arXiv subjects

Magnus C. Ørke

Publications and source records attributed to Magnus C. Ørke.

5 recordsLinked to original sources

arXiv2026

Stochastic perturbation and zero noise limit for scalar conservation laws

Scalar conservation laws sit at the intersection between being simple enough to study analytically, while being complex enough to exhibit a wide range of nonlinear phenomena. We introduce a novel stochastic perturbation of scalar conservation laws, inspired by mean field games. We prove well-posedness of the stochastically perturbed equation; prove that it converges as the noise parameter is sent to $0$; and that the limit is the unique entropy solution of the conservation law. Thus, the noise acts as a selection criterion for (deterministic) conservation laws. This is the first such result for nonlinear hyperbolic conservation laws.

math.AP↗

arXiv2025

Stochastic flows for Hölder drifts and transport/continuity equations with noise

We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, α}_x$ for any $q \in [2, \infty)$ and $α\in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $BV_\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, α}_x$. For both equations, solutions may fail to be unique in the deterministic setting.

math.PR↗

arXiv2025

Convergence rate for a semidiscrete approximation of scalar conservation laws

We propose a semidiscrete scheme for approximation of entropy solutions of one-dimensional scalar conservation laws with nonnegative initial data. The scheme is based on the concept of particle paths for conservation laws and can be interpreted as a finite-particle discretization. A convergence rate of order $1/2$ with respect to initial particle spacing is proved. As a special case, this covers the convergence of the Follow--the--Leader model to the Lighthill--Whitham--Richards model for traffic flow.

math.AP↗

arXiv2024

The particle paths of hyperbolic conservation laws

Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is $1/2$-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.

math.AP↗

arXiv2023

Highest waves for fractional Korteweg--De Vries and Degasperis--Procesi equations

We study traveling waves for a class of fractional Korteweg--De Vries and fractional Degasperis--Procesi equations with a parametrized Fourier multiplier operator of order $-s \in (-1, 0)$. For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal $s$-Hölder regularity, attained in the cusp.

math.AP↗