08:30
MS28: Analysis and numerical methods for conservation laws with stochastic terms (Part 1)
Chair: Haakon Hoel
08:30
25 mins
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Qualitative properties of stochastic Hamilton-Jacobi equations
Paul Gassiat
Abstract: Stochastic Hamilton-Jacobi equations with multiplicative noise appear naturally in several contexts, in particular in the level set formulation of motion of interfaces, when the dynamics are perturbed by a stochastic noise. Lions and Souganidis have shown in the late 90's that viscosity solutions methods could be applied successfully to obtain well-posedness for these equations. In this talk, we will present recent progress on the analysis of several qualitative properties of the solutions (regularity, finite speed of propagation of initial data, long-time behaviour).
This is joint work with B. Gess, P. Souganidis and P.L. Lions.
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08:55
25 mins
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Statistical solutions of hyperbolic conservation laws
Ulrik Skre Fjordholm
Abstract: Recent theoretical and numerical results have shown that inviscid models in gas dynamics, such as the (in)compressible Euler equations, are unstable with respect to initial data or even ill-posed. Going back to the roots of turbulence theory, we interpret instead these hyperbolic conservation laws in a probabilistic manner. In this talk I will survey some recent developments in so-called statistical solutions, both theoretical and numerical. These include well-posedness for scalar conservation laws; energy conservation for regular solutions of the incompressible Euler equations; and numerical evidence for the convergence of the mean flow, structure functions etc. for the compressible Euler equations.
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09:20
25 mins
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Computing statistical solutions of hyperbolic conservation laws
Kjetil Lye Lye, Siddhartha Mishra, Ulrik Fjordholm, Franziska Weber
Abstract: We review the theory of statistical solutions for conservation laws. Afterward, we introduce a convergent
numerical method for computing the statistical solution of conservation laws. In the case of systems of equations in multiple space dimensions, we prove a compactness result together with a version of the Lax-Wendroff theorem for statistical solutions. We test our theory against the compressible Euler equations in two and three spatial dimensions.
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09:45
25 mins
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Numerical methods for stochastic conservation laws with multiplicative rough drivers in the flux
Haakon Hoel, Henrik Risebro, Kenneth Karlsen, Erlend Briseid Storrøsten
Abstract: Abstract. Stochastic conservation laws (SCL) with quasilinear multiplicative “rough” path dependence in the flux arise in modeling of mean field games. An impressive collection of theoretical results has been developed for SCL in recent years by Gess, Lions, Perthame, and Souganidis. We present the first fully computable numerical methods for pathwise solutions of scalar SCL with, for instance, ”rough” paths in the form of Wiener processes. Convergence rates are derived for the numerical methods and we show that for strictly convex flux functions, ”rough” path oscillations lead to cancellations in the flow map solution; a property we take advantage of to develop more efficient numerical methods.
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