19 papers
Stochastic perturbation and zero noise limit for scalar conservation laws
Ulrik S. Fjordholm, Magnus C. Ørke
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 in…
Semi-discrete heat equations with variable coefficients and the parametrix method
Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang
We develop a parametrix approach for constructing solutions and establishing grid-size independent estimates for semi-discrete heat equations with variable coefficients. While the…
Transport equations for Osgood velocity fields
Ulrik Skre Fjordholm, Ola Isaac Høgåsen Mæhlen
We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual…
Convergent finite difference schemes for stochastic transport equations
Ulrik S. Fjordholm, Kenneth H. Karlsen, Peter H. C. Pang
We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish stability and convergence of the difference approximations…
The particle paths of hyperbolic conservation laws
Ulrik S. Fjordholm, Ola H. Mæhlen, Magnus C. Ørke
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 v…
The zero-noise limit of SDEs with drift
Ulrik Skre Fjordholm, Markus Musch, Andrey Pilipenko
We study the zero-noise limit for autonomous, one-dimensional ordinary differential equations with discontinuous right-hand sides. Although the deterministic equation might have in…