February 22 - 26, 2010
Mathematical models of cavitation in fluids and quick and accurate numerical predictions are
essential at different stages in the design process and the evaluation of performance and cavitation
patterns. A question of particular interest is how to extend the traditional hodograph method used
for simply connected flows to more complicated cavitation models. These models are nonlinear and
include flows in multiply connected Riemann surfaces. We present two approaches for the
construction of a conformal map from a canonical parametric domain into an n-connected Riemann
surface of supercavitating flow. The first approach works when n is not greater than 3, and the
canonical domain is the exterior of n slits along the real axis. The map is given by quadratures and
expressed through the solutions of two Riemann-Hilbert problems on a symmetric hyperelliptic
Riemann surface. The second method is applicable for any n, and the parametric domain is the
exterior of n circles. To reconstruct the map, it is required to solve two Riemann-Hilbert problems
for piece-wise meromorphic, G-automorphic functions (G is a Schottky group). This method leads to
a series-form solution and does not need the solution to a Jacobi inversion problem (for the first
method, it cannot be bypassed).
We regularize the 3-D Navier-Stokes equations with hyperviscosity of
degree alpha, but applied only to the high wavenumbers past a cutoff m;
for now we are on a periodic box. Attractor estimates stay within the
Landau-Lifschitz degrees-of-freedom estimates even for very large m. An
inertial manifold exists for m large enough whenever alpha is at or
above 3/2. Galerkin-convergence and inviscid-limit results are optimized
for the high wavenumbers; the latter case is defined to mean that nu
goes to zero while the spectral hyperviscous term stays fixed.
Computational studies over many runs produce parameter choices that
facilitate close-to-parallel agreement (over a good-sized portion of the
inertial range) with the Kolmogorov energy-spectrum power law for high
(up to 10^{7}) Reynolds numbers.
Keywords: Navier-Stokes flow, Stokes flow, boundary integral,
stiff equations, fractional stepping, immersed interface,
immersed boundary, semigroups of operators,
finite difference methods, parabolic equations, diffusion,
regularity, stability, L-stable, A-stable, maximum norm
Abstract: We will discuss two related projects. Work with A. Layton has the goal
of designing a second-order accurate numerical method for viscous fluid
flow with a moving elastic interface with zero thickness, the original
problem for which Peskin introduced the immersed boundary method. We
will discuss some of the background for such numerical
methods. In our approach, we decompose the velocity in
the Navier-Stokes equations at each time into a part determined by the
(equilibrium) Stokes equations, with the interfacial force, and a
"regular" remainder which can be calculated without special treatment
at the interface. For the "Stokes" part we use the immersed interface
method or boundary integrals; for the regular part we use the
semi-Lagrangian method to
advance in time. Simple test problems indicate second-order accuracy
despite a first-order truncation error near the interface, as has come
to be expected with certain interfacial methods. We will describe
analytical results which partially justify this expectation. For a
fully discrete parabolic equation, we have proved a
regularizing effect: If we solve a nonhomogeneous heat equation with a
finite difference method, with L-stable temporal discretization, using
large time steps, then the solution and its first differences are
bounded uniformly by the maximum of the nonhomogeneity, and the second
differences are almost bounded. The proof uses the point of view of
analytic semigroups of operators.
Joint work with C. Foias.
We augment the (3d) Navier-Stokes system with an equation
governing the space analyticity
radius of the solutions and identify a suitable domain where
this equation is well-posed. This
allows us to study the maximal analyticity radius of the
(regular) solutions. We also obtain certain estimates of the analyticity radius of the
solution on the entire regularity interval.
Keywords: Darcy´s law, surface quasi-geostrophic equation.
Abstract: Recent work with collaborators has been focused on understanding the
different features regarding well-posedness and regularity issues of the incompressible porous media and the surface quasi-geostrophic equation. In this talk we will discuss solutions with infinite energy and contour dynamics of patches.
