arXiv subjects

Walter A. Strauss

Publications and source records attributed to Walter A. Strauss.

15 recordsLinked to original sources

arXiv2026

Transverse Instability of Stokes Waves at Finite Depth

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves are unstable with respect to transverse perturbations. In \cite{CreNguStr} for the case of infinite depth we proved rigorously that the spectrum of the water wave system linearized at small Stokes waves, with respect to transverse perturbations, contains unstable eigenvalues lying approximately on an ellipse. In this paper we consider the case of finite depth and prove that the same spectral instability result holds for all but finitely many values of the depth. The computations and some aspects of the theory are considerably more complicated in the finite depth case.

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arXiv2025

High-Voltage Ionized Gas with Spherical Cathode Emission

We consider a plasma that is created by a high voltage difference, which is known as a Townsend gas discharge. The plasma is confined to the region between two concentric spheres, one of which is a cathode and the other an anode. Ion-electron pairs are created by collisions inside the plasma. Additional electrons enter the plasma by collisions of ions with the cathode. We prove under certain conditions that there are many steady states exhibiting gas discharge, beginning with a `sparking' voltage. In fact, there is an analytic one-parameter family of them that connects the non-ionized gas to a plasma with arbitrarily high ionization or arbitrarily high potential, or else the family ends at an `anti-sparking' voltage.

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arXiv2025

Ionized Gas in an Annular Region

We consider a plasma that is created by a high voltage difference $λ$, which is known as a Townsend discharge. We consider it to be confined to the region $Ω$ between two concentric spheres, two concentric cylinders, or more generally between two star-shaped surfaces. We first prove that if the plasma is initially relatively dilute, then either it may remain dilute for all time or it may not, depending on a certain parameter $κ(λ, Ω)$. Secondly, we prove that there is a connected one-parameter family of steady states. This family connects the non-ionized gas to a plasma, either with a sparking voltage $λ^*$ or with very high ionization, at least in the cylindrical or spherical cases.

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arXiv2024

Amplitude bounds of steady rotational water waves

We consider classical steady water waves with a free surface, a flat bottom and constant vorticity $γ$. In the adverse case $γ>0$ we prove that there is an absolute upper bound on the amplitude, independent of the physical constants, provided that $γ$ is sufficiently small. In any favorable case $γ\leq0$ we present a new proof of such an absolute bound on the amplitude and prove that the amplitude tends to zero as $γ$ tends to negative infinity.

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arXiv2023

Existence of Rotating Magnetic Stars

We consider a star as a compressible fluid subject to gravitational and magnetic forces. This leads to an Euler-Poisson system coupled to a magnetic field, which may be regarded as an MHD model together with gravity. The star executes steadily rotating motion about a fixed axis. We prove, for the first time, the existence of such stars provided that the rotation speed and the magnetic field are sufficiently small.

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arXiv2022

Existence of Rotating Stars with Variable Entropy

We model a rotating star as a compressible fluid subject to gravitational forces. In almost all the mathematical literature the entropy is considered to be constant. Here we allow it to be variable. We consider a star that steadily rotates differentially around a fixed axis, say the $z$-axis. We prove the existence of a family of such stars with small angular velocity $ω$ and small entropy variation $s$ and with an equation of state $p=Ke^sρ^γ$. Our analysis reduces to a hyperbolic equation for the modified entropy coupled to an elliptic equation for the modified density, together with a mass constraint. Due to the variable entropy and the consequent loss of both regularity and variational structure, all the methods in the previous literature fail. We develop a new ad hoc perturbative strategy that allows us to construct rotating stars that bifurcate from the non-rotating ones.

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arXiv2022

Global Continuation of a Vlasov Model of Rotating Galaxies

A typical galaxy consists of a huge number of stars attracted to each other by gravity. For instance, the Milky Way has about $10^{11}$ stars. Thus it is typically modeled by the Vlasov-Poisson system. We prove an existence theorem for axisymmetric steady states of galaxies that may rotate rapidly. Such states are given in terms of a fairly general function $ϕ$ of the particle energy and angular momentum. The set $\mathcal K$ of such states form a connected set in an appropriate function space. Along the set $\mathcal K$, we prove under some conditions that either (a) the supports of the galaxies become unbounded or (b) both the rotation speeds and the densities somewhere within the galaxy become unbounded.

