq system

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 $θ$.

Global Regularity for the Two-dimensional Boussinesq Equations Without Diffusivity in Bounded Domains

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.

关于Oberbeck-Boussinesq方程弱解的正则性

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。

Regularity Criterion for the Three-dimensional Boussinesq Equations

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 $u$.

Global regularity of non-diffusive temperature fronts for the 2D viscous Boussinesq system

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 $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ …

轴对称MHD-Boussinesq系统的单分量正则性准则与无粘极限

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。

A note on global regularity results for 2D Boussinesq equations with fractional dissipation

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}.

On the striated regularity for the 2D anisotropic Boussinesq system

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…

Well-posedness for the Boussinesq system in critical spaces via maximal regularity

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.

Large time decay and growth for solutions of a viscous Boussinesq system

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to\infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from $(u_0,θ_0)$ with zero-mean for the initial temperature $θ_0$ have a special behavior as $|x|$ or $t$ tends to infi…

On the local well-posedness and a Prodi-Serrin type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion

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-…

A regularity criterion to the 3d Boussinesq equations

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.

Global regularity of solutions for the 3D non-resistive and non-diffusive MHD-Boussinesq system with axisymmetric data

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.

Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index $s=-1$

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{u}_0,θ_0)$ ($\nabla\cdot\vec{u}_0=0$) either in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times{B}^{-1}_{p,r}$ or in ${B^{-1,1}_{\infty,\infty}}\times{B}^{-1,ε}_{p,\infty}$ if $r\in[1,\infty]$, $ε>0$ and $p\in(\frac{n}{2},\infty)$, where $B^{s,ε}_{p,q}$ ($s\in\mathbb{R}$, $1\leq p,q\leq\infty$, $ε>0$) is the logarithmically modified Besov space to the standard Besov space $B^{s}_{p,q}$. We also prove that this…

One component regularity criteria for the axially symmetric MHD-Boussinesq system: criteria on the swirl component of the velocity

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.

On echo chains in the linearized Boussinesq equations around traveling waves

We consider the 2D Boussinesq equations with viscous but without thermal dissipation and observe that in any neighborhood of Couette flow and hydrostatic balance (with respect to local norms) there are time-dependent traveling wave solutions of the form $ω=-1+f(t)\cos(x-ty)$, $θ=αy + g(t)\sin(x-ty)$. As our main result we show that the linearized equations around these waves for $α=0$ exhibit echo chains and norm inflation despite viscous dissipation of the velocity. Furthermore, we construct initial data in a critical Gevrey 3 class, for which temperature and vorticity diverge to infinity in …

Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation

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 the global regularity of the 2D critical Boussinesq system with $α>2/3$

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…

具有临界正则性与色散效应的二维无粘Boussinesq方程的长时间可解性

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。

A global regularity result for the 2D Boussinesq equations with critical dissipation

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…

Higher Order Regularity and Blow-up Criterion for Semi-dissipative and Ideal Boussinesq Equations

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.

Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain

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…

Global regularity for 2D Boussinesq temperature patches with no diffusion

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…

On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces

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.

Evolution Boussinesq model with nonmonotone friction and heat flux boundary conditions

In this paper we prove the existence and regularity of a solution to a two-dimensional system of evolutionary hemivariational inequalities which describes the Boussinesq model with nonmonotone friction and heat flux. We use the time retardation and regularization technique, combined with a regularized Galerkin method, and recent results from the theory of hemivariational inequalities.

Oberbeck-Boussinesq近似的严格推导揭示意外项

Rigorous derivation of the Oberbeck-Boussinesq approximation revealing unexpected term

本文在低Mach和Froude数机制下严格推导可压缩粘性导热流体的渐近极限，发现其表现为带温度偏差非局部边界条件的Oberbeck-Boussinesq系统。

负指标齐次 Besov 空间中三维 Boussinesq 方程的一个正则性准则

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。

Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

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…

Derivation of a viscous KP equation including surface tension, and related equations

The aim of this article is to derive surface wave models in the presence of surface tension and viscosity. Using the Navier-Stokes equations with a free surface, flat bottom and surface tension, we derive the viscous 2D Boussinesq system with a weak transverse variation. The assumed transverse variation is on a larger scale than along the main propagation direction. This Boussinesq system is proved to be consistent with the Navier-Stokes equations. This system is only an intermediate result that enables us to derive the Kadomtsev-Petviashvili (KP) equation which is a 2D generalization of the K…

广义改进Boussinesq方程的正则性无限损失的范数膨胀

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。

On the global solvability of the axisymmetric Boussinesq system with critical regularity

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_…

层状多孔介质中 Darcy-Boussinesq 系统的适定性与正则性

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。

On the regularity of weak solutions of the Boussinesq equations in Besov spaces Dedicated to Enrique Zuazua on the occasion of his sixtieth birthday

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…

具有减弱初始温度的3D-Boussinesq系统的温和解

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。

The integral Cauchy problem for generalized Boussinesq equations with general leading parts

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.

有界域上Boussinesq方程的分数阶正则性、全局持续性和渐近性质

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。

具有次临界耗散的二维分数阶Boussinesq方程的整体正则性

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。

粘性Boussinesq系统轨迹的小时间全局精确能控性

Small-time global exact controllability to the trajectories for the viscous Boussinesq system

本文针对二维和三维光滑有界域内带特定边界条件的粘性Boussinesq系统，通过区域延拓、借鉴Navier-Stokes方程能控性策略及线性化加Carleman估计三步法，证明了边界小区域控制下的轨迹小时间全局精确能控性。

The 2D incompressible Boussinesq equations with general critical dissipation

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…

On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations

We improve and extend some known regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony's para-product decomposition.

Boussinesq系统的适定性

Wellposedness of Boussinesq system

本文研究二维和三维粘性不可压缩非线性Boussinesq系统在非齐次边界条件下的适定性，建立了无需小性假设的三维全局弱解存在性。

A Note on Regularity and Uniqueness of Natural Convection with Effects of Viscous Dissipation in 3D Open Channels

We prove the existence of unique regular solutions of steady-state buoyancy-driven flows of viscous incompressible heat-conducting fluids in 3D open channels under mixed boundary conditions. The model takes into account the phenomena of the viscous energy dissipation.

Boussinesq方程的Korteweg-de Vries近似的长时间有效性

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。

Lower regularity solutions of non-homogeneous boundary value problems of the sixth order Boussinesq equation in a quarter plane

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}}(…

全空间中稳态三维Boussinesq系统$L^p$解的存在性、正则性与唯一性研究

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。

The classical Boussinesq system revisited

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…