Keywords: Incompressible Euler equations, Arnold
distance, Relaxation
Abstract: According to Arnold's interpretation, the Euler equations for
incompressible fluids can be seen as a geodesic equation in the space of
measure preserving diffeomorphism. Hence, one can look for solutions by
minimizing the geodesic action functional with fixed endpoints. Since this
problem is in general ill-posed, Brenier introduced at the end of the 80's
a relaxed version of Arnold's approach. The aim of the talk is to describe
some recent results on this problem.
Reacting to various new forms of lagrangian and local geometric
criteria for finite time blowup of the three-dimensional
incompressible Euler equations (J. Deng, T. Hou and X. Yu
(2005), P. Constantin (2001)) numerical simulations are carried
out. High resolution is achieved by utilizing an adaptive mesh
refinement technique. Lagrangian tracer particles are injected
into the flow to analyze the evolution of both vortex lines and
the inverse flowmap. Preliminary results regarding relevant
blowup criteria are presented.
Keywords: 3D incompressible Navier-Stokes equations,
finite time blow-up, and global regularity,
and stabilizing effect of convection.
Abstract: We study the singularity formation of a recntly proposed 3D model
for the incompressible Euler and Navier-Stokes equations. This
3D model is derived from the axisymmetric Navier-Stokes equations
with swirl using a set of new variables. The model preserves
almost all the properties of the full 3D Euler or Navier-Stokes
equations except for the convection term which is neglected. If we add the convection term back to our model, we would recover the
full Navier-Stokes equations. We will present numerical evidence
which supports that the 3D model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new 3D model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. Finally, we prove rigorously that the 3D model develops finite time singularities for a large class of initial data with finite energy and appropriate bounadry conditions. This work may shed interesting light into the stabilizing effect of convection for 3D incompressible Euler and Navier-Stokes equations.
Keywords: Navier Stokes, stochastic-Lagrangian, particle method.
Abstract: I will introduce an exact stochastic representation for certain
non-linear transport equations (e.g. 3D-Navier-Stokes, Burgers)
based on noisy Lagrangian paths, and use this to construct a
(stochastic) particle system for the Navier-Stokes equations. On any
fixed time interval, this particle system converges to the
Navier-Stokes equations as the number of particles goes to infinity.
Curiously, a similar system for the (viscous) Burgers equations
shocks in finite time, and solutions can not be continued past these
shocks using classical methods. I will describe a resetting
procedure by which these shocks can (surprisingly!) be avoided, and
thus obtain convergence to the viscous Burgers equations on long
time intervals.
Keywords: Euler equations, quantum fluids
Abstract: Circulation is an often neglected conservation law in
developing
the mathematics of ideal fluids, meaning the classical 3D
Euler
equations and the quantum defocussing Gross-Pitaevskii
equations.
Recent Euler calculations demonstrated that the numerics
must conserve
circulation and when this is satisfied, it appears that
circulation
controls the growth of enstrophy in a manner consistent
with a
finite-time singularity of these equations.
In a quantum fluid the circulation, that is defects in
phase,
is inherently conserved. These equations allow
reconnection without
dissipation and without singularities.
Nonetheless, when compared with Navier-Stokes reconnection
there are
strong similarities, at least for a new Navier-Stokes
initial
condition which considers the role of circulation more
carefully.
Following reconnection in GP, waves form on vortex lines,
vortex
rings detach, and an inertial subrange develops, all in a
manner that
could explain experimental observations of the decay of
vortex line
length, a proxy for kinetic energy, despite the absence of
viscosity.
Co-author Miguel D. Bustamante (UCD Dublin).
While characterization of singular behaviour of simulations of the Euler
equations using the localized spatio-temporal growth of the vorticity
modulus would be preferred, because numerical simulations are discrete,
this approach cannot reliably follow its position and value
at times and positions very close to the potential singularity.
With this paradigm in mind, the natural alternative is methods that
identify possible singular evolution in terms of global quantities
determined over suitably-defined areas and volumes of the dynamical system.
