Fast Communication Burgers’ Equation with Vanishing Hyper-viscosity∗
نویسندگان
چکیده
We prove that bounded solutions of the vanishing hyper-viscosity equation, ut + f(u)x + (−1)sε∂2s x u = 0 converge to the entropy solution of the corresponding convex conservation law ut +f(u)x = 0, f ′′ > 0. The hyper-viscosity case, s > 1, lacks the monotonicity which underlines the Krushkov BV theory in the viscous case s = 1. Instead we show how to adapt the Tartar-Murat compensated compactness theory together with a weaker entropy dissipation bound to conclude the convergence of the vanishing hyper-viscosity.
منابع مشابه
Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods
We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf-Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the co...
متن کاملIs multiscaling an artifact in the stochastically forced Burgers equation?
We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum approximately 1/k, where k is the wave number. From very high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be falsified by standard tests. We find a new a...
متن کاملOptimal Control and Vanishing Viscosity for the Burgers Equation
We revisit an optimization strategy recently introduced by the authors to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the one-dimensional (1-D) Burgers equation. This new descent strategy, called the alternating descent method, in the inviscid case, distinguishes and alternates descent di...
متن کاملEntropic Burgers equation via a minimizing movement scheme based on the Wasserstein metric
As noted by the second author in the context of unstable two-phase porous medium flow, entropy solutions of Burgers’ equation can be recovered from a minimizing movement scheme involving the Wasserstein metric in the limit of vanishing time step size [4]. In this paper, we give a simpler proof by verifying that the anti-derivative is a viscosity solution of the associated Hamilton Jacobi equati...
متن کاملOn a transport equation with nonlocal drift
In [16], Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocaldrift variant of the 1D Burgers equation does not have global classical solutions ∂tθ + u ∂xθ = 0, u = Hθ, where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004