On the motion of incompressible inhomogeneous Euler-Korteweg fluids
Miroslav Bulíček Eduard Feireisl Josef Málek Roman Shvydkoy
Discrete & Continuous Dynamical Systems - S 2010, 3(3): 497-515 doi: 10.3934/dcdss.2010.3.497
We study a system of equations governing evolution of incompressible inhomogeneous Euler-Korteweg fluids that describe a class of incompressible elastic materials. A local well-posedness theory is developed on a bounded smooth domain with no-slip boundary condition on velocity and vanishing gradient of density. The cases of open space and periodic box are also considered, where the local existence and uniqueness of solutions is shown in Sobolev spaces up to the critical smoothness $\frac{n}{2}+1$.
keywords: local-in-time well-posedness smooth solution. Korteweg stress Korteweg fluid inhomogeneous Euler fluid
Lectures on the Onsager conjecture
Roman Shvydkoy
Discrete & Continuous Dynamical Systems - S 2010, 3(3): 473-496 doi: 10.3934/dcdss.2010.3.473
These lectures give an account of recent results pertaining to the celebrated Onsager conjecture. The conjecture states that the minimal space regularity needed for a weak solution of the Euler equation to conserve energy is $1/3$. Our presentation is based on the Littlewood-Paley method. We start with quasi-local estimates on the energy flux, introduce Onsager criticality, find a positive solution to the conjecture in Besov spaces of smoothness $1/3$. We illuminate important connections with the scaling laws of turbulence. Results for dyadic models and a complete resolution of the Onsager conjecture for those is discussed, as well as recent attempts to construct dissipative solutions for the actual equation.
   The article is based on a series of four lectures given at the 11th school "Mathematical Theory in Fluid Mechanics" in Kácov, Czech Republic, May 2009.
keywords: Navier-Stokes equation Onsager conjecture Euler equation dyadic models. weak solutions Besov spaces turbulence
Eulerian dynamics with a commutator forcing Ⅱ: Flocking
Roman Shvydkoy Eitan Tadmor
Discrete & Continuous Dynamical Systems - A 2017, 37(11): 5503-5520 doi: 10.3934/dcds.2017239

We continue our study of one-dimensional class of Euler equations, introduced in [11], driven by a forcing with a commutator structure of the form $[{\mathcal L}_φ, u](ρ)$, where $u$ is the velocity field and ${\mathcal L}_φ$ belongs to a rather general class of convolution operators depending on interaction kernels $φ$.

In this paper we quantify the large-time behavior of such systems in terms of fast flocking, for two prototypical sub-classes of kernels: bounded positive $φ$'s, and singular $φ(r) = r^{-(1+α)}$ of order $α∈ [1, 2)$ associated with the action of the fractional Laplacian ${\mathcal L}_φ=-(-\partial_{xx})^{α/2}$. Specifically, we prove fast velocity alignment as the velocity $u(·, t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|{u_x}( \cdot ,t){|_\infty } + |{u_{xx}}( \cdot ,t){|_\infty }\lesssim{e^{ - \delta t}}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $ρ(·, t) -{ρ_{∞}}(x -\bar{u} t) \to 0$.

keywords: Flocking alignment fractional dissipation Cucker-Smale Motsch-Tadmor critical thresholds

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