arXiv subjects

Kenneth H. Karlsen

Publications and source records attributed to Kenneth H. Karlsen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Weak convergence of stochastic integrals

The convergence of stochastic integrals driven by a sequence of Wiener processes $W_n\to W$ (with convergence in $C_t$) is crucial in the analysis of stochastic partial differential equations (SPDEs). The convergence we focus on in this paper is of the form $\int_0^T V_n\, {\rm d} W_n \to \int_0^T V\,{\rm d} W$, where $V_n$ takes values in $L^p([0,T];X)$ for some finite $p\ge 2$ and a Banach space $X$. Standard methods do not directly apply when $V_n$ only converges weakly in the temporal variable to $V$. We provide (weak) convergence results that address the need to take limits of stochastic integrals when only weak temporal convergence is available. This is particularly relevant for SPDEs with singular behaviour.

math.PR↗

arXiv2023

Quantitative compactness estimates for stochastic conservation laws

We present a quantitative compensated compactness estimate for stochastic conservation laws, which generalises a previous result of Golse & Perthame (2013) for deterministic equations. With a stochastic modification of Kruzkov's interpolation lemma, this estimate provides bounds on the rate at which a sequence of vanishing viscosity solutions becomes compact. This contribution is for the Proceedings of HYP2022.

math.AP↗

arXiv2022

Global well-posedness of the viscous Camassa--Holm equation with gradient noise

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa--Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna--Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

math.AP↗

arXiv2022

Homogenization of Stochastic Conservation Laws with Multiplicative Noise

We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.

math.AP↗

arXiv2021

Renormalization of stochastic continuity equations on Riemannian manifolds

We consider the initial-value problem for stochastic continuity equations of the form $$ \partial_t ρ+ \text{div}_h \left[ρ\left(u(t,x) + \sum_{i=1}^N a_i(x)\circ \frac{dW^i}{dt}\right)\right] = 0, $$ defined on a smooth closed Riemanian manifold $M$ with metric $h$, where the Sobolev regular velocity field $u$ is perturbed by Gaussian noise terms $\dot{W}_i(t)$ driven by smooth spatially dependent vector fields $a_i(x)$ on $M$. Our main result is that weak ($L^2$) solutions are renormalized solutions, that is, if $ρ$ is a weak solution, then the nonlinear composition $S(ρ)$ is a weak solution as well, for any "reasonable" function $S:\mathbb{R}\to\mathbb{R}$. The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna-Lions type commutators $\mathcal{C}_\varepsilon (ρ,D)$ between (first/second order) geometric differential operators $D$ and the regularization device ($\varepsilon$ is the scaling parameter). This work, which is related to the "Euclidean" result in Punshon-Smith (2017), reveals some structural effects that noise and nonlinear domains have on the dynamics of weak solutions.

math.AP↗

arXiv2021

Strong solutions of a stochastic differential equation with irregular random drift

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$\mathrm{d} X= u(ω,t,X)\, \mathrm{d} t + \frac12 σ(ω,t,X)σ'(ω,t,X)\,\mathrm{d} t + σ(ω,t,X) \, \mathrm{d}W(t), $$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $σ$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(Ω; L^\infty([0,T];\dot{H}^1(\mathbb{R})))$ for any finite $p\ge 1$, $\mathbb{E}\left|u(t)-u(0)\right|_{\dot{H}^1(\mathbb{R})}^2 \to 0$ as $t\downarrow 0$, and $u$ satisfies the one-sided gradient bound $\partial_x u(ω,t,x) \le K(ω, t)$, where the process $K(ω,t )>0$ exhibits an exponential moment bound of the form $\mathbb{E} \exp\Big(p\int_t^T K(s)\,\mathrm{d} s\Big) \lesssim {t^{-2p}}$ for small times $t$, for some $p\ge1$. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter--Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation (SPDE).

math.PR↗

arXiv2020

The Hunter-Saxton equation with noise

In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter-Saxton equatio, and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.

math.AP↗

arXiv2019

Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Ito) noise. The Cauchy problem defined on a Riemannian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ($L^1$ contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch (2007), who worked with Kruzkov-DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle (2010).

math.AP↗

arXiv2018

The cardiac bidomain model and homogenization

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.

math.AP↗

arXiv2018

Stochastically forced cardiac bidomain model

The bidomain system of degenerate reaction-diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with "reaction" linked to the cellular action potential and "diffusion" representing current flow between cells. The purpose of this paper is to introduce a "stochastically forced" version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo-Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod-Jakubowski a.s.~representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.

math.AP↗

arXiv2016

Analysis of a splitting method for stochastic balance laws

We analyze a semi-discrete splitting method for conservation laws driven by a semilinear noise term. Making use of fractional $BV$ estimates, we show that the splitting method produces a compact sequence of approximate solutions converging to the exact solution, as the time step $Δt \rightarrow 0$. Under the assumption of a homogenous noise function, and thus the availability of $BV$ estimates, we provide an $L^1$ error estimate. Bringing into play a generalization of Kruzkov's entropy condition, permitting the "Kruzkov constants" to be Malliavin differentiable random variables, we establish an $L^1$ convergence rate of order $\frac13$ in $Δt$.

math.AP↗

arXiv2014

Conservation laws driven by Lévy white noise

We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itô-Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the $L^1$-contraction principle. Finally, the $L^1$ contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.

math.AP↗

arXiv2013

L1 Error Estimates for Difference Approximations Of Degenerate Convection-Diffusion Equations

We analyze monotone difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1 difference between the approximate solutions and the unique entropy solution is bounded above by a constant times the cube root of the spatial discretization parameter.

math.AP↗

arXiv2012

Operator splitting for well-posed active scalar equations

We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar unknown itself and hence the equations are so-called active scalar equations. Examples are provided by the surface quasi-geostrophic and aggregation equations. In addition, Burgers-type equations with fractional diffusion as well as the KdV and Kawahara equations are covered. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data is sufficiently regular.

math.NA↗

arXiv2011

On Nonlinear Stochastic Balance Laws

We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial $BV$ bounds for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in $L^1$ of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed.

math.AP↗

arXiv2011

Operator splitting for two-dimensional incompressible fluid equations

We analyze splitting algorithms for a class of two-dimensional fluid equations, which includes the incompressible Navier-Stokes equations and the surface quasi-geostrophic equation. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data are sufficiently regular.

math.NA↗

arXiv2011

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (Lévy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments.

math.NA↗

arXiv2010

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

math.NA↗