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Publication# Blow-up, partial regularity and turbulence in incompressible fluid dynamics

Abstract

Weak solutions arise naturally in the study of the Navier-Stokes and Euler equations both from an abstract regularity/blow-up perspective and from physical theories of turbulence. This thesis studies the structure and size of singular set of such weak solutions to equations of incompressible fluid dynamics from two opposite directions. First, it aims to single-out new mechanisms which allow to break the typically supercritical scaling of the equations and, in this way, prevent the formation of singularities either globally or locally in spacetime. Second, in the absence of such mechanisms, we seek to quantify how singular (in terms of dimension of the singular set, for instance) the solutions that we are actually able to construct are. This thesis collects four results pointing in the two directions outlined above which have been obtained in several collaborations during the Ph.D. studies:- a global regularity result for the fractional Navier-Stokes equation slightly blow the critical fractional order,- a global well-posedness result for the defocusing wave equation with slightly supercritical power nonlinearity,- an a.e. smoothness / partial regularity result for the supercritical surface quasigeostrophic (SQG) equation,- an estimate (and a discussion of its sharpness) on the dimension of the singular set of wild Hölder continuous solutions of the incompressible Euler equations.All results presented in the thesis have either been published or are submitted for publication.

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Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, i

Equation

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages

Navier–Stokes equations

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier an

We address the free boundary problem that consists in finding the shape of a three dimensional glacier over a given period and under given climatic conditions. Glacier surface moves by sliding, internal shear and external exchange of mass. Ice is modelled as a non Newtonian fluid. Given the shape of the glacier, the velocity of ice is obtained by solving a stationary non-linear Stokes problem with a sliding law along the bedrock-ice interface. The shape of the glacier is updated by computing a Volume Of Fluid (VOF) function, which satisfies a transport equation. Climatic effects (accumulation and ablation of ice) are taken into account in the source term of this equation. A decoupling algorithm with a two-grid method allows the velocity of ice and the VOF to be computed using different numerical techniques, such that a Finite Element Method (FEM) and a characteristics method. On a theoretical level, we prove the well-posedness of the non-linear Stokes problem. A priori estimates for the convergence of the FEM are established by using a quasi-norm technique. Eventually, convergence of the linearisation schemes, such that a fixed point method and a Newton method, is proved. Several applications demonstrate the potential of the numerical method to simulate the motion of a glacier during a long period. The first one consists in the simulation of Rhone et Aletsch glacier from 1880 to 2100 by using climatic data provided by glaciologists. The glacier reconstructions over the last 120 years are validated against measurements. Afterwards, several different climatic scenarios are investigated in order to predict the shape the glaciers until 2100. A dramatic retreat during the 21st century is anticipated for both glaciers. The second application is an inverse problem. It aims to find a climate parametrization allowing a glacier to fit some of its moraines. Two other aspects of glaciology are also addressed in this thesis. The first one consists in modeling and in simulating ice collapse during the calving process. The previous ice flow model is supplemented by a Damage variable which describes the presence of micro crack in ice. An additional numerical scheme allows the Damage field to be solved and a two dimensional simulation of calving to be performed. The second problem aims to prove the existence of stationary ice sheet when considering shallow ice model and a simplified geometry. Numerical investigation confirms the theoretical result and shows physical properties of the solution.

The simulation of flows of viscoelastic fluids is a very challenging domain from the theoretical as well as the numerical modelling point of view. In particular, all the existing methods have failed to solve the high Weissenberg number problem (HWNP). It is therefore clear that new tools are necessary. In this thesis we propose to tackle the problem of the simulation of viscoelastic fluids presenting memory effects, which is the first attempt of applying the lattice Boltzmann method (LBM) to this field for non-trivial flows. A theoretical development of the discrete models corresponding to the equations of mass, momentum conservation and of the constitutive equation is presented as well as the particular treatment of the associated boundary conditions. We start by presenting a simplified case where no memory but shear-thinning or shear-thickening effects are present : simulating the flow of generalized Newtonian fluids. We test the corresponding method against two-dimensional benchmarks : the 2D planar Poiseuille and the 4:1 contraction flows. Then we propose a new model consisting in solving the constitutive equations that account for memory effects, by explicitly including an upper-convected derivative, using the lattice Boltzmann method. In particular, we focus on the polymer dumbbell models, with infinite or finite spring extension (Oldroyd-B and FENE-P models). Using our model, we study the periodic (simplified) 2D four-roll mill and the 3D Taylor-Green decaying vortex cases. Finally, we propose an approach for simulating flat walls and show the applicability of the method on the 2D planar Poiseuille case. Two of the advantages of the proposed method are the ease of implementation of new viscoelastic models and of an algorithm for parallel computing.

In a region D in R-2 or R-3, the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by partial derivative(t)v + (v . del(x))v = -del(xP), div(x)v = 0, where v(t, x) is the velocity of the particle located at x is an element of D at time t and p(t, x) is an element of R is the pressure. Solutions v and p to the Euler equation can be obtained by solving {del x {partial derivative t phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2)} = 0 at a = kappa(t, x), v(t, x) = del(x)phi(t, x, a) at a = kappa(t, x), partial derivative(t)kappa(t, x) | (v. del(x))kappa(t, x) = 0 div(x)v(t, x) = 0 where phi : R x D x R-l -> R and kappa : R x D -> R-l are additional unknown mappings (l >= 1 is prescribed). The third equation in the system says that kappa is an element of R-l is convected by the flow and the second one that phi can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition kappa(0, x) = x on D (and thus l = 2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52: 411-452, 1999) in his Eulerian-Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross. D and that carry each "particle" at time t = 0 at a prescribed location at time t = T > 0, that is, kappa(T, x) is prescribed in D for all x is an element of D. We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary partial derivative D of the bounded region D (a two-or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through partial derivative D of particles labelled by each value of. at each point of partial derivative D. One of the main novelties is the introduction of a prescribed "generalized" Bernoulli's function H : R-l -> R, namely, we add to (0.1) the requirement that partial derivative(t)phi(t, x, a) + p(t, x) + (1/2)|del(x)phi(t, x, a)|(2) = H(a) at a = kappa(t, x) with phi, p, kappa periodic in time of prescribed period T > 0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of "Lamb's surfaces" and "isotropic manifolds" in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier's formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional (kappa, v) -> integral(T)(0)integral(D) {1/2|v(t, x)|(2) + H(kappa(t, x))} dtdx defined for kappa and v that are T-periodic in t, such that partial derivative(t)kappa(t, x) + (v . del(x))kappa(t, x) = 0, div(x)v(t, x) = 0, and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize integral(]0, L[x]0,1[) {(1/2)|del psi|(2) + H(psi)}dx for psi is an element of W-1,W-2 (]0, L[x]0,1[) under appropriate boundary conditions, where. is the stream function. For a minimizer, corresponding functions phi and kappa are given in terms of the stream function psi.