# <span style=color:#ff0000><br/><br/><b>Coffee Break and Poster Session</b></span><br><br/><br/>Poster submissions welcome from all participants<br><br/><br/><a<br/><br/>href=/visitor-folder/contents/workshop.html#poster>Instructions</a>

Monday, July 20, 2009 - 10:00am - 11:00am

EE/CS 3-176

**Elastodynamics, differential forms, and weak solutions**

David Wagner (University of Houston)

We offer an alternate derivation for the symmetric-hyperbolic formulation of the equations of motion for a hyperelastic material with

polyconvex stored energy. The derivation makes it clear that the expanded system is equivalent, for weak solutions, to the original system. We consider motions with variable as well as constant temperature. In addition, we present equivalent Eulerian equations of motion, which are also symmetric-hyperbolic.**Transonic flows in nozzles**

Hairong Yuan (East China Normal University)

We introduce some

results obtained by the author and his collaborators on existence,

stability and uniqueness of transonic shocks and subsonic-supersonic

flows in nozzles by using various different models.**Stability of noncharacteristic viscous boundary layers**

Toan Nguyen (Indiana University)

We present our recent results on one- and multi-dimensional

asymptotic stability of noncharacteristic boundary layers in gas dynamics (or a more general class of

hyperbolic--parabolic systems) with suction- or blowing-type boundary conditions. The linearized and nonlinear stability is established for layers with arbitrary amplitudes, under the assumption of strong spectral, or uniform Evans, stability. The latter assumption has been verified for small-amplitude layers in one-d case by various authors using energy estimates and in multi-d case by Guès, Métivier, Williams, and Zumbrun. For large-amplitude layers, it may be efficiently checked numerically by a combination of asymptotic ODE estimates and numerical Evans function computations.

This is a joint work with Kevin Zumbrun.**On an inhomogeneous slip-inflow boundary value problem for a steady**

flow of a viscous compressible fluid in a cylindrical domain

Tomasz Piasecki (Polish Academy of Sciences)

We investigate a steady flow of a viscous compressible fluid with inflow

boundary condition on the density

and inhomogeneous slip boundary conditions on the velocity in a

cylindrical domain.

We show existence of a strong solution that is a small perturbation

of a constant flow (v^{*}= (1,0,0), ρ^{*}= 1).

We also show that this solution is unique in the class of small

perturbations of (v^{*},ρ^{*}).

The nonlinear term in the continuity makes it impossible to apply

a fixed point argument. Therefore in order to show the existence

of the solution we use a method of successive approximations.**Central schemes for a new class of entropy solutions of the modified**

Buckley-Leverett equation

Ying Wang (The Ohio State University)

Joint work with Chiu-Yen Kao.

Buckley-Leverett (BL) equation arises in two-phase flow problem in porous

media. It models the oil recovery by water-drive in one-dimension. Here we

propose a central scheme for an extension of the BL equation which

includes the dynamic effects in the pressure difference between the two

phases and results in a third order mixed derivatives term in the modified

BL equation. The numerical scheme is able to capture the admissible shocks

which is the so-called nonclassical shock due to the violation of the

Oleinik entropy condition.**Modulational instability of periodic waves**

Mathew Johnson (Indiana University)

Using periodic Evans function expansions, we derive geometric criterion for the modulational instability, i.e. spectral instability to long-wavelength perturbations, of periodic traveling waves of the generalized KdV equation. Such techniques have also recently proven useful for analyzing spectral stability to long-wavelength transverse perturbations, as well as studying nonlinear stability to periodic perturbations. Using standard elliptic function techniques, we can explicitly calculate the necessary geometric information for polynomial nonlinearities: we will only present the results for KdV and modified KdV equations here.

This is joint work with Jared. C. Bronski (University of Illinois at Urbana-Champaign).**Non-uniqueness of entropy solutions and the carbuncle phenomenon**

Volker Elling (University of Michigan)

In one space dimension, the Godunov scheme can only converge to entropy solutions, for

which uniqueness theorems in various classes are known. We present an example in 2d where

the Godunov scheme converges, depending on the grid, either to an exact entropy solution or

to a second, numerical solution, which is also produced by all other numerical schemes tested.

