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Abstracts and Talk Materials
Analysis and Computation of Incompressible Fluid Flow
February 22 - 26, 2010

Supercavitating flow in multiply connected domains
December 31, 1969

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).

The spectrally-hyperviscous Navier-Stokes equations
December 31, 1969

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 107) 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.

Spatial analyticity for the Navier-Stokes equations
December 31, 1969

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.

Dynamical constraints for 3D Euler and possible blow-up
December 31, 1969

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.

December 31, 1969

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 H1 norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H1 norm. Moreover, we showed that in (vorticity) H2 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) H2 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 H1, 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 R3.

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 Lp, 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|uk|)-1, (k|vk|)-1, or (k√ (uk, vk) )-1, where u_k and vk 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 |vk|)-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√ (uk, vk) )-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.

Three dimensional stability of the Burgers vortex
December 31, 1969

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 v0 satisfies curlv0/r is a finite Randon measure with compact support. Furthermore, the global existence and uniqueness, is also established in this case provided that curlv0/r belongs to Lpc(R3) 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.

Geometric aspects of hydrodynamic blowup
February 26, 2010

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 L2 measure-preserving transformations.

Crossover in coarsenig rates in demixing binary viscous fluids
December 31, 1969

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 ∼ t1/3 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.

On stationary anomalous solutions to the Euler equations
February 26, 2010

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.

Weak and measure-valued solutions of the Euler equations
February 26, 2010

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 L3. 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.

The surface quasi-geostrophic equation
February 25, 2010

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.

On fluid-rigid body interaction problem
December 31, 1969

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.

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