Abstract
We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.
Highlights
In this paper we consider the Cauchy problem for conservative solutions of the (integrated) Hunter–Saxton (HS) equation [12]
In this paper we consider the Cauchy problem for conservative solutions of the Hunter–Saxton (HS) equation [12] (1.1)1 ut + uux = 4 x −∞ u2x(y)dy − ∞u2x(y)dy, x u|t=0 = u0.The equation has been extensively studied, starting with [13, 14]
The initial value problem is not well-posed without further constraints: Consider the trivial case u0 = 0 which clearly has as one solution u(t, x) = 0
Summary
In this paper we consider the Cauchy problem for conservative solutions of the (integrated) Hunter–Saxton (HS) equation [12] It turns out that the solution u of the HS equation may develop singularities in finite time in the following sense: Unless the initial data is monotone increasing, we find inf(ux) → −∞ as t ↑ t∗ = 2/ sup(−u0). Bressan and Constantin [1], using a clever rewrite of the equation in terms of new variables, showed global existence of conservative solutions without the assumption of compactly supported initial data.
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