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Thomas C. Sideris

Publications and source records attributed to Thomas C. Sideris.

13 recordsLinked to original sources
arXiv2026

Lifespan Bounds and Super-Classical Propagation for Weak Solutions of the Three-Dimensional Compressible Euler Equations

We establish an upper bound on the lifespan of bounded weak solutions to the three-dimensional isentropic compressible Euler equations on $[0,T)\times\mathbb R^3$, under a localized positivity condition on the initial data and the assumption that the essential support of the disturbance propagates into an undisturbed background state no faster than predicted by classical local well-posedness. We further show that any entropy-admissible bounded weak solution with bounded inverse density that persists beyond this upper bound must propagate into the undisturbed region at a strictly super-classical speed and that this accelerated propagation is necessarily accompanied by a jump discontinuity in an associated one-sided $L^\infty$-profile of the solution.

math.AP↗
arXiv2023

Expansion and collapse of spherically symmetric isotropic elastic bodies surrounded by vacuum

A class of isotropic and scale invariant strain energy functions is given for which the corresponding spherically symmetric elastic motion includes bodies whose diameter becomes infinite with time or collapses to zero in finite time, depending on the sign of the residual pressure. The body is surrounded by vacuum so that the boundary surface forces vanish, while the density remains strictly positive. The body is subject only to internal elastic stress.

math.AP↗
arXiv2020

Affine motion of 2d incompressible fluids surrounded by vacuum and flows in ${\rm SL}(2,{\mathbb R})$

The affine motion of two-dimensional (2d) incompressible fluids surrounded by vacuum can be reduced to a completely integrable and globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. A complete classification of the dynamics is given including rigid motions, rotating eddies with stable and unstable manifolds, and solutions with vanishing pressure. For perfect fluids, the displacement generically becomes unbounded, as $t\to\pm\infty$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.

math.AP↗
arXiv2019

Global existence of near-affine solutions to the compressible Euler equations

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Hadžić \& Jang with the pressure-density relation $p = ρ^γ$ with the constraint that $1< γ\le {\frac{5}{3}} $. They asked if a different approach could go beyond the $γ> {\frac{5}{3}} $ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $γ>1$, thus also establishing global existence for the shallow water equations when $γ=2$.

math.AP↗
arXiv2017

Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum

The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $\rm{GL}^+(3,\mathbb R)$. The evolution of the fluid domain is described by a family ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank $r=1$, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along $3-r$ of its principal axes. In the compressible case, the asymptotic limit has rank $r=3$, and asymptotic completeness holds, when the adiabatic index $γ$ satisfies $4/3<γ<2$. The number of possible degeneracies, $3-r$, increases with the value of the adiabatic index $γ$. In the incompressible case, affine motion reduces to geodesic flow in $\rm{SL}(3,\mathbb R)$ with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes.

math.AP↗
arXiv2016

Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons

We study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the "peakon solutions". In particular, we shall show that the local regularity is similar to that exhibited by the solution of the inviscid Burger's equation with the same initial datum. The second goal is to prove that the decay results obtained in a paper of Himonas, Misiołek, Ponce, and Zhou extend to the class of solutions considered here.

math.AP↗
arXiv2012

Almost Global Existence for 2-D Incompressible Isotropic Elastodynamics

We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $[0,{T}/ε]$, for initial data of the form $εU_0$, where $T$ depends only on some Sobolev norm of $U_0$. We show that for such data there exists a unique solution on a time interval $[0, \exp{T}/ε]$, provided that $ε$ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted $L^2$ estimates and the ghost weight introduced by Alinhac.

math.AP↗
arXiv2012

Turbulence properties and global regularity of a modified Navier-Stokes equation

We introduce a modification of the Navier-Stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit. This new equation is studied numerically and turbulence properties are analyzed concerning energy spectra and scaling of structure functions. The dissipative structures arising in this new equation are curled vortex sheets contrary to vortex tubes arising in Navier-Stokes turbulence. The numerically calculated scaling of structure functions is compared with a phenomenological model based on the She-Lévêque approach. Finally, for this equation we demonstrate global well-posedness for sufficiently smooth initial conditions in the periodic case and in $\mathbb R^3$. The key feature is the availability of an additional estimate which shows that the $L^4$-norm of the velocity field remains finite.

physics.flu-dyn↗
arXiv2009

Stability in $L^1$ of circular vortex patches

The motion of incompressible and ideal fluids is studied in the plane. The stability in $L^1$ of circular vortex patches is established among the class of all bounded vortex patches of equal strength without any restriction on the size of the initial perturbation.

math.AP↗
arXiv2001

Convergence of Finite Difference Methods for Poisson's Equation with Interfaces

In this paper, a weak formulation of the discontinuous variable coefficient Poisson equation with interfacial jumps is studied. The existence, uniqueness and regularity of solutions of this problem are obtained. It is shown that the application of the Ghost Fluid Method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the Ghost Fluid Method is proven to be convergent.

math.NA↗
arXiv2000

Nonresonance and global existence of prestressed nonlinear elastic waves

The nonlinear hyperbolic system of pde's governing the evolution of the deformation of isotropic hyperelastic materials is considered. In the absence of boundaries and with an additional nonresonance or null condition, the system has global smooth solutions starting close to a one-parameter family of homogeneous dilations. The proof combines energy estimates with new decay estimates for the linear problem.

math.AP↗