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arXiv2021

Proof of modulational instability of Stokes waves in deep water

It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases.

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arXiv2020

Steady States of Gas Ionization with Secondary Emission

We consider the steady states of a gas between two parallel plates that is ionized by a strong electric field so as to create a plasma. There can be a cascade of electrons due both to the electrons colliding with the gas molecules and to the ions colliding with the cathode (secondary emission). We use global bifurcation theory to prove that there is a one-parameter family $\mathscr{K}$ of such steady states with the following property. The curve $\mathscr{K}$ begins at the sparking voltage and either the particle density becomes unbounded or $\mathscr{K}$ ends at an anti-sparking voltage. These critical voltages are characterized explicitly.

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arXiv2015

Bound on the slope of steady water waves with favorable vorticity

We consider the angle $θ$ of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. Although $θ$ may surpass 30$^\circ$ for some irrotational waves close to the extreme wave, Amick [Ami87] proved that for any irrotational wave the angle must be less than 31.15$^\circ$. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps ($θ= 90^\circ$). Moreover, numerical calculations show that even waves of finite depth can overturn if the vorticity is adverse. In this paper, on the other hand, we prove an upper bound of 45$^\circ$ on $θ$ for a large class of waves with favorable vorticity and finite depth. In particular, the vorticity can be any constant with the favorable sign. We also prove a series of general inequalities on the pressure within the fluid, including the fact that any overturning wave must have a pressure sink.

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arXiv2014

Global bifurcation theory for periodic traveling interfacial gravity-capillary waves

We consider the global bifurcation problem for spatially periodic traveling waves for two-dimensional gravity-capillary vortex sheets. The two fluids have arbitrary constant, non-negative densities (not both zero), the gravity parameter can be positive, negative, or zero, and the surface tension parameter is positive. Thus, included in the parameter set are the cases of pure capillary water waves and gravity-capillary water waves. Our choice of coordinates allows for the possibility that the fluid interface is not a graph over the horizontal. We use a technical reformulation which converts the traveling wave equations into a system of the form "identity plus compact." Rabinowitz' global bifurcation theorem is applied and the final conclusion is the existence of either a closed loop of solutions, or an unbounded set of nontrivial traveling wave solutions which contains waves which may move arbitrarily fast, become arbitrarily long, form singularities in the vorticity or curvature, or whose interfaces self-intersect.

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arXiv2014

Global magnetic confinement for the 1.5D Vlasov-Maxwell system

We establish the global-in-time existence and uniqueness of classical solutions to the "one and one-half" dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studied the Cauchy problem without boundaries.

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arXiv2013

Linear Stability Analysis of a Hot Plasma in a Solid Torus

This paper is a first step toward understanding the effect of toroidal geometry on the rigorous stability theory of plasmas. We consider a collisionless plasma inside a torus, modeled by the relativistic Vlasov-Maxwell system. The surface of the torus is perfectly conducting and it reflects the particles specularly. We provide sharp criteria for the stability of equilibria under the assumption that the particle distributions and the electromagnetic fields depend only on the cross-sectional variables of the torus.

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arXiv2011

Stability analysis of collisionless plasmas with specularly reflecting boundary

In this paper we provide sharp criteria for linear stability or instability of equilibria of collisionless plasmas in the presence of boundaries. Specifically, we consider the relativistic Vlasov-Maxwell system with specular reflection at the boundary for the particles and with the perfectly conducting boundary condition for the electromagnetic field. Here we initiate our investigation in the simple geometry of radial and longitudinal symmetry.

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arXiv2006

Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as $\d_{tt}u -Δu= f(x,u)$ where $f$ is convex in $u$. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as $\d_t u -Δu= f(x,u)$. Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.

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