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We examine the blow-up claims of the incompressible Euler equations for several specific flow-fields, (1) the columnar eddies in the vicinity of stagnation; (2) a quasi-three-dimensional structure for illustrating oscillations and…
This note provides a detailed proof of the fact that a linear vector field on a vector bundle has a flow by vector bundle isomorphisms. It implies then easily the existence of global solutions to linear non-autonomous ODE's, with a standard…
We show the uniqueness of particle paths of a velocity field, which solves the compressible isentropic Navier-Stokes equations in the half-space $\mathbb{R}_+^3$ with the Navier boundary condition. In particular, by means of energy…
In this work a result of existence and uniqueness for a plane cavity driven steady flow is deduced using an analytical method for the resolution of a linear partial differential problem on a triangular domain. The solution admits a symbolic…
We develop several efficient numerical schemes which preserve exactly the global constraints for constrained gradient flows. Our schemes are based on the SAV approach combined with the Lagrangian multiplier approach. They are as efficient…
We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients. Asking only boundedness of the divergence of the coefficients (a classical condition in both the…
Based on the concept of manifold valued generalized functions we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right hand sides in a global setting. After establishing several…
We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few…
We are interested in the gradient flow of a general first order convex functional with respect to the $L^1$-topology. By means of an implicit minimization scheme, we show existence of a global limit solution, which satisfies an…
A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both…
We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.
The statistical properties of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics is investigated in a time range spanning…
Mean-field-based Lagrangian framework is developed for the fluid turbulence theory. The space- time vector flow is naturally introduced from the mean velocity, which provides the Lagrangian picture based on the mean field in totally…
We present a new application of Lagrangian Perturbation Theory (LPT): the stability analysis of fluid flows. As a test case that demonstrates the framework we focus on the plane Couette flow. The incompressible Navier-Stokes equation is…
We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral…
We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.
We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this…
It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal…
We investigate the incompressible Navier-Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous ini- tial density. In dimension n = 2, 3, assuming only that the initial…
The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (kinetic energy), and the regularization of density paths (potential energy). These combinations yield different…