This poster presents methods based on the conservation of integrals
defined on symmetry planes of a commonly used class of initial
conditions. These are:
(i) strong tools designed to validate any numerical simulation,
irrespective of the conclusions on the potentially singular behaviour, and
(ii) sharp inequalities and exact results to estimate more precisely the
vorticity growth exponents in a blow-up scenario.
This is a
joint work
with Pawel Konieczny. We study the steady flow of a second
grade fluid
past an obstacle in 3D. Arising equations can be split into a
transport
equation and an Oseen equation for the velocity. We use the
theory of
fundamental solutions to the Oseen equation in weighted
Lebesgue spaces
together with results of Herbert Koch to prove the existence of
a wake
region behind the obstacle, i.e. a region where the solution
decays
slower to the prescribed velocity at infinity.
Keywords: Navier-Stokes equations, partial regularity,
Hausdorff dimension, fractal dimension.
Abstract: A classical result of Caffarelli, Kohn, and Nirenberg states
that the one dimensional Hausdorff measure of singularities
of a suitable weak solution of the Navier-Stokes system is
zero. We present a short proof of the partial regularity
result which allows the force to belong to a singular Morrey
space. We also provide a new upper bound for the fractal
dimension of the singular set.
Keywords: Couette flow, Sommerfeld paradox, inviscid damping
Abstract: Couette flows are shear flows with a linear velocity profile. First, They are known to be linearly stable for any Reynolds number, but become turbulent for large Reynolds numbers (Sommerfeld, 1908). With Charles Y. Li, we proposed an explanation of this paradox by constructing unstable shear flows arbitrarily close to Couette flows, for both inviscid and slightly viscous cases. Such unstable shears are possible seeds for the turbulent behaviors near Couette flows, as also supported by numerical work. Second, starting from the work of Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time. Such inviscid damping is open in the nonlinear level. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H^{1} norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H^{1} norm. Moreover, we showed that in (vorticity) H^{2} neighborhood of Couette flows, the only steady structures (including traveling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H^{2} space might be much simpler. We also obtained similar results for the problem of nonlinear Landau damping in 1D electrostatic plasmas.
Keywords: Helical flow, swirl, vortex stretching, energy method, Delort type symmetrization
Abstract: Helical flows are flows which are covariant with respect to translation along helices. In recent work, B. Ettinger and E. Titi established global existence and uniqueness of helical flow solutions of the incompressible 3D Euler equations with bounded vorticity and no "helical swirl" (the component of velocity along helices). In earlier work, A. Mahalov, E. Titi and S. Leibovich had established well-posedness, in H^{1}, of viscous helical flows (solutions of the Navier-Stokes equations) with no restriction on helical swirl. Absence of helical swirl prevents vortex stretching, but, although a conserved quantity for Euler, it is not conserved by the Navier-Stokes evolution. In both sets of results, the fluid domain was a bounded, smooth, helical subset of R^{3}.
In this talk we will review these results and discuss the problem of taking the vanishing viscosity limit of helical flows with small, with respect to viscosity, helical swirl. We work with bounded domain flows, assuming Navier boundary conditions, but we discuss the full-space problem as well. Our results assume that the flow has finite enstrophy. We also discuss an alternative way to obtain global existence of helical solutions of the 3D Euler equations, without helical swirl, with vorticity only in L^{p}, p>3/2.
The collaborators involved are Anne Bronzi, Quansen Jiu, Milton Lopes Filho and Dongjuan Niu.
The Navier-Stokes-α model of turbulence is a
mollification of the Navier-Stokes equations in which the
vorticity is advected and stretched by a smoothed velocity
field. The smoothing is performed by filtering the velocity
field over spatial scales of size smaller than -α. The
statistical properties of the smoothed velocity field are
expected to match those of Navier-Stokes turbulence for scales
larger than α.
For wavenumbers k such that kα»1,
corresponding to spatial scales smaller than α, there
are three candidate power laws for the energy spectrum,
corresponding to three possible characteristic time scales in
the model equations. The three
possibilities depend on whether the
time scale of an eddy of size k^{-1} is determined by
(k|u_{k}|)^{-1}, (k|v_{k}|)^{-1}, or (k√
(u_{k},
v_{k}) )^{-1}, where u_{_k} and
v_{k} are the
components of the filtered velocity field u and unfiltered
velocity field v, respectively, for wavenumber k.
