On a Lyapunov Functional Relating Shortening Curves and Viscous Conservation Laws
نویسندگان
چکیده
We study a non linear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the “change of shape” of a BV solution to a scalar conservation law.
منابع مشابه
On the Convergence Rate of Vanishing Viscosity Approximations
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, · )− uε(t, · )‖L1 = O(1)(1 + t) · √ ε |ln ε| on the distance between an exact BV solution u and a viscous approximation uε , letting the viscosity coefficient ε → 0. In the proof, starting from u we construct an approximation of the viscous solution uε by taking a mollification u ∗ φ√...
متن کاملSparse + low-energy decomposition for viscous conservation laws
For viscous conservation laws, solutions contain smooth but high-contrast features, which require the use of fine grids to properly resolve. On coarse grids, these high-contrast jumps resemble shocks rather than their true viscous profiles, which could lead to issues in the numerical approximation of their underlying dynamics. In many cases, the equations of motion emit traveling wave solutions...
متن کاملOn the Convergence Rate of Vanishing Viscosity Approximations for Nonlinear Hyperbolic Systems
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ∥u(t, ·) − u(t, ·) ∥∥ L = O(1)(1 + t) · √ε| ln ε| on the distance between an exact BV solution u and a viscous approximation u, letting the viscosity coefficient ε → 0. In the proof, starting from u we construct an approximation of the viscous solution u by taking a mollification u ∗ φ√ ε a...
متن کاملViscous Conservation Laws, Part I: Scalar Laws
Viscous conservation laws are the basic models for the dissipative phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: th...
متن کاملDissipative boundary conditions for 2 2 hyperbolic systems of conservation laws for entropy solutions in BV
In this article, we investigate the BV stability of 2×2 hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions. More precisely, we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in BV. Our proof is based on a front tracking algorithm used to construct approximate...
متن کامل