arXiv subjects

Robin Østern Lien

Publications and source records attributed to Robin Østern Lien.

3 recordsLinked to original sources

arXiv2026

Higher-order estimates of highest waves of the Whitham equation

The Whitham equation has given its name to a wider family of nonlocal, nonlinear equations with very weak dispersion, which all feature highest waves. Recent work by Ehrnström, Maehlen and Varholm establishes leading-order asymptotics at the crest of solutions to such Whitham-type equations, and conjectures similar expansions for all derivatives of the solution. By extending their techniques we confirm the conjecture for a range of equations with highest waves of Hölder regularity $C^{s}$ for $s\in[0.35,1)$. We do this by strong induction, redistributing difference operators to deal with singularities in higher-order derivatives arising from integral convolution kernels. The restriction $s\geq0.35$ arises solely from the separate argument establishing the zeroth-order asymptotics, which requires a uniform sign estimate that we verify using rigorous interval arithmetic. The higher-order induction itself applies for every $s\in(0,1)$, so any extension of the zeroth-order result immediately yields the corresponding higher-order expansions.

math.AP↗

arXiv2026

A Schauder regularity theory for nonlocal and mixed local-nonlocal viscous Hamilton$\unicode{x2013}$Jacobi equations

We prove space-time Schauder estimates $\unicode{x2013}$ optimal regularity estimates in Hölder spaces $\unicode{x2013}$ and well-posedness results for mild and classical solutions of viscous Hamilton$\unicode{x2013}$Jacobi equations with subcritical nonlocal and mixed local-nonlocal diffusions in $\mathbb{R}^d$. Our spatial Schauder estimates hold under mild assumptions on the nonlocal/mixed operators and Hamiltonians. The Laplacian, fractional Laplacians, nonsymmetric, spectrally one-sided, and strongly anisotropic integral operators, as well as sums of such operators are covered. We observe an interplay between the regularity of the initial data and the growth of the Hamiltonian in the gradient, and develop a spatial Schauder theory for two canonical cases: (i) Lipschitz initial data and general Hamiltonians that are Hölder in space and merely locally Lipschitz in the gradient, and (ii) Hölder initial data and Hamiltonians that are Hölder in space and locally Lipschitz with power growth in the gradient. We compute explicit blow-up rates for $C^1$ and higher order Hölder norms as $t\to 0$. The results include short and long time existence of mild solutions, optimal regularity in Hölder spaces and corresponding Schauder a priori estimates, and that spatially smooth mild solutions are regular in time and pointwise classical solutions. Under further assumptions on the diffusion operator, we then prove time and space-time Schauder regularity estimates in optimal Hölder spaces which respect the natural fractional parabolic scaling. These results generalize classical linear local and fractional Schauder estimates to our non-linear fractional, possibly anisotropic and nonsymmetric setting.

math.AP↗

arXiv2024

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $σ\in(0,2)$ since they involve fractional Laplace operators $(-Δ)^{σ/2}$. They arise e.g. in control and game theory as dynamic programming equations -- HJB and Isaacs equation -- and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $σ$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $σ$ and are worse when $σ$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

math.NA↗