Determining the actual
scaling requires resolved numerical simulations.
We measure the scaling of the energy spectra from
high-resolution simulations of the two-dimensional
Navier-Stokes-α model, in the limit as
α→∞. The energy spectrum of
the smoothed
velocity field scales as k^{-7} in the direct enstrophy
cascade regime, consistent with the dynamics dominated by the
time scale given by (k |v_{k}|)^{-1}. We are
thus able to
deduce that the dynamics of the dominant cascading
conserved quantity, namely the enstrophy of the rough velocity
v, determines the power law for small scales.
For the two-dimensional Leray-α model, the time scale
given by (k√ (u_{k},
v_{k}) )^{-1} is understood to
characterize the dynamics of the conserved enstrophy. Indeed,
our numerical simulation of this model gives a k^{-5}
power
law in the enstrophy inertial subrange. This result supports
our claim regarding the characteristic time scale of the
two-dimensional NS-α model for wavenumbers kα»1.
This is a joint work with Thierry Gallay. Burgers vortices are explicit
stationary solutions of the Navier-Stokes equations which are often used
to describe the vortex tubes observed in numerical simulations of
three-dimensional turbulence. In this model, the velocity field is a
two-dimensional perturbation of a linear straining flow with axial
symmetry. The only free parameter is the Reynolds number Re =
Γ/ν, where Γ is the total circulation of the vortex and
ν is the kinematic viscosity. We will show that the Burgers vortex is
asymptotically stable with respect to general three-dimensional
perturbations, for all values of the Reynolds number.
Keywords: 3D Navier-Stokes equations,
multiscale
atmospheric dynamics and turbulence, high resolution
simulations.
Abstract: High resolution multi-scale simulations of limited area atmospheric environments are one of the major frontiers of atmospheric sciences and environmental sustainability. The latest developments in high performance computing technologies represent an opportunity to advance and improve real time atmospheric characterization and forecasting over limited areas. Among the new capabilities required are improved physics based sub-grid parameterizations and one- or two-way nesting to integrate models of disparate scales. We present high resolution horizontally
and vertically nested mesoscale/microscale simulations for effective resolution of multiscale atmospheric physics phenomena in regional atmospheres (A. Mahalov and M. Moustaoui, Journal of Computational Physics, vol. 228, p. 1294-1311, 2009).
Keywords: existence results, weak solutions, strong solutions, micro-macro models.
Abstract: Systems coupling fluids and polymers
are of great interest in many branches of applied physics, chemistry
and biology. There are many models to describe them.
We will present here several existence results of weak or strong solutions. In particular we will consider the FENE, the Doi
and FENE-P models.
This is a joint work with Yoshikazu Giga.
We give a geometric nonblow up criterion on the direction of the vorticity
for the 3D Navier-Stokes flow. We prove that under a restriction on
behavior in time (type I condition) the solution does not blow up if the
vorticity direction is uniformly continuous in the region where vorticity
is large.
The behavior of solutions to the Navier-Stokes equations with no-slip
boundary conditions when the viscosity goes to zero
has been a long standing mathematical problem since its formulation by
Prandtl.
The main difficulty lies in the possible production of extreme velocity
gradients near boundaries.
We have undertaken a series of numerical experiments using a Fourier
mode expansion of the solution along with a volume penalization method
to impose the no-slip condition.
The results support a scenario in which the energy dissipation rate
remains strictly positive in the inviscid limit, due to a boundary layer
of tickness orders of magnitude smaller than the classical Re^{-1/2}
estimate.
When the initial condition is a Gaussian noise, a wavelet analysis of
the vorticity field after some time suggests that it has organized into
"dissipative structures", that are recycled by detachment from the
boundary but visit the whole domain.
Some implications for modeling of boundary layer detachment phenomena
are briefly discussed.
