arXiv subjects

Robert L. Pego

Publications and source records attributed to Robert L. Pego.

44 records · Page 3Linked to original sources

arXiv2005

Divorcing pressure from viscosity in incompressible Navier-Stokes dynamics

We show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. We illustrate the advantages of this view in a number of ways. In particular, we provide simple proofs of (i) local-in-time existence and uniqueness of strong solutions for an unconstrained formulation of the Navier-Stokes equations, and (ii) the unconditional stability and convergence of difference schemes that are implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions.

math.AP↗

arXiv2004

Dynamical scaling in Smoluchowski's coagulation equations: uniform convergence

We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's coagulation equations for the solvable kernels K(x,y)=2, x+y and xy. We prove the uniform convergence of densities to the self-similar solution with exponential tails under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For the discrete equations we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution by the method of characteristics for the Laplace transform.

nlin.AO↗

arXiv2003

Approach to self-similarity in Smoluchowski's coagulation equations

We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels $K(x,y)=2$, $x+y$ and $xy$. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K=2 the size distribution is Mittag-Leffler, and for $K=x+y$ and $K=xy$ it is a power-law rescaling of a maximally skewed $α$-stable Levy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.

nlin.AO↗

arXiv2002

Spectrally stable encapsulated vortices for nonlinear Schroedinger equations

A large class of multidimensional nonlinear Schroedinger equations admit localized nonradial standing wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (non-localized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeros of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.

nlin.PS↗

arXiv2001

A host of traveling waves in a model of three-dimensional water-wave dynamics

We describe traveling waves in a basic model for three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation (or orthogonal to it) form an infinite-dimensional family. We characterize these solutions through spatial dynamics, by reducing a linearly ill-posed mixed-type initial-value problem to a center manifold of infinite dimension and codimension. A unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). A dispersive, nonlocal, nonlinear wave equation governs the spatial evolution of bottom velocity.

nlin.PS↗

arXiv1999

Non-self-similar behavior in the LSW theory of Ostwald ripening

The classical Lifshitz-Slyozov-Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.

cond-mat.mtrl-sci↗

arXiv1998

On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening

The LSW theory of Ostwald ripening concerns the time evolution of the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. We prove global existence, uniqueness and continuous dependence on initial data for measure-valued solutions with compact support in particle size. These results are established with respect to a natural topology on the space of size distributions, one given by the Wasserstein metric which measures the smallest maximum volume change required to rearrange one distribution into another.

math.AP↗

arXiv1998

Models of low-speed flow for near-critical fluids with gravitational and capillary effects

We study low-speed flows of a highly compressible, single-phase fluid in the presence of gravity, for example in a regime appropriate for modeling recent space-shuttle experiments on fluids near the liquid-vapor critical point. In the equations of motion, we include forces due to capillary stresses that arise from a contribution made by strong density gradients to the free energy. We derive formally simplified sets of equations in a low-speed limit analogous to the zero Mach number limit in combustion theory. When viscosity is neglected and gravity is weak, the simplified system includes: a hyperbolic equation for velocity, a parabolic equation for temperature, an elliptic equation related to volume expansion, an integro-differential equation for mean pressure, and an algebraic equation (the equation of state). Solutions are determined by initial values for the mean pressure, the temperature field, and the divergence-free part of the velocity field. To model multidimensional flows with strong gravity, we offer an alternative to the anelastic approximation, one which admits stratified fluids in thermodynamic equilibrium, as well as gravity waves but not acoustic waves.

physics.flu-dyn↗