Conference on PDEs and Free Boundary Problems. Department of Mathematics, University of Pittsburgh. March 11-14, Program

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1 Conference on PDEs and Free Boundary Problems Department of Mathematics, University of Pittsburgh March 11-14, 2015 Program The Conference Program includes two mini-courses, March 11; and twenty-four research talks, March All talks are in Thackerary Hall 704 in the Department of Mathematics, Pittsburgh, PA Sponsors: National Science Foundation (NSF), Institute for Mathematics and its Applications (IMA), Mathematics Research Center (MRC) of the University of Pittsburgh. Organizers: Ming Chen and Dehua Wang, on behalf of the Applied Analysis Group.

2 Conference on PDEs and Free Boundary Problems University of Pittsburgh, March 11-14, 2015 Schedule for Mini-Courses, March 11, 2015 The mini-courses will be in Thackeray Hall 704 in the Department of Mathematics. Morning Session Feimin Huang Mini-Course on Compensated Compactness Method 9:00-10:30am Compensated compactness method and applications (1) 10:30-11:10am Coffee break K 11:10am-12:30pm Compensated compactness method and applications (2) 12:30-2:00pm Lunch break Afternoon Session Deane Yang Mini-Course on Nash Moser Method 2:00-3:30pm The Nash-Moser implicit function theorem and its applications to PDE s (1) 3:30-4:00pm Coffee break K 4:00-5:30pm The Nash-Moser implicit function theorem and its applications to PDE s (2)

3 Conference on PDEs and Free Boundary Problems University of Pittsburgh, March 11-14, 2015 Schedule for Research Talks, March 12-14, 2015 All talks are in Thackeray Hall 704 in the Department of Mathematics. Morning Session Thursday, 3/12 Friday, 3/13 Saturday, 3/14 Chair: M. Slemrod Chair: A. Scheel Chair: E. Feireisl 9:00-9:50am C. Dafermos C. Zeng Y. Zheng 9:50-10:40am M. Torres J. Clelland P. Miller 10:40-11:10am Coffee break K 11:10am-12:00pm G.-Q. Chen M. Slemrod F. Huang 12:00-2:00pm Lunch break Afternoon Session Chair: C. Dafermos Chair: P. Miller Chair: G.-Q. Chen 2:00-2:50pm A. Ionescu A. Scheel E. Feireisl 2:50-3:40pm A. Vasseur K. Trivisa F. Li 3:40-4:10pm Coffee break K 4:10-5:00pm M. Feldman D. Yang S. Walsh 5:00-5:30pm* M. Wheeler F. Yu C. Yu 5:30-6:00pm* G. Jaramillo S. Li H. Ying Banquet, 6:30pm, 2nd floor, University Club *Talks by postdocs/students.

4 4 Thursday, March 12 8:50-9:00am: Welcome and opening remarks Morning Session Chair: Marshall Slemrod 9:00-9:50: Constantine Dafermos, Brown University BV solutions to hyperbolic systems of balance laws with relaxation in the absence of conserved quantities 9:50-10:40: Monica Torres, Purdue University Characterizations of measures in the dual of BV 10:40-11:10: Coffee Break K 11:10-12:00: Gui-Qiang Chen, University of Oxford, UK Some free boundary problems in shock reflection/diffraction and related transonic flow problems 12:00-2:00pm: Lunch Break Afternoon Session Chair: Constantine Dafermos 2:00-2:50: Alexandru Ionescu, Princeton University On the regularity of certain water wave models in 2D 2:50-3:40: Alexis Vasseur, University of Texas at Austin Global weak solutions to the inviscid 3D quasi-geostrophic equation 3:40-4:10pm: Coffee Break K 4:10-5:00: Mikhail Feldman, University of Wisconsin at Madison Shock reflection and von Neumann conjectures: existence and properties of solutions 5:00-5:30: Miles Wheeler, New York University The slope of steady water waves with vorticity 5:30-6:00: Gabriela Jaramillo, University of Minnesota Pacemakers in a large array of oscillators 6:30pm: Banquet, Gold Room, 2nd floor, University Club.