The behavior of solutions to the Navier-Stokes equations with no-slip
boundary conditions when the viscosity goes to zero
has been a long standing mathematical problem since its formulation by
Prandtl.
The main difficulty lies in the possible production of extreme velocity
gradients near boundaries.
We have undertaken a series of numerical experiments using a Fourier
mode expansion of the solution along with a volume penalization method
to impose the no-slip condition.
The results support a scenario in which the energy dissipation rate
remains strictly positive in the inviscid limit, due to a boundary layer
of tickness orders of magnitude smaller than the classical Re^{-1/2}
estimate.
When the initial condition is a Gaussian noise, a wavelet analysis of
the vorticity field after some time suggests that it has organized into
"dissipative structures", that are recycled by detachment from the
boundary but visit the whole domain.
Some implications for modeling of boundary layer detachment phenomena
are briefly discussed.
Keywords: vortex sheet motion, vortex blob method, Euler-alpha model
Abstract: The vortex sheet is a mathematical model for a shear layer
in which the layer is approximated by a surface. Vortex
sheet evolution has been shown to approximate the motion
of shear layers well, both in the case of free layers and
of separated flows at sharp edges.
Generally, the evolving sheets develop singularities
in finite time. To approximate the fluid past this time,
the motion is regularized and the sheet defined as the
limit of zero regularization. However, besides weak existence
results in special cases, very little is
known about this limit. In particular, it is not known whether
the limit is unique or whether it depends on the regularization.
I will discuss several regularizing mechanisms, including physical
ones such as fluid viscosity, and purely numerical ones such
as the vortex blob and the Euler-alpha methods. I will show
results for a model problem and discuss some of the unanswered
questions of interest.
The global existence of weak solutions for the
three-dimensional
axisymmetric Euler-alpha (also known as Lagrangian-averaged
Euler-α) equations, without swirl, is established, whenever
the initial unfiltered velocity v_{0} satisfies
curlv_{0}/r is a
finite Randon measure with compact support. Furthermore, the
global existence and uniqueness, is also
established in this case provided that curlv_{0}/r belongs to
L^{p}_{c}(R^{3}) with p>3/2. It is worth mention that
no such results are known to be available, so far, for the
three-dimensional Euler equations of ideal incompressible
flows.
We study some of the key quantities arising in the Arnold's theory (1966)
of the incompressible Euler equations both in two and three dimensions.
The sectional curvatures for the Taylor-Green vortex and ABC flows
initial conditions are calculated exactly in three dimensions.
We trace the time evolution of the Jacobi fields by direct numerical
simulations and, in particular, see how the sectional curvatures get more and
more negative in time. The spatial structure of the the Jacobi fields is
compared with the vorticity fields by visualizations.
The Jacobi fields are found to grow exponentially in time for the flows with
negative sectional curvatures.
In two dimensions, a family of initial data proposed by Arnold (1966)
is considered. The sectional curvature is observed to change its sign quickly
even if it starts from a positive value. The Jacobi field is shown to be
correlated with the passive scalar gradient in spatial structure.
On the basis of Rouchon's expression (1984) for the sectional curvature
(in physical space), the origin of negative curvature is investigated.
It is found that a 'potential' $alpha_{bm{xi}}$ appearing in the
definition of covariant time derivative plays an important role,
in that a rapid growth in its gradient makes a major contribution to
the negative curvature.
Keywords: maximum principle, blow-up, nonlocality
Abstract: We consider nonlocal versions of Burgers equations in a preliminary attempt to
1) generalize Hopf-Cole transforms for incompressible flows, and 2) to assess
the effect of nonlocality on the breakdown of maximum principle leading
to blow-up.
It is well-known that by the Forsyth-Florin-Hopf-Cole transform 1D Burgers
equation is integrable through a Hamilton-Jacobi-like equation for
the velocity potential. In higher spatial dimensions, on top of a potential
component, there appears a solenoidal component. For the former, a similar
method of solution works for multi-dimensional Burgers equation. To treat the
latter, we recast e.g. 2D incompressible Navier-Stokes equations as a nonlocal
Hamilton-Jacobi-like equation using the stream function. This form apparently
exhibits nontrivial cancellations of nonlinear terms, known as nonlinearity
depletion.
