arXiv subjects

Walter Strauss

Publications and source records attributed to Walter Strauss.

10 recordsLinked to original sources

arXiv2021

Traveling water waves -- the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.

math.AP↗

arXiv2019

Rapidly Rotating White Dwarfs

A rotating star may be modeled as a continuous system of particles attracted to each other by gravity and with a given total mass and prescribed angular velocity. Mathematically this leads to the Euler-Poisson system. A white dwarf star is modeled by a very particular, and rather delicate, equation of state for the pressure as a function of the density. We prove an existence theorem for rapidly rotating white dwarfs that depend continuously on the speed of rotation. The key tool is global continuation theory, combined with a delicate limiting process. The solutions form a connected set $\mathcal K$ in an appropriate function space. As the speed of rotation increases, we prove that either the supports of white dwarfs in $\mathcal K$ become unbounded or their densities become unbounded. We also discuss the polytropic case with the critical exponent $γ=4/3$.

math.AP↗

arXiv2018

Large-amplitude steady downstream water waves

We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.

math.AP↗

arXiv2018

Rapidly Rotating Stars

A rotating star may be modeled as a continuous system of particles attracted to each other by gravity and with a given total mass and prescribed angular velocity. Mathematically this leads to the Euler-Poisson system. We prove an existence theorem for such stars that are rapidly rotating, depending continuously on the speed of rotation. This solves a problem that has been open since Lichtenstein's work in 1933. The key tool is global continuation theory, combined with a delicate limiting process. The solutions form a connected set $\mathcal K$ in an appropriate function space. As the speed of rotation increases, we prove that {\it either the supports of the stars in $\mathcal K$ become unbounded or the density somewhere within the stars becomes unbounded}. We permit any equation of state of the form $p=ρ^γ,\ 6/5<γ<2$, so long as $γ\ne4/3$. We consider two formulations, one where the angular velocity is prescribed and the other where the angular momentum per unit mass is prescribed.

math.AP↗

arXiv2017

Steady States of Rotating Stars and Galaxies

A rotating continuum of particles attracted to each other by gravity may be modeled by the Euler-Poisson system. The existence of solutions is a very classical problem. Here it is proven that a curve of solutions exists, parametrized by the rotation speed, with a fixed mass independent of the speed. The rotation is allowed to vary with the distance to the axis. A special case is when the equation of state is $p=ρ^γ,\ 6/5<γ<2,\ γ\ne4/3$, in contrast to previous variational methods which have required $4/3 < γ$. The continuum of particles may alternatively be modeled microscopically by the Vlasov-Poisson system. The kinetic density is a prescribed function. We prove an analogous theorem asserting the existence of a curve of solutions with constant mass. In this model the whole range $(6/5,2)$ is allowed, including $γ=4/3$.

math.AP↗

arXiv2015

Convergence to Equilibrium of a Body Moving in a Kinetic Sea

We consider a continuum of particles that are acted upon by an external force $\mathbf{G}(t,\mathbf{x})$ and that collide with a rigid body. The body itself is subject to a constant force $E$ as well as to the collective force of interaction with the particles. We assume that the particles that collide with the body reflect probabilistically with some probablility distribution $K(v,u)$. Under certain conditions on $\mathbf{G}(t,\mathbf{x})$ and $K(v,u)$, we identify an equilibrium velocity $V_{\infty }$ of the body and we prove that this equilibrium is asymptotically stable.

math.AP↗

arXiv2014

Velocity Reversal Criterion of a Body Immersed in a Sea of Particles

We consider a rigid body colliding with a continuum of particles. We assume that the body is moving at a velocity close to an equilibrium velocity V_{\infty} and that the particles colliding with the body reflect diffusely, that is, probabilistically with some probablility distribution K. We find a condition that is sufficient and almost necessary that the collective force of the colliding particles reverses the relative velocity V(t) of the body, that is, changes the sign of V(t)-V_{\infty}, before the body approaches equilibrium. Examples of both reversal and irreversal are given. This is in strong contrast with the pure specular reflection case in which only reversal happens.

math.AP↗

arXiv2014

Global bifurcation of steady gravity water waves with critical layers

We construct large families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. A Riemann-Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudo-differential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler-Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.

math.AP↗

arXiv2013

Approach to Equilibrium of a Body Colliding Specularly and Diffusely with a Sea of Particles

We consider a rigid body acted upon by two forces, a constant force and the collective force of interaction with a continuum of particles. We assume that some of the particles that collide with the body reflect elastically (specularly), while others reflect probabilistically with some probablility distribution K. We find that the rate of approach of the body to equilibrium is O(t^{-3-p}) in three dimensions where p can take any value from 0 to 2, depending on K.

math.AP↗

arXiv2007

A sharp stability criterion for the Vlasov-Maxwell system

We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov-Maxwell system that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically with the particle energy, we obtained a linear stability criterion in our previous paper. Here we prove that this criterion is sharp; that is, there would otherwise be an exponentially growing solution to the linearized system. Therefore for the class of symmetric Vlasov-Maxwell equilibria, we establish an energy principle for linear stability. We also treat the considerably simpler periodic 1.5D case. The new formulation introduced here is applicable as well to the nonrelativistic case, to other symmetries, and to general equilibria.

physics.plasm-ph↗