We propose that entropy solutions are not unique in 2d. The observation also yields an

explanation for the carbuncle phenomenon, an instability in numerical calculations of shock

waves. Thus fundamental consequences both for numerics and theory.**Compressible turbulence modelling in astrophysics**

Christian Klingenberg (Bayerische-Julius-Maximilians-Universität Würzburg)

We present a numerical method for the compressible Euler

equations in 3 space dimensions where we combine adaptive mesh refinement with subgrid scale modelling.

This increases efficiency and accuracy when computing turbulent flows. We apply this to astrophysical

problems.**On the motion of several rigid bodies in an incompressible non-Newtonian**

and heat-conducting fluid

Sarka Necasova (Czech Academy of Sciences (AVČR))

We will consider the problem of the motion of several rigid bodies in

viscous non-Newtonian heat fluid in bounded domain in three dimensional

situation.Using penalization method developed by Conca, San Martin,

Tucsnak and Starovoitov we have shown the existence of weak solution and

moreover we show that for certain non-newtonian fluids there are no

collisions among bodie or body and boundary of domain.**Elasticity of thin shells and Sobolev spaces of isometries**

Reza Pakzad (University of Pittsburgh)

We describe the approach through which various elastic shell models are

rigorously derived from the 3D nonlinear elasticity theory. Some results

and a conjecture on the limiting theories are presented. Spaces of weakly

regular isometries or infinitesimal isometries of surfaces arise in this

context. Important problems regarding these spaces include rigidity,

regularity and density of smooth mappings.

This project is partly a collaboration with Marta Lewicka and Maria-Giovanna Mora.**A new fourth-order non-oscillatory central scheme for**

hyperbolic conservation laws

Arshad Peer (University of Mauritius)

We propose a new fourth-order non-oscillatory central scheme for computing approximate solutions of hyperbolic conservation laws. A piecewise cubic polynomial is used for the spatial reconstruction and for the numerical derivatives we choose genuinely fourth-order accurate non-oscillatory approximations. The solution is advanced in time using natural continuous extension of Runge-Kutta methods. Numerical tests on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and yields sharp results when solving profiles with discontinuities. Experiments on nonlinear Burgers’ equation indicate that our scheme is superior to existing fourth-order central schemes in the sense that the total variation (TV) of the computed solutions are closer to the total variation of the exact solution.**On the classical solutions of two dimensional inviscid**

rotating shallow water system

Bin Cheng (University of Michigan)

Joint work with with Chunjing Xie. We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. Such global existence is in contrast with the finite time breakdown of compressible Euler equations in two and three dimensions. A key step in our proof is reformulation of the problem into a symmetric quasilinear Klein-Gordon system, which is then studied with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity.**Strong waves and vaccums in isentropic gas dynamics**

Robin Young (University of Massachusetts)

We give a complete description of nonlinear waves and their pairwise

interactions in isentropic gas dynamics. Our analysis includes

rarefactions, compressions and shock waves. We describe the

interaction in terms of a reference state and incident wave

strengths, and give explicit estimates of the outgoing wave

strengths. Our estimates are global in that they apply to waves of

arbitrary strength, and they are uniform in the incoming reference

state. In particular, the estimates continue to hold as this state

approaches vacuum. We also consider composite interactions, which

can be regarded as a degenerate superposition of pairwise

interactions. We construct a class of exact weak solutions which

demonstrate some interesting and surprising features of

interactions, and use these to demonstrate the collapse of a vacuum:

in most cases two shocks will emerge from the vacuum, but in certain

asymmetric cases a single shock and a rarefaction may emerge.**Global strong solutions to density-dependent viscoelasticity and liquid crystal**

Xianpeng Hu (University of Pittsburgh)Dehua Wang (University of Pittsburgh)

