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Helge Kristian Jenssen

Publications and source records attributed to Helge Kristian Jenssen.

24 records · Page 2Linked to original sources
arXiv2011

Extensions for Systems of Conservation Laws

Extensions (entropies) play a central role in the theory of hyperbolic conservation laws by providing intrinsic selection criteria for weak solutions. For a given hyperbolic system u_t+f(u)_x=0, a standard approach is to analyze directly the second order PDE system for the extensions. Instead we find it advantageous to consider the equations satisfied by the lengths beta^i of the right eigenvectors r_i the Jacobian matrix Df, as measured with respect to the inner product defined by an extension. Our geometric formulation provides a natural and systematic approach to existence of extensions. By prescribing the eigen-fields r_i our results automatically apply to all systems with the same eigen-frame. The equations for the lengths beta^i form a first order algebraic-differential system (the beta-system) to which standard integrability theorems can be applied. The size of the set of extensions follows by determining the number of free constants and functions present in the general solution to the beta-system. We provide a complete breakdown of the various possibilities for systems of three equations, as well as for rich hyperbolic systems of any size.

math.AP↗
arXiv2009

1D compressible flow with temperature dependent transport coefficients

We establish existence of global-in-time weak solutions to the one dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas (pressure $p=Kθ/τ$, internal energy $e=c_v θ$), when the viscosity $μ$ is constant and the heat conductivity $κ$ depends on the temperature $θ$ according to $κ(θ) = \bar κθ^β$, with $0\leqβ<{3/2}$. This choice of degenerate transport coefficients is motivated by the kinetic theory of gasses. Approximate solutions are generated by a semi-discrete finite element scheme. We first formulate sufficient conditions that guarantee convergence to a weak solution. The convergence proof relies on weak compactness and convexity, and it applies to the more general constitutive relations $μ(θ) = \bar μθ^α$, $κ(θ) = \bar κθ^β$, with $α\geq 0$, $0 \leq β< 2$ ($\bar μ, \bar κ$ constants). We then verify the sufficient conditions in the case $α=0$ and $0\leqβ<{3/2}$. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.

math.AP↗
arXiv2009

Systems of hyperbolic conservation laws with prescribed eigencurves

We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan-Kahler). We give a complete analysis of the possible scenarios, including examples, for systems of three equations. As an application we characterize conservative systems with the same eigencurves as the Euler system for 1-dimensional compressible gas dynamics. The case of general rich systems of any size (i.e. when the given eigenvector fields are pairwise in involution; this includes all systems of two equations) is completely resolved and we consider various examples in this class.

math.AP↗
arXiv2005

An Instability of the Godunov Scheme

We construct a solution to a $2\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes.

math.AP↗
arXiv2005

BV instability for the Lax-Friedrichs scheme

It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation.

math.AP↗