Pressure, Intermittency, Singularity
Peter Constantin
本文研究三维不可压缩 Navier-Stokes 方程解的正则性，利用压力与结构函数提出新的判据条件。
我们基于压力和结构函数，给出了三维不可压缩 Navier-Stokes 方程解的正则性条件。
We give conditions for regularity of solutions of three dimensional incompressible Navier-Stokes equations based on the pressure and on structure functions.
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In this paper, the singularity formation of classical solutions for the compressible Euler equations with general pressure law is considered. The gradient blow-up of classical solutions is shown without any smallness assumption by the delicate analysis on the decoupled Riccati type equations. The proof also relies on a new estimate for the upper bound of density.
Intermittency and Dissipation Regularity in Turbulence
该研究建立几何分析框架捕捉不可压缩欧拉方程弱解能量耗散特性，证明杜雄 - 罗伯特分布相关正则性，对耗散集维度等施加约束，还恢复了一些“昂萨格奇点”结果。
We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions $u\in C([0, T); W^{2,q} (\mathbb R^3))$, $q>3$ of the incompressible Euler equations. We show that a blow up at $t=T$ happens only if $$\int_0 ^T \int_0 ^t \left\{\int_0 ^s \|D^2 p (τ)\|_{L^\infty} dτ\exp \left( \int_{s} ^t \int_0 ^{\s} \|D^2 p (τ)\|_{L^\infty} dτd\s \right) \right\}dsdt \, = +\infty.$$ As consequences of this criterion we show that there is no blow up at $t=T$ if $ \|D^2 p(t)\|_{L^\infty} \le \frac {c}{(T-t)^2}$ with $c<1$ as $t\nearrow T$. Under the additional assumption of $\int_0 ^T \|…
We analyze nonnegative solutions of the nonlinear elliptic problem $Δu=\frac{λf(x)}{u^2}+P$, where $λ>0$ and $P\geq0$, on a bounded domain $Ω$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This equation models an electrostatic--elastic membrane system with an external pressure $P\geq 0$, where $λ>0$ denotes the applied voltage. First, we completely address the existence and nonexistence of positive solutions. The classification of all possible singularities at $|x|=0$ for nonnegative solutions $u(x)$ satisfying $u(0)=0$ is then analyzed for the special case where $Ω=B_1(0)…
A sufficient condition is derived for a finite-time $L_2$ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $\lim_{t \to T_*} \sup | \frac{D ø} {Dt} |_{L_2(\vO)} = \infty$, where $~ \vO \subset \R3$ moves with the fluid. In particular, $|ø|$, $|§_{ij}| , and $|¶_{ij}|$ all become unbounded at one point $(x_1,T_1)$, $T_1$ being the first blow-up time in $L_2$.
We consider the singularity formation of strong solutions to the two-dimensional full compressible Navier-Stokes equations with zero heat conduction in a bounded domain. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density and the pressure are bounded from above. Critical Sobolev inequalities of logarithmic type play a crucial role in the proof.
We study the singularity formation of strong solutions to the two-dimensional (2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic equations without heat conductivity. It is proved that the strong solution exists globally if the density and the pressure are bounded from above. In particular, the criterion is independent of the magnetic field and is just the same as that of the compressible Navier-Stokes equations. Our method relies on weighted energy estimates and a Hardy-type inequality.
The formation of singularity and breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction are considered. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density $ρ$ and the pressure $P$ satisfy $\|ρ\|_{L^{\infty}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In addition, the initial density can even have compact support. The logarithm-type estimate for the Lam{é} system and some weighted estimates play a crucial role in the proof.
In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier-Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak Lebesgue norm in the axisymmetric case. This type of results is inspired in particular by a work of Neustupa (2014), which handles certain non endpoint critical norms. Our work enables to have a better understanding of the nonlocal effect of the pressure on the regularity of the solutions.
Intermittency and lower dimensional dissipation in incompressible fluids
该研究针对不可压缩流体湍流间歇性问题，基于几何假设证明无粘情形下低维耗散会使p>3的p阶结构函数偏离K41预测，其得到的ζ_p上界与β-模型吻合，为该模型提供了严格解析基础。
Cascade mechanisms for Navier-Stokes blow-up
本文通过Obukhov二进模型阐明三维Navier-Stokes方程的逆能量级联导致瞬时I型爆破的机制，并证明在高间歇性下有限能量解中同样发生，同时给出混合模型的有限时间爆破定理。
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial dat…
Critical Morrey Rigidity and Removable Singularities for Five-Dimensional Stationary Navier-Stokes Flows
该研究证明了五维定常Navier-Stokes方程的临界Morrey刚性定理，推导了五维可去奇点准则，还通过有限能量论证证明了四维对应的三次Morrey刚性定理。
We classify all $(-1)-$homogeneous axisymmetric no swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of $(-1)-$homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of …
Splash singularity for the free boundary incompressible viscous MHD
本文证明二维自由边界不可压缩粘性磁流体动力学方程存在光滑初始数据，使界面在有限时间内产生飞溅奇异性（自相交），且磁场不阻止该现象。
The Cauchy problem for the system of equations of two-dimensional rotational gas dynamics is considered. It is assumed that the Cauchy data are a smooth compact perturbation of a constant state. Integral conditions for the data sufficient for the loss of smoothness by a solution within a finite time are found. We analyze the possibility of fulfilling these conditions and compare them with the criterion of singularity formation, known for rotational gas dynamics without pressure.
A new formulation of the Navier-Stokes equation, in terms of the gradient of the total mechanical energy, is derived for the time-averaged flows, and the singular point possibly existing in the Navier-Stokes equation is exactly found. Transition of a laminar flow to turbulence must be implemented via this singular point. For pressure driven flows, this singular point corresponds to the inflection point on the velocity profile. It is found that the stability of a flow depends on the direction of the gradient of the total mechanical energy for incompressible pressure-driven flow. When this direc…
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