On this basis, we propose a nonlocal model equation in 1D and study its
behavior numerically. It is shown that this model is equivalent to another
model equation which is known to blow up. We derive a Hopf-Cole-like transform
to recast the model as close as a heat diffusion equation, but with an
additional dangerous term. Attempts are made to explore possible transforms
for the 2D Navier-Stokes equations. Time permitting, we may describe yet
another model (joint work with M. Dowker) where a nonlocal term, mimicking the
pressure, leads apparently to blow-up in finite time.
We study the disappearance of criticality of a reactive fully
developed flow of an
incompressible, thermodynamically compatible fluid of grade
three with
viscous heating and heat generation between two horizontal flat
plates, where the
top is moving with uniform speed and the bottom plate is fixed
in the presence of
imposed pressure gradient. This is a natural continuation of
earlier work on
rectilinear shear flows. The governing coupled ordinary
differential equations
are transformed into dimensionless forms using an appropriate
transformation
and then solved numerically for thermal transition
(disappearance of criticality)
using Maple based shooting method. Attention is focused upon
the disappearance of
criticality of the solution set for various values of the
physical parameters and
the numerical computations are presented graphically
to show salient features of the solution set.
Keywords: Gross-Pitaevskii hierarchy
Abstract: In this talk we will discuss joint work with Thomas Chen on the
dynamics of a boson gas with three-body interactions in dimensions d=1,2. We prove that in the limit as the particle number N tends to infinity, the
BBGKY hierarchy of k-particle marginals converges to a limiting
Gross-Pitaevskii (GP) hierarchy for which we prove existence and uniqueness of solutions. For factorized initial data, the solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrodinger equation.
Time permitting, we will briefly describe our new approach for studying
well-posedness of the Cauchy problem for focusing and defocusing GP
hierarchy.
Keywords: Time-dependent incompressible viscous flow, stable discretization, time splitting, Stokes pressure, Leray projection.
Abstract: How to properly specify boundary conditions for pressure for
no-slip incompressible viscous flow has been a longstanding issue
in analysis and computation. A recent analytical resolution of
this issue is based on a local well-posedness theorem for an
extended Navier-Stokes dynamics, in which the zero-divergence
condition is replaced by a pressure formula that involves the
commutator of the Laplacian and Leray projection operators.
I'll indicate progress on some related analytical questions
(domains with corners, MHD), but will focus on improvements in
numerical schemes that involve projection methods in time and
finite elements in space. We find schemes that involve simple
kinds of finite elements (Lagrange of equal order for velocity
and pressure, including piecewise-linear, for example) that are
stable in tests with large time steps at low Reynolds number,
with up to 3rd-order accuracy in time for both velocity and
pressure. Notably, these schemes do not include projection
methods that update the pressure from previous steps.
The geometric approach to hydrodynamics was developed by Arnold
to study Lagrangian stability of ideal fluids. It identifies a
Lagrangian fluid flow with a geodesic on the Riemannian
manifold of volume-preserving diffeomorphisms. The curvature of
this manifold is typically negative but sometimes positive, and
positivity leads to conjugate points (where initially close
geodesics spread apart and come together again).
In this talk we suppose a fluid in
^{3} satisfies a
pointwise version of the Beale-Kato-Majda criterion for blowup
at a finite time T. I will describe a theorem which states
that either the geodesic experiences an infinite sequence of
consecutive conjugate pairs approaching the blowup time, or the
deformation tensor has a fairly special form at the blowup
time. The first possibility suggests that one could "see"
blowup geometrically in a weak space, such as the space of
L
^{2} measure-preserving transformations.
We consider the demixing process of a binary mixture of two
liquids after a temperature quench. In viscous liquids, the
demixing is mediated by diffusion and convection. The typical
particle size
grows as a function of time t, a
phenomenon called coarsening. Simple scaling arguments based on
the assumption of statistical self-similarity of the domain
morphology suggest the coarsening rate: from
∼
t
^{1/3}
for diffusion-mediated to
∼ t for flow-mediated.
