Related papers: Delayed parabolic regularity for curve shortening …
In this paper we establish some regularity results concerning the behavior of weak solutions and very weak solutions of the degenerate wave equation near the boundary. For the nondegenerate case, the correponding results were originally…
Parabolic geometric flows have the property of smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of this paper is that, by bringing in…
In this paper we give rates of convergence for the $p$-curve shortening flow for $p\geq 1$ an integer, which improves on the known estimates and which are probably sharp.
We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time…
We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of…
In this paper, we study curve shortening flows on rotational surfaces in $\mathbb{R}^3$. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve…
We study the evolution of a weakly convex surface $\Sigma_0$ in $\R^3$ with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the…
In this paper, we deal with the boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domain of the form ($\alpha (t)<x<\beta (t)$). By using the characteristics method, we prove under a natural…
For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the…
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the…
We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that…
In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in C^{\infty}…
This paper focuses on the optimal control of weak (i.e. in general non smooth) solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows.…
This paper aims to investigate the evolution problem for planar curves with singularities. Motivated by the inverse curvature flow introduced by Li and Wang (Calc. Var. Partial Differ. Equ. 62 (2023), No. 135), we intend to consider the…
The transitional regime of plane channel flow is investigated {above} the transitional point below which turbulence is not sustained, using direct numerical simulation in large domains. Statistics of laminar-turbulent spatio-temporal…
We establish local regularity theory for parabolic systems of Uhlenbeck type with $\varphi$-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local H\"older continuity of the gradients,…
We study the controllability of linearized shape-dependent operators for flow problems. The first operator is a mapping from the shape of the computational domain to the tangential wall velocity of the potential flow problem and the second…
In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique…
We consider the evolution of a quantity advected by a compressible flow and subject to diffusion. When this quantity is scalar it can be, for instance, the temperature of the flow or the concentration of some pollutants. Because of the…
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain…