Related papers: The elastic flow with obstacles: small obstacle re…
In this paper, we prove the existence and uniqueness of subsonic solutions to the steady Euler flows past a smooth, axisymmetric obstacle. Specifically, for a broad class of prescribed positive axial velocities in the upstream, the subsonic…
We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are applied to homogenize convection dominated parabolic equations with rapidly…
This paper is devoted to classical variational problems for planar elastic curves of clamped endpoints, so-called Euler's elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain…
We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraint in time and natural regularity assumptions. We provide first a notion of weak solution, inspired by the theory of curves of maximal slope, and…
We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach…
We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same…
The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The…
In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time…
We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants.…
For a given $p\in[2,+\infty)$, we define the $p$-elastic energy $\mathscr{E}$ of a closed curve $\gamma:\mathbb{S}^1\to M$ immersed in a complete Riemannian manifold $(M,g)$ as the sum of the length of the curve and the $L^p$--norm of its…
The 1D flow of a continuous beam of Bose-Einstein condensed atoms in the presence of an obstacle is studied as a function of the beam velocity and of the type of perturbing potential (representing the interaction of the obstacle with the…
We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of…
In non-linear incompatible elasticity, the configurations are maps from a non-Euclidean body manifold into the ambient Euclidean space, $\mathbb{R}^k$. We prove the $\Gamma$-convergence of elastic energies for configurations of a converging…
In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave…
We consider a class of parabolic variational inequalities with time dependent obstacle of the form $|{\boldsymbol u}(x,t)| \le p(x,t)$, where ${\boldsymbol u}$ is the velocity field of a fluid governed by the Navier--Stokes variational…
Edges are abundant when elastic solids glide in guiding rails or fluids are contained in vessels. We here address induced displacements in elastic solids or small-scale flows in viscous fluids in the vicinity of one such edge. For this…
This paper investigates the lift force exerted on an elliptical obstacle immersed in a granular flow through analytical calculations and computer simulations. The results are shown as a function of the obstacle size, orientation with…
We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered…
Effects of geometric constraints on a steady flow potential are described by an elliptic-hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality.
We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.