James C. Robinson

activity

20062021

most citedLocal existence for the non-resistive MHD equations in nearly optimal Sobolev spaces

128 citations · 387 across the 40 of their papers we have counts for

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30 papers · 1 filter

math.AP2021

On 2D Harmonic Extensions of Vector Fields and Stellarator Coils

Adam J. Golab, James C. Robinson, José L. Rodrigo

We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investi…

math.AP2020★ 8 cited

Using periodic boundary conditions to approximate the Navier-Stokes equations on $R^{3}$ and the transfer of regularity

James C. Robinson

This paper considers solutions $u_{α}$ of the three-dimensional Navier--Stokes equations on the periodic domains $Q_{α} := (- α, α)^{3}$ as the domain size $α \to \infty$ , and compares them to…

math.AP2019

Robustness of Regularity for the $3$ D Convective Brinkman-Forchheimer Equations

Karol W. Hajduk, James C. Robinson, Witold Sadowski

We prove a robustness of regularity result for the $3$ D convective Brinkman-Forchheimer equations $$ \partial_tu -μΔu + (u \cdot \nabla)u + \nabla p + αu + β\abs{u}^{r - 1}u = f, $…

math.AP2018

Optimal existence classes and nonlinear--like dynamics in the linear heat equation in $R^{d}$

James C. Robinson, Aníbal Rodíguez-Bernal

We analyse the behaviour of solutions of the linear heat equation in $R^{d}$ for initial data in the classes $M_{ε} (R^{d})$ of Radon measures with $\int_{{\…

math.AP2018

Energy conservation for the Euler equations on $T^{2} \times R_{+}$ for weak solutions defined without reference to the pressure

James C. Robinson, José L. Rodrigo, Jack W. D. Skipper

We study weak solutions of the incompressible Euler equations on $T^{2} \times R_{+}$ ; we use test functions that are divergence free and have zero normal component, t…

math.AP2017★ 17 cited

Partial regularity for a surface growth model

W. S. Ożański, J. C. Robinson

We prove two partial regularity results for the scalar equation $u_{t} + u_{xxxx} + \partial_{xx} u_{x}^{2} = 0$ , a model of surface growth arising from the physical process of molecular epitax…

math.AP2017★ 9 cited

A Necessary and Suffcient Condition for Uniqueness of the Trivial Solution in Semilinear Parabolic Equations

R. Laister, J. C. Robinson, M. Sierzega

In their 1968 paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negat…

math.AP2016★ 71 cited

Energy equality for the 3D critical convective Brinkman-Forchheimer equations

Karol W. Hajduk, James C. Robinson

In this paper we give a simple proof of the existence of global-in-time smooth solutions for the convective Brinkman-Forchheimer equations (also called in the literature the tamed…

math.AP2016★ 4 cited

Energy conservation in the 3D Euler equations on $T^{2} \times R_{+}$

James C. Robinson, José L. Rodrigo, Jack W. D. Skipper

The aim of this paper is to prove energy conservation for the incompressible Euler equations in a domain with boundary. We work in the domain $T^{2} \times R_{+}$ , wher…

math.AP2016★ 128 cited

Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces

Charles L. Fefferman, David S. McCormick, James C. Robinson +1

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $R^{d}$ , $d = 2, 3$ , with i…

math.AP2016★ 1 cited

Well-posedness for the diffusive 3D Burgers equations with initial data in $H^{1/2}$

Benjamin C. Pooley, James C. Robinson

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}…

math.AP2015★ 1 cited

Lower bounds on blowing-up solutions of the 3D Navier--Stokes equations in $H ˙^{3/2}$ , $H ˙^{5/2}$ , and $B ˙_{2, 1}^{5/2}$

David S. McCormick, Eric J. Olson, James C. Robinson +3

If $u$ is a smooth solution of the Navier--Stokes equations on $R^{3}$ with first blowup time $T$ , we prove lower bounds for $u$ in the Sobolev spaces $H ˙^{3/2}$ , $\dot…

math.AP2015

Local existence for the non-resistive MHD equations in Besov spaces

Jean-Yves Chemin, David S. McCormick, James C. Robinson +1

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of $R^{n}$ , $n = 2, 3$ , for divergence-free…

math.AP2014★ 3 cited

On the continuity of global attractors

Luan Hoang, Eric J. Olson, James C. Robinson

Let $Λ$ be a complete metric space, and let ${S_{λ} (\cdot) : λ \in Λ}$ be a parametrised family of semigroups with global attractors $A_{λ}$ . We assume that there exists a f…

math.AP2014★ 3 cited

Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain

Michel Chipot, Jerome Droniou, Gabriela Planas +2

In this paper we treat three problems on a two-dimensional `punctured periodic domain': we take $Ω_{r} = (- L, L)^{2} ∖ D_{r}$ , where $D_{r} = B (0, r)$ is the disc of radius $r$ centred at…

math.AP2014★ 30 cited

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

Robert Laister, James C. Robinson, Mikolaj Sierzega +1

We consider the scalar semilinear heat equation $u_{t} - Δ u = f (u)$ , where $f : [0, \infty) \to [0, \infty)$ is continuous and non-decreasing but need not be convex. We completely charact…

math.AP2014★ 2 cited

An Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces

Benjamin C. Pooley, James C. Robinson

In 2000 Constantin showed that the incompressible Euler equations can be written in an "Eulerian-Lagrangian" form which involves the back-to-labels map (the inverse of the trajecto…

math.AP2014★ 1 cited

Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

C. L. Fefferman, D. S. McCormick, J. C. Robinson +1

This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n /2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in…

math.AP2013

Rigorous Numerical Verification of Uniqueness and Smoothness in a Surface Growth Model

Dirk Blömker, Christian Nolde, James C. Robinson

Based on numerical data and a-posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities…

math.AP2013★ 1 cited

Gaussian lower bounds on the Dirichlet heat kernel and non-existence of local solutions for semilinear heat equations of Osgood type

Robert Laister, James C. Robinson, Mikolaj Sierzega

We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equat…

math.AP2013★ 20 cited

Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation

David S. McCormick, James C. Robinson, Jose L. Rodrigo

We prove existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in two dimensions; we consider the standard equations of magnetohydrodynamics…

math.AP2013

Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO

David S. McCormick, James C. Robinson, Jose L. Rodrigo

Using elementary arguments based on the Fourier transform we prove that for $1 \leq q < p < \infty$ and $s \geq 0$ with $s > n (1/2 - 1/ p)$ , if $f \in L^{q,\infty}(\R^n) \cap \dot{H}^…

math.AP2013★ 18 cited

Non-existence of local solutions for semilinear heat equations of Osgood type

Robert Laister, James C. Robinson, Mikolaj Sierzega

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, nonnegative source term $f$ . Globa…

math.AP2012

Minimal periods of semilinear evolution equations with Lipschitz nonlinearity revisited

James C. Robinson, Alejandro Vidal-López

We obtain a lower bound for the period of periodic solutions of semilinear evolution equations for the full range of nonlinear terms for which standard local existence theory appli…