Robert M. Strain
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View article: The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit
The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit Open
In the paper we study the Boltzmann equation in the diffusive limit in a channel domain $\mathbb{T}^2\times (-1,1)$ for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by…
View article: On Nonlinear Stability of Muskat Bubbles
On Nonlinear Stability of Muskat Bubbles Open
In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain a new approach to improve our result in [25]. Due to a new bubble-adapted formulation, the improvement is two fold. We significantly condense the proof and we no…
View article: The Peskin problem with viscosity contrast
The Peskin problem with viscosity contrast Open
The Peskin problem models the dynamics of a closed elastic filament immersed in an incompressible fluid. In this paper, we consider the case when the inner and outer viscosities are possibly different. This viscosity contrast adds further …
View article: Well-Posedness of the 3D Peskin Problem
Well-Posedness of the 3D Peskin Problem Open
This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral …
View article: Critical local well-posedness for the fully nonlinear Peskin problem
Critical local well-posedness for the fully nonlinear Peskin problem Open
We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosi…
View article: Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus
Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus Open
Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathem…
View article: Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off
Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off Open
This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness, and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case …
View article: Frequency multiplier estimates for the linearized relativistic Boltzmann operator without angular cutoff
Frequency multiplier estimates for the linearized relativistic Boltzmann operator without angular cutoff Open
This paper is concerned with the relativistic Boltzmann equation without angular cutoff. The non-cutoff theory for the relativistic Boltzmann equation has been rarely studied even under a smallness assumption on the initial data due to the…
View article: The Peskin Problem with Viscosity Contrast
The Peskin Problem with Viscosity Contrast Open
The Peskin problem models the dynamics of a closed elastic filament immersed in an incompressible fluid. In this paper, we consider the case when the inner and outer viscosities are possibly different. This viscosity contrast adds further …
View article: Global Mild Solutions of the Landau and <scp>Non‐Cutoff</scp> Boltzmann Equations
Global Mild Solutions of the Landau and <span>Non‐Cutoff</span> Boltzmann Equations Open
This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, …
View article: Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation
Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation Open
In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To acco…
View article: Uniqueness of bounded solutions for the homogeneous relativistic Landau equation with Coulomb interactions
Uniqueness of bounded solutions for the homogeneous relativistic Landau equation with Coulomb interactions Open
We prove the uniqueness of weak solutions to the spatially homogeneous\nspecial relativistic Landau equation under the conditional assumption that the\nsolution satisfies $(p^0)^7 F(t,p) \\in L^1 ([0,T]; L^\\infty)$. The existence of\nstan…
View article: Global Regularity for Gravity Unstable Muskat Bubbles
Global Regularity for Gravity Unstable Muskat Bubbles Open
In this paper, we study the dynamics of fluids in porous media governed by Darcy's law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one…
View article: Decay Estimates for the Muskat Equation
Decay Estimates for the Muskat Equation Open
We prove time decay of solutions to the Muskat equation in 2D and in 3D. In \cite{JEMS} and \cite{CCGRPS}, the authors introduce the norms $\|f\|_{s}(t)= \int_{\mathbb{R}^{2}} |\xi|^{s}|\hat{f}(\xi)| d\xi$ in order to prove global existen…