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.
The primitive equations of large-scale oceanic dynamics form a fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by the hydrostatic approximation. On the other hand, the mathematically rigorous derivation of the primitive equations without coupling with the temperature is also known. In this paper, we generalize the above result from the mathematical point of view. More precisely, we prove that the scaled Boussinesq equations with rotation converge to the full primitive equations in a strong sense, globally in time, with the convergen…
Derivation of dissipative Boussinesq equations using the Dirichlet-to-Neumann operator approach
水波理论传统上假设流体是完美的，因此忽略了粘性的所有影响。然而，几个实验数据集的解释需要明确包括耗散效应。为了解决这些实际问题，发展了粘势流理论（见P.-P. F. Liu & A. Orfila（2004）和D. Dutykh & F. Dias（2007））。然后，通常通过在垂直坐标中展开整个系列的势并引入长波近似来进一步简化该公式。在本研究中，我们提出了一个推导的耗散Boussinesq方程是基于渐近展开的Dirichlet-to-Neumann（D2 N）运营商。这两种方法通过不同的方式产生相同的系统。
Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,α}$ initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with $C^{1,α}$ initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodyn…
In this paper, we rigorously derive the governed equations describing the motion of stable stratified fluid, from the mathematical point of view. Specially, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ration parameter goes to zero, and the rate of convergence is of the same order as the aspect ratio parameter. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of strong solutions to the viscous primitive equations with density stratification.
We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of approximate Riemann invariants for a Whitham-Boussinesq system and is adapted to unidirectional waves. The second one is based on a generalisation of Birkhoff's normal form algorithm for almost smooth Hamiltonians and is adapted to bidirectional propagation. In both cases we clarify the improved accuracy on the fully dispersive Whitham model with respect to the lo…
Coupled Boussinesq equations describe long weakly-nonlinear longitudinal strain waves in a bi-layer with a soft bonding between the layers (e.g. a soft adhesive). From the mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations, for the right- and left-propagating waves in each layer. However, the models impose a ``zero-mass constraint'' i.e. the initial conditions should necessarily have zero…
We derive the 3D spatially homogeneous Boltzmann's equation with moderately soft potentials and singular angular interaction, from an interacting particles system. The collision kernel is of the form $B(z,σ)=|z|^γb\left( \frac{z}{|z|}\cdot σ\right)$ and for $K>0$, $\sin(θ)b\left(\cos(θ)\right)\sim Kθ^{-1-ν}$, with $γ\in (-2,-1)$ and $ν\in(1,2)$ satisfying $γ+ν>0$. We use at the particle level the regularizing effects of the grazing collisions, in order to control the singularity of the soft potential. This enables to use a classical compactness argument, and provide a qualitative convergence r…
Hölder regularity for the parabolic perturbed fractional 1-Laplace equations
Harbin Institute of Technology（哈尔滨工业大学）；School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)