In joint works with Yann Brenier, Felix Otto, and Dejan
Slepcev, we derive the crossover of both scaling regimes in
form of time-averaged upper bounds. The mathematical model is a
Cahn-Hilliard equation with convection term, where the fluid
velocity is determined by a Stokes equation. The analysis
follows closely a method proposed by Kohn and Otto, which is
based on the gradient flow structure of the evolution.
Joint work with Doug Dokken and Kurt Scholz.
In this work we consider a modification of the model developed by J. Serrin where velocity, in spherical coordinates, decreases in proportion to the reciprocal of the distance from the vortex line.
Serrin’s model has three distinct solutions, depending on the kinematic viscosity and the value of a “pressure” parameter. These are: down-draft core with radial outflow, downdraft core with a
compensating radial inflow, and updraft core with radial inflow (single-cell vortex). Recent studies, based on radar data of selected severe weather events show that the ratio of natural log of vorticity
and natural log of grid spacing have a linear relationship, suggesting a fractal-like phenomenon with a constant ratio. In two of the cases the ratio is close to 0.6. We have attempted Serrin’s approach seeking solution with the assumption that the velocity decreases in proportion to the reciprocal of the distance to the power 0.6. This ansatz leads to a substantially more complicated boundary-value problem for sixth order system of nonlinear differential equations. However, the spherical variable has not been eliminated from the right-hand side of the system That contradicts the original
assumption that the velocity would be represented by ratio of a function of the angle with the vortex line to the corresponding reciprocal of the distance to the power not equal to one. We discuss some
specific cases of the system and applications of a shooting method. Also dependence on the radial variable is studied as small variations in the solutions would indicate that our system
is essentially independent of the parameter.
Keywords: Energy conservation, Onsager conjecture, turbulence, Besov spaces.
Abstract: In this talk we will discuss a possibility of constructing solutions to the forced stationary Euler equations with limit regularity.
The problem is motivated by finding vector fields that enjoy the properties of a turbulent flow, i.e. anomalous energy dissipation,
smooth forcing, and regularity 1/3 in a certain Besov norm. The time-dependent version of this problem is known as the Onsager conjecture.
We will exhibit a number of conditions which rule out existence of such solutions. Those include, for instance, conditions on the singularity set. An example of a field with smoothness 1/3, but integrability 18/11, which is Onsager-supercritical will
be presented.
Keywords: Boussinesq equations, wave-vortical interactions, quasi-geostrophic approximation,
intermediate models
Abstract: Starting from the rotating Boussinesq equations, we show
how to derive a hierarchy of new models intermediate
between the quasi-geostrophic (QG) approximation and
the full equations. The new PDEs progressively include
more effects of inertia-gravity waves. We explain how to
derive the new models in two ways: (i) by eigenmode projection,
and (ii) directly in physical space. We illustrate how the
new reduced PDEs can be used to identify the nonlinear interactions primarily responsible for observed non-QG phenomena, such as
cyclone/anticyclone asymmetry in geophysical flows.
A waves-only model gives insight into the growth of
horizontal shear flows. Simulations of the models highlight
the practical implications of under-resolving wave-mode
interactions in numerical calculations.
Keywords: weak solutions, turbulence, h-principle
Abstract: In 1993 V. Scheffer produced a nontrivial weak solution of the 2D incompressible Euler equations
with compact support in space-time. Subsequently A.Shnirelman gave different constructions
for solutions with (i) compact support in time and (ii) strictly decreasing energy. Such "wild" solutions
seemingly contradict the idea of an evolution equation. In this
talk we will discuss a recent approach to such constructions in joint work with Camillo De Lellis. Moreover,
we show that the underlying phenomenon has a striking similarity to the h-principle, a well known
phenomenon of flexibility in underdetermined geometric problems. In such situations the underlying
PDE seems to represent no constraint at all, the only restrictions on the space of solutions come from
topology. We look at the Euler equations in this light and show that there are indeed nontrivial restrictions
arising from the initial data.
