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 $\gamma$ satisfies $4/3<\gamma<2$. The number of possible degeneracies, $3-r$, increases with the value of the adiabatic index $\gamma$. 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

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 Had\v{z}i\'{c} \& Jang with the pressure-density relation $p = \rho^\gamma$ with the constraint that $1< \gamma\le {\frac{5}{3}}$. They asked if a different approach could go beyond the $\gamma > {\frac{5}{3}}$ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $\gamma >1$, thus also establishing global existence for the shallow water equations when $\gamma=2$.Comment: 51 pages, details added to Section 4.7, to appear in Arch. Rational Mech. Ana

Affine motion of 2d incompressible fluids surrounded by vacuum and flows in SL(2,R){\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.Comment: 60 pages, 7 figure

Stability in L1L^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

Global existence of small displacement solutions for Hookean incompressible viscoelasticity in 3D

The initial value problem for Hookean incompressible viscoelastictic motion in three space dimensions has global strong solutions with small displacements

Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t-3/2. While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data

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-L\'ev\^eque 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