The global strong solutions to the multi-dimensional viscoelastic fluids and liquid crystals are discussed. Here, a strong solution is a solution in W^{2,q}satisfying the equations almost everywhere.**Gasdynamic regularity and nondegeneracy: some classifying remarks**

Liviu Dinu (Romanian Academy of Sciences)

We consider two gasdynamic contexts: an isentropic context and, respectively, an anisentropic context of a special type. Some [multidimensional] classes of regular solutions [simple waves solutions and regular interactions of simple waves solutions] can be constructively associated to the isentropic context, in presence of an ad hoc requirement of genuine nonlinearity, via a Burnat type [algebraic] approach [centered on a duality connection between the hodograph character and the physical character]. The algebraic construction fails generally [for geometrical and/or physical reasons] in the mentioned anisentropic context. Incidentally, in this case the algebraic construction can be replaced by a Martin type [differential] approach [centered on a Monge-Ampere type representation]. Some classes [unsteady 1D; steady supersonic] of regular solutions [pseudo simple waves solutions, regular interactions of pseudo simple waves solutions] can be constructed in this case via a linearized version of the differential approach. Some contrasts and consonances are considered, in a classifying parallel, between the two approaches [algebraic, differential] and the two classes [isentropic, anisentropic] of regular solutions respectively associated.

This is a joint work with Marina Ileana Dinu.**On the approximation of conservation laws by vanishing viscosity**

Carlotta Donadello (Northwestern University)

We consider a piecewise smooth solution to a scalar conservation law, with

possibly interacting shocks. After the interactions have taken place,

vanishing viscosity approximations can still be represented by

a regular expansion on smooth regions and by a singular perturbation

expansion near the shocks, in terms of powers of the viscosity

coefficient.**L**^{2}stability estimates for shock solutions of scalar

conservation laws using the relative entropy method

Nicholas Leger (The University of Texas at Austin)

We consider scalar nonviscous conservation laws with strictly

convex flux in one space dimension, and we investigate the

behavior of bounded L^{2}perturbations of shock wave solutions

to the Riemann problem using the relative entropy method. We

show that up to a time-dependent translation of the shock, the

L^{2}norm of a perturbed solution relative to the shock wave is

bounded above by the L^{2}norm of the initial perturbation.**Optimal nodal control of networked systems of conservation laws**

Michael Herty (RWTH Aachen)

In applications of flow in gas and water networks

the 1d p-system is used to describe the flow in pipes.

The dynamics of each pipe is coupled at a node through

coupling conditions inducing boundary conditions.

We study the well-posedness of these

initial boundary value problems of 2×2 balance laws for a

set

of general possibly time-dependent coupling conditions.**The Riemann problem for the stochastically perturbed non-viscous**

Burgers equation and the pressureless gas dynamics model

Olga Rozanova (Moscow State University)

Proceeding from the method of stochastic perturbation of a

Langevin system associated with the non-viscous Burgers equation

we construct a solution to the Riemann problem for the

pressureless gas dynamics and sticky particles system. We analyze

the difference in the behavior of discontinuous solution for these

two models and relations between them.**Periodic solutions to the Euler equations**

Robin Young (University of Massachusetts)

Joint work with Blake Temple.

In this ongoing collaboration with Blake Temple, we attempt

to prove

the existence of periodic solutions to the Euler equations of

gas

dynamics. Such solutions have long been thought not to exist

due to

shock formation, and this is confirmed by the celebrated

Glimm-Lax

decay theory for 2×2 systems. However, in the full

3×3

system,

multiple interaction effects can combine to slow down and

prevent

shock formation. We describe the physical mechanism

supporting

periodicity, analyze combinatorics of simple wave

interactions, and

develop periodic solutions to a linearized problem.

These

linearized solutions have a beautiful structure and exhibit

several

surprising and fascinating phenomena. We then consider

perturbing

these as a bifurcation problem, which leads us to problems of

small

divisors and KAM theory. As an intermediate step, we find

solutions

which are periodic to within arbitrarily high Fourier modes.