Keywords: Asymptotics, Exterior flows, Navier-Stokes equations, self-similar
Abstract: We prove the unique existence of solutions of the 3D
incompressible
Navier-Stokes equations in an exterior domain with small
non-decaying boundary data, for all t ∈ R or t > 0. In
the case t > 0 it is coupled with a small initial data in
weak L^{3}. If the boundary data is time-periodic, the spatial
asymptotics of the time-entire solution is given by a Landau
solution which is the same for all time. If the boundary data
is
time-periodic and the initial data is asymptotically discretely
self-similar, the solution is asymptotically the sum of a
time-periodic vector field and a forward discretely
self-similar
vector field as time goes to infinity. This is a joint work
with
Kyungkuen Kang and Hideyuki Miura.
Keywords: Complex Fluids, Complactness
Abstract: A classical result of P. Lax states that ``a (linear) numerical
scheme converges if and only if it is stable and consist''. For nonlinear
problems this statement needs to augmented to include a compactness hypotheses
sufficient to guarantee convergence of the nonlinear terms. This talk will
focus on the development of numerical schemes for parabolic equations that are
stable and inherit compactness properties of the underlying partial
differential equations. I will present a discrete analog of the classical
Lions-Aubin compactness theorem and use it to establish convergence of
numerical schemes for fluids transporting membranes, and the Ericksen Leslie
equations for (nematic) liquid crystals. The talk will finish with some open
problems that arise in the numerical simulation of this class of problems.
Keywords: fractional Laplace, global regularity, the surface quasi-geostrophic equation
Abstract:Fundamental mathematical issues concerning the
surface quasi-geostrophic (SQG) equation have recently
attracted the attention of many researchers and important
progress has been made. This talk focuses on the existence,
uniqueness and regularity of solutions to the SQG equation
and covers both the inviscid and dissipative cases. We will
summarize some existing work and report very recent
numerical and theoretical results.
Keywords: water wave problem
Abstract: We consider the question of global in time existence and uniqueness of solutions of the infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial data that is small in its kinetic energy and height, we show that the 2-D full water wave equation is uniquely solvable almost globally in time. And for any initial interface that is small in its steepness and velocity, we show that the 3-D full water wave equation is uniquely solvable globally in time.
We will discuss the classical solutions of two dimensional inviscid rotating shallow water equations with small initial data. The global existence and asymptotic behavior are obtained when the initial data has zero relative vorticity, where rotating shallow water system can be transformed to a symmetric quasilinear Klein-Gordon system. We also give the lower bound for the lifespan of classical solutions with general initial data. This is a joint work with Bin Cheng.
We study the motion of a rigid body immersed in an incompressible ideal (or viscous) fluids. We first establish an existence of global (in time) existence of weak solution with natural far field condition for 2-dimensional Euler system. Then for viscous case, we prove that the corresponding generalized Stokes operator is the infinitesimal generator of an analytic semigroup on some appropriate spaces so that we can obtain local existence of strong solutions in such spaces.
Keywords: Viscous boundary layers, Prandtl's boundary layer system,
Navier-slip bounadry conditions, incompressible Navier-Stokes system.
Abstract: In this talk, I will discuss some issues and recent results related to the viscous boundary layer theory. In particular, I will present a global existence result on classical soluition to the 2-dimensional unsteady Prandtl's boundary system; some convergence results on viscous solutions when the viscosity becomes small. Both non-slip and Navier-type slip boundary conditions will be studied.
Modeling and numerical approximation of two-phase incompressible flows
with different densities and viscosities are considered using the diffusive
phase-field model. A physically consistent phase-field model that admits
an energy law is proposed, and several energy stable, efficient and accurate
time discretization schemes for the coupled nonlinear phase-field model
are constructed and analyzed. Ample numerical experiments are carried
out to validate the correctness of these schemes and their accuracy for
problems with large density and viscosity ratios.