The Navier-Stokes regularity problem

There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier–Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted

Well-posedness for the diffusive 3D Burgers equations with initial data in H1/2H^{1/2}

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski and A. Vidal-L\'opez, Cambridge University Press, 201

Parametrization of global attractors experimental observations and turbulence

This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method. Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory. Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence. Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself

Embedding Properties of sets with finite box-counting dimension

In this paper we study the regularity of embeddings of finite--dimensional subsets of Banach spaces into Euclidean spaces. In 1999, Hunt and Kaloshin [Nonlinearity 12 1263-1275] introduced the thickness exponent and proved an embedding theorem for subsets of Hilbert spaces with finite box--counting dimension. In 2009, Robinson [Nonlinearity 22 711-728] defined the dual thickness and extended the result to subsets of Banach spaces. Here we prove a similar result for subsets of Banach spaces, using the thickness rather than the dual thickness. We also study the relation between the box-counting dimension and these two thickness exponents for some particular subsets of $\ell_{p}$.Comment: Submitted, Referres comments addresse

Rigorous Numerical Verification of Uniqueness and Smoothness in a Surface Growth Model

Based on numerical data and a-posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities to the 3D Navier--Stokes equations, for certain initial data for which analytical approaches fail. The key point is the derivation of a scalar ODE controlling the norm of the solution, whose coefficients depend on the numerical data. Instead of solving this ODE explicitly, we explore three different numerical methods that provide rigorous upper bounds for its solutio

Procedure for implementation of temperature-dependent mechanical property capability in the Engineering Analysis Language (EAL) system

A procedure is presented to allow the use of temperature dependent mechanical properties in the Engineering Analysis Language (EAL) System for solid structural elements. This is accomplished by including a modular runstream in the main EAL runstream. The procedure is applicable for models with multiple materials and with anisotropic properties, and can easily be incorporated into an existing EAL runstream. The procedure (which is applicable for EAL elastic solid elements) is described in detail, followed by a description of the validation of the routine. A listing of the EAL runstream used to validate the procedure is included in the Appendix

Robustness of Regularity for the 33D Convective Brinkman-Forchheimer Equations

We prove a robustness of regularity result for the $3$D convective Brinkman-Forchheimer equations \partial_tu -\mu\Delta u + (u \cdot \nabla)u + \nabla p + \alpha u + \beta\abs{u}^{r - 1}u = f, for the range of the absorption exponent $r \in [1, 3]$ (for $r > 3$ there exist global-in-time regular solutions), i.e. we show that strong solutions of these equations remain strong under small enough changes of the initial condition and forcing function. We provide a smallness condition which is similar to the robustness conditions given for the $3$D incompressible Navier-Stokes equations by Chernyshenko et al. (2007) and Dashti & Robinson (2008).Comment: 22 page