Research Papers of Robert Strain

Below are downloadable preprints of Robert Strain in reverse chronological order. They study the analysis of nonlinear partial differential equations from the Boltzmann theory of dilute gases and plasmas. But, recently, I have been working on the incompressible Navier-Stokes equations. You may additionally find my recent (e.g. post-Ph.D.) papers posted on the ArXiv.

Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II, with Chiun-Chuan Chen, Tai-Peng Tsai, and Horng-Tzer Yau	Submitted, 25 pages	-arXiv
Abstract. Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $\|v (x,t)\| \le C_{\|t\|^{-1/2}} $ or, for some $\e > 0$, $\|v (x,t)\| \le C_ r^{-1+\epsilon} \|t\|^{-\epsilon /2}$ for $-T_0\le t < 0$ and $0 < C_* < \infty$ allowed to be large. We prove that $v$ is regular at time zero.

Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations, with Chiun-Chuan Chen, Tai-Peng Tsai, and Horng-Tzer Yau	Int. Math. Res. Not., (2008) Vol. 2008 published on March 14, 2008	-arXiv
Abstract. Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $\|v (x,t)\| \le C_{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and $0 < C_ < \infty$ allowed to be large, we then prove that $v$ is regular at time zero.

Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, with Clément Mouhot	J. Math. Pures Appl. 87 (2007), no. 5, 515-535	-arXiv -HAL
Abstract. In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials or the so-called moderately soft potentials (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [46]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [34] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators.

On the Linearized Balescu-Lenard Equation	Comm. P.D.E. 32 (2007), no. 10, 1551-1586	-arXiv
Abstract. The Balescu-Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation.

The Vlasov-Maxwell-Boltzmann System in the Whole Space	Comm. Math. Phys. 268 (2006), no. 2, 543-567	-arXiv
Abstract. The Vlasov-Maxwell-Boltzmann system is a very fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We construct global in time classical solutions to the Cauchy problem near Maxwellians.

Exponential Decay for Soft Potentials Near Maxwellian, with Yan Guo	Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287-339.	-PDF -PS
Abstract. Consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches to zero at the rate of $e^{-\lambda\| t^p}$ for some $\lambda > 0$ and $0 < p < 1$. Our method is based on a unified energy estimate with appropriate exponential velocity weight. Our results extend the classical Caflisch result [2] to the case of very soft potential and Coulomb interactions, and also improve the recent “almost exponential” decay results by [4, 12].

Almost Exponential Decay Near Maxwellian, with Yan Guo	Comm. P.D.E. 31 (2006), no. 3, 417-429	-PDF -PS
Abstract. By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in [7–9, 11], we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results [6] for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.

Stability of the Relativistic Maxwellian in a Collisional Plasma, with Yan Guo	Comm. Math. Phys. 251 (2004), no. 2, 263-320	-PDF -PS
Abstract. The relativistic Landau-Maxwell system is one of the most fundamental models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field. We construct the first global in time classical solutions. Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the Juttner solution.

Last updated: January 27, 2007