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 regularity and uniqueness of solutions for the case when $σ=0$ and $γ\ge 0$. In addition, the vorticity $ω$ is shown to be globally bounded in several functional settings such as $L^2$ for $σ>0$ in a suitable range.
The present paper is dedicated to the global well-posedness issue for the Boussinesq system with the temperature-dependent viscosity in $\mathbb{R}^2.$ We aim at extending the work by Abidi and Zhang ( Adv. Math. 2017 (305) 1202--1249 ) to a supercritical dissipation for temperature.
We prove the global regularity of the solution pair to the N-dimensional logarithmically supercritical magnetohydrodynamics system with zero diffusivity. This is the endpoint case omitted in the work of [24]; it also improves some previous results logarithmically.
In this paper we study a transport-diffusion model with some logarithmic dissipations. We look for two kinds of estimates. The first one is a maximum principle whose proof is based on Askey theorem concerning characteristic functions and some tools from the theory of $C_0$-semigroups. The second one is a smoothing effect based on some results from harmonic analysis and sub-Markovian operators. As an application we prove the global well-posedness for the two-dimensional Euler-Boussinesq system where the dissipation occurs only on the temperature equation and has the form $\frac{\DD}{\log^α(e^4+…
We prove global existence of smooth solutions for a slightly supercritical dyadic model. We consider a generalized version of the dyadic model introduced by Katz-Pavlovic [2005] and add a viscosity term with critical exponent and a supercritical correction. This model catches for the dyadic a conjecture that for Navier-Stokes equations was formulated by Tao [2009].
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）
Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+ \times \R^d \to \R$, where $u_0: \R^d \to \R^d$ is smooth and divergence free, and $D$ is a Fourier multiplier whose symbol $m: \R^d \to \R^+$ is non-negative; the case $m(ξ) = |ξ|$ is essentially Navier-Stokes. It is folklore (see e.g. \cite{kp}) that one has global regularity in the critical and subcritical hyperdissipation regimes $m(ξ) = |ξ|^α$ for $α\geq \f…
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.
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…
The Boussinesq $abcd$ system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters $(a,b,c,d)$ satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to $b \ne d$). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the glo…
We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol defined as a power of logarithm is added to both velocity and dissipation terms to penalize the equation's criticality. Our primary tool is the nonlinear maximum principle which provides transparent proofs of global regularity for nonlinear dissipative equations. Combining the nonlinear maximum principle with a modulus of continuity, we prove a uniform-in-time gr…
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the sup…
We consider a damped wave equation in a bounded domain. The damping is nonlinear and is homogeneous with degree p -- 1 with p > 2. First, we show that the energy of the strong solution in the supercritical case decays as a negative power of t; the rate of decay is the same as in the subcritical or critical cases, provided that the space dimension does not exceed ten. Next, relying on a new differential inequality, we show that if the initial displacement is further required to lie in L p , then the energy of the corresponding weak solution decays logarithmically in the supercritical case. Thos…
We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high freq…
This article studies the global regularity problem of the two-dimensional zero thermal diffusion tropical climate model with fractional dissipation, given by $(-Δ)^αu$ in the barotropic mode equation and by $(-Δ)^βv$ in the first baroclinic mode of the vector velocity equation. More precisely, we show that the global regularity result holds true as long as $α+β\geq2$ with $1<α<2$. In addition, with no dissipation from both the temperature and the first baroclinic mode of the vector velocity, we also establish the global regularity result with the dissipation strength at the logarithmically sup…
