Thomas C. Sideris

Formation of singularities in three-dimensional compressible fluids

Presented are several results on the formation of singularities in solutions to the three-dimensional Euler equations for a polytropic, ideal fluid under various assumptions on the initial data. In particular, it is shown that a localized fluid which is initially compressed and outgoing, on average, will develop singularities regardless of the size of the initial disturbance.

On almost global existence for nonrelativistic wave equations in 3D

Almost global solutions are constructed to three-dimensional, quadratically nonlinear wave equations. The proof relies on generalized energy estimates and a new decay estimate. The method applies to equations that are only classically invariant, such as the nonlinear system of hyperelasticity.

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

Abstract 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.

Nonresonance and global existence of prestressed nonlinear elastic waves

This article considers the existence of global classical solutions to the Cauchy problem in nonlinear elastodynamics. The unbounded elastic medium is assumed to be homogeneous, isotropic, and hyperelastic. As in the theory of 3D nonlinear wave equations in three space dimensions, global existence hinges on two basic assumptions. First, the initial deformation must be a small displacement from equilibrium, in this case a prestressed homogeneous dilation of the reference conflguration, and equally important, the nonlinear terms must obey a type of nonresonance or null condition. The omission of either of these assumptions can lead to the breakdown of solutions in flnite time. In particular, nonresonance complements the genuine nonlinearity condition of F. John, under which arbitrarily small spherically symmetric displacements develop singularities (although one expects this to carry over to the nonsymmetric case, as well), [4]. John also showed that small solutions exist almost globally [5] (see also [10]). Formation of singularities for large displacements was illustrated by Tahvildar-Zadeh [16]. The nonresonance condition introduced here represents a substantial improvement over our previous work on this topic [13]. To explain the difierence roughly, our earlier version of the null condition forced the cancellation of all nonlinear wave interactions to flrst order along the characteristic cones. Here, only the cancellation of nonlinear wave interactions among individual wave families is required. The di‐culty in realizing this weaker version is that the decomposition of elastic waves into their longitudinal and transverse components involves the nonlocal Helmholtz projection, which is ill-suited to nonlinear analysis. However, our decay estimates make clear that only the leading contribution of the resonant interactions along the characteristic cones is potentially dangerous, and this permits the usage of approximate local decompositions.

Global Stability of Large Solutions to the 3D Navier-Stokes Equations

We prove the stability of mildly decaying global strong solutions to the Navier-Stokes equations in three space dimensions. Combined with previous results on the global existence of large solutions with various symmetries, this gives the first global existence theorem for large solutions with approximately symmetric initial data. The stability of unforced 2D flow under 3D perturbations is also obtained.

GLOBAL EXISTENCE FOR SYSTEMS OF NONLINEAR WAVE EQUATIONS IN 3D WITH MULTIPLE SPEEDS

Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.

The null condition and global existence of nonlinear elastic waves

The equations of motion for the displacement of an isotropic, homogeneous, hyperelastic material form a quasilinear hyperbolic system, Lu = ∂ t u− c2∆u− (c1 − c2)∇(∇ · u) = F (∇u)∇u, in three space dimensions, with wave speeds 0 < c2 < c1 and a nonlinearity, the precise form of which will be spelled out in later sections. We shall prove that for certain classes of materials, small initial disturbances give rise to global smooth solutions. These special materials are distinguished by a null condition imposed on the quadratic portion of the nonlinearity. It is known from the work of John [6], that the equations possess almost global solutions for small initial values. Moreover, John [4] has identified a genuine nonlinearity condition which, at least in the spherically symmetry case, leads to formation of singularities even for small data, see also [3]. The null condition, presented below, is the complementary case to genuine nonlinearity. Thus, the situation is entirely analogous to the case of scalar nonlinear wave equations. Small solutions exist almost globally [7], [8]. Examples suggest that small solutions break down in finite time [4], [11], unless the quadratic terms in the nonlinearity satisfy a null condition, in which case they exist globally [1], [9]. In the present case, however, such results are not immediate generalizations of the scalar case. The original proofs of the existence results for the wave equation depended upon the Lorentz invariance of the equations, a property not shared by the equations of elasticity. Nevertheless, using various extensions of the ideas in [10], we obtain global existence with the null condition without the benefit of Lorentz invariance. There are two new observations which lead to the global existence result. First, within the class of physically meaningful nonlinearities arising from the hyperelasticity assumption, there exists a null condition. The null condition leads to enhanced decay (i.e. (1 + t)−2 instead of (1 + t)−1) of the nonlinear terms along the light cones. Secondly, it is shown how to combine the weighted L∞ − L and L − L estimates obtained in [10] in order to obtain enhanced decay inside the light cones.

Convergence of the ghost fluid method for elliptic equations with interfaces

This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the existence and uniqueness of a solution to the problem by classical methods. 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 proved to be convergent.

Delayed singularity formation in 2D compressible flow

The initial value problem for the compressible Euler equations in two space dimensions is studied. Of interest is the lifespan of classical solutions with initial data that is a small perturbation from a constant state. The approach taken is to regard the compressible solution as a nonlinear superposition of an underlying incompressible flow and an irrotational compressible flow. This viewpoint yields an improvement for the lifespan over that given by standard existence theory. The estimate for the lifespan is further improved when the initial data possesses certain symmetry. In the case of rotational symmetry, a result of S. Alinhac is reconsidered. The approach is also applied to the study of the incompressible limit. The analysis combines energy and decay estimates based on vector fields related to the natural invariance of the equations.

Finite time singularities in ideal fluids with swirl

Three-dimensional ideal, incompressible fluids with swirl are studied numerically using two different methods: standard finite differences and a projection method based on upwind differencing. Both methods give quantitatively similar results, leading to the conclusion that singularities form in finite time in a manner consistent with known theoretical criteria. The effect of singularities in incompressible flows on nearby compressible flows is discussed.