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We review recent developments in the field of mathematical fluid dynamics which utilize techniques that go under the umbrella name convex integration. In the hydrodynamical context, these methods produce paradoxical solutions to the fluid equations which defy physical laws. These counterintuitive solutions have a number of properties that resemble predictions made by phenomenological theories of fluid turbulence. The goal of this review is to highlight some of these similarities while maintaining an emphasis on rigorous mathematical statements. We focus our attention on the construction of weak solutions for the incompressible Euler, Navier-Stokes, and magneto-hydrodynamic equations which violate these systems' physical energy laws. [ABSTRACT FROM AUTHOR]

We consider the Stokes and Navier-Stokes equations in a bounded domain with analytic boundary. We present a direct, robust, energy-type approach for establishing the instantaneous gain of space-time analyticity of the solution from any Sobolev smooth initial datum, with analyticity radius which is uniform up to the curved boundary. [ABSTRACT FROM AUTHOR]

In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions ***where is the Hilbert transform. We H provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion ***where Λ = (−Δ)1/2, and 1/2 ≤ γ < 1. Our results also apply to the model with velocity field u = ΛsHθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the H¨older class in C(s+1)/2, for all positive time. [ABSTRACT FROM AUTHOR]

We consider the convergence in the L² norm, uniformly in time, of the Navier-Stokes equations with Dirichlet boundary conditions to the Euler equations with slip boundary conditions. We prove that if the Oleinik conditions of no back-flow in the trace of the Euler flow, and of a lower bound for the Navier-Stokes vorticity is assumed in a Kato-like boundary layer, then the inviscid limit holds. [ABSTRACT FROM AUTHOR]