139 citations · 534 across the 31 of their papers we have counts for
31 papers
Uniqueness of dissipative solutions for the Camassa-Holm equation
Katrin Grunert
We show that the Cauchy problem for the Camassa-Holm equation has a unique, global, weak, and dissipative solution for any initial data , such that $u_{0,x}…
A Convergent Numerical Algorithm for -Dissipative Solutions of the Hunter-Saxton Equation
Thomas Christiansen, Katrin Grunert, Anders Nordli +1
A convergent numerical method for -dissipative solutions of the Hunter-Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initi…
A Lipschitz metric for -dissipative solutions to the Hunter-Saxton equation
Katrin Grunert, Matthew Tandy
We explore the Lipschitz stability of solutions to the Hunter-Saxton equation with respect to the initial data. In particular, we study the stability of -dissipative solutions…
Shock interactions for the Burgers-Hilbert Equation
Alberto Bressan, Sondre T. Galtung, Katrin Grunert +1
This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the…
Uniqueness of conservative solutions for the Hunter-Saxton equation
Katrin Grunert, Helge Holden
We show that the Hunter-Saxton equation and has a unique, global, weak, and conservativ…
Stumpons are non-conservative traveling waves of the Camassa-Holm equation
Sondre Tesdal Galtung, Katrin Grunert
It is well-known that by requiring solutions of the Camassa--Holm equation to satisfy a particular local conservation law for the energy in the weak sense, one obtains what is know…
Lipschitz Stability for the Hunter-Saxton Equation
Katrin Grunert, Matthew Tandy
We study the Lipschitz stability in time for -dissipative solutions to the Hunter-Saxton equation, where is a constant. We define metrics in both Lagrangian and Eul…
Traveling Waves for the Nonlinear Variational Wave Equation
Katrin Grunert, Audun Reigstad
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classi…
A Regularized System for the Nonlinear Variational Wave Equation
Katrin Grunert, Audun Reigstad
We derive a new generalization of the nonlinear variational wave equation. We prove existence of local, smooth solutions for this system. As a limiting case, we recover the nonline…
A numerical study of variational discretizations of the Camassa-Holm equation
Sondre Tesdal Galtung, Katrin Grunert
We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an…
Numerical conservative solutions of the Hunter--Saxton equation
Katrin Grunert, Anders Nordli, Susanne Solem
In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by…
A Lipschitz metric for the Camassa--Holm equation
J. A. Carrillo, K. Grunert, H. Holden
We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa--Holm (CH) equation. Generically, the solutions of the CH equation develop singular…
Symmetries and multipeakon solutions for the modified two-component Camassa-Holm system
Katrin Grunert, Xavier Raynaud
Compared with the two-component Camassa-Holm system, the modified two-component Camassa-Holm system introduces a regularized density which makes possible the existence of solutions…
On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa-Holm system
Markus Grasmair, Katrin Grunert, Helge Holden
The Camassa-Holm equation and its two-component Camassa-Holm system generalization both experience wave breaking in finite time. To analyze this, and to obtain solutions past wave…
A Lipschitz metric for the Hunter-Saxton equation
José Antonio Carrillo, Katrin Grunert, Helge Holden
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 dev…
Existence and Lipschitz stability for -dissipative solutions of the two-component Hunter-Saxton system
Katrin Grunert, Anders Nordli
We establish the concept of -dissipative solutions for the two-component Hunter-Saxton system under the assumption that either or for all $x\in \mathbb{R…
A Lagrangian view on complete integrability of the two-component Camassa-Holm system
Jonathan Eckhardt, Katrin Grunert
We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa-Holm system can be understood in terms of certain reparametrizations of the underlying…
On the Burgers-Poisson Equation
Katrin Grunert, Khai T. Nguyen
In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers-Poisson equation for initial data in L^1(R). Additional an Oleinik type estimate is es…
Solutions of the Camassa-Holm equation with accumulating breaking times
Katrin Grunert
We present two initial profiles to the Camassa-Holm equation which yield solutions with accumulating breaking times.
The general peakon-antipeakon solution for the Camassa-Holm equation
Katrin Grunert, Helge Holden
We compute explicitly the peakon-antipeakon solution of the Camassa-Holm equation in the non-symmetric and -dissipative case. The solut…
A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system
Katrin Grunert, Helge Holden, Xavier Raynaud
We introduce a novel solution concept, denoted -dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy prob…
Blow-up for the two-component Camassa-Holm system
Katrin Grunert
Following conservative solutions of the two-component Camassa-Holm system , along characteristics, we determine if wave…
Periodic conservative solutions for the two-component Camassa-Holm system
Katrin Grunert, Helge Holden, Xavier Raynaud
We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa-Holm system, an…
Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics
Katrin Grunert, Helge Holden, Xavier Raynaud
We show existence of a global weak dissipative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line with nonvanishing and distinct spatial asy…