5 5 Friday, March 13 Morning Session Chair: Arnd Scheel 9:00-9:50: Chongchun Zeng, Georgia Institute of Technology Wind-driven water waves and instabilities of the Euler equation 9:50-10:40: Jeanne Clelland, University of Colorado at Boulder Isometric embedding via strongly symmetric positive systems 10:40-11:10: Coffee Break K 11:10-12:00: Marshall Slemrod, University of Wisconsin at Madison Friedrichs positive symmetric systems and isometric embedding the case of higher space dimensions 12:10-2:00pm: Lunch Break Afternoon Session Chair: Peter Miller 2:00-2:50: Arnd Scheel, University of Minnesota Pinning and unpinning in nonlocal equations 2:50-3:40: Konstantina Trivisa, University of Maryland On a nonlinear model for tumor growth with drug application 3:40-4:10pm: Coffee Break K 4:10-5:00: Deane Yang, New York University School of Engineering The logarithmic Minkowski problem 5:00-5:30: Fang Yu, Pennsylvania State University Stability of vortex sheets for three dimensional compressible steady Euler flows 5:30-6:00: Siran Li, University of Oxford A generalised div-curl lemma and its application to isometric immersions of higher dimensional Riemannian manifolds

6 6 Saturday, March 14 Morning Session Chair: Eduard Feireisl 9:00-9:50: Yuxi Zheng, Penn State University No blow-up to a variational wave equation 9:50-10:40: Peter Miller, University of Michigan Semiclassical initial-boundary value problems for the defocusing nonlinear Schrödinger equation 10:40-11:10: Coffee Break K 11:10-12:00: Feimin Huang, Chinese Academy of Sciences Sonic-subsonic limit of approximate solutions to multidimensional steady Euler equations 12:10-2:00pm: Lunch Break Afternoon Session Chair: Gui-Qiang Chen 2:00-2:50: Eduard Feireisl, Czech Academy of Sciences Weak and strong solutions to problems arising in fluid mechanics 2:50-3:40: Fucai Li, Nanjing University, China Low Mach number limit to the compressible magnetohydrodynamic equations 3:40-4:10pm: Coffee Break K 4:10-5:00: Samuel Walsh, University of Missouri Instability of traveling water waves with compact vorticity 5:00-5:30: Cheng Yu, University of Texas at Austin Existence of global weak solutions for the compressible Navier-Stokes equations with degenerate viscosity 5:30-6:00: Hao Ying, Ohio State University Shock formation at the sonic line THE END.

7 7 Conference on PDEs and Free Boundary Problems University of Pittsburgh, March 11-14, 2015 Abstracts Mini-Course on Compensated Compactness Method by Feimin Huang Title: Compesated compactness in fluid dynamics Abstract: In this mini-course, I will briefly introduce the thoery of compensated compactness and its application in fluid dynamics, in particular the hyperbolic system of conservation laws. Mini-Course on Nash-Moser Method by Deane Yang Title: The Nash-Moser implicit function theorem and its applications to PDE s Abstract: The Nash-Moser implicit function theorem was originally part of Nash s proof of his isometric embedding theorem. Later, Moser stated and proved a standalone version that could be applied to other problems. The implicit function theorem demonstrates the existence of smooth solutions to a nonlinear PDE, when the linearized equation can be solved but without the regularity estimates required for the standard Banach space implicit function theorem. A simplified statement and proof due to J. T. Schwartz and F. Sergeraert will be presented. Applications of the theorem to nonlinear PDE s will also be discussed. Jeanne Clelland, University of Colorado Title: Isometric embedding via strongly symmetric positive systems Abstract: (Joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang) In this talk, I will give an outline of our new proof for the local existence of a smooth isometric embedding of a smooth 3-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6-dimensional Euclidean space. Our proof avoids the sophisticated microlocal analysis used in earlier proofs by Bryant-Griffiths-Yang and Nakamura- Maeda; instead, it is based on a new local existence theorem for a class of nonlinear, first-order PDE systems that we call strongly symmetric positive. These are a subclass of the symmetric positive systems, which were introduced by Friedrichs in order to study certain PDE systems that do not fall under one of the standard types (elliptic, hyperbolic, and parabolic). As in earlier proofs, we construct solutions via the Nash-Moser implicit function theorem, which requires showing that the linearization of the isometric embedding PDE system near an approximate embedding has a smooth solution that satisfies smooth tame estimates. We accomplish this in two steps: (1) Show that the approximate embedding can be chosen so that the reduced linearized system becomes strongly symmetric positive after a carefully chosen change of variables. (2) Show that any such system has local solutions that satisfy smooth tame estimates.

8 8 The main advantage of our approach is that step (2) is much more straightforward than similar results for other classes of PDE systems used in prior proofs, while step (1) requires only linear algebra. The talk will focus on the main ideas of the proof; technical details will be kept to a minimum. Gui-Qiang Chen, University of Oxford, UK Title: Some free boundary problems in shock reflection/diffraction and related transonic flow problems Abstract: Shock waves are steep wave fronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this talk, we will first show how several longstanding shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches, and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann s problem for shock reflection-diffraction by two-dimensional wedges with concave corner, Lighthill s problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer s problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws. This talk is based on joint works with Mikhail Feldman, as well as Myoungjean Bae and Wei Xiang. Constantine Dafermos, Brown University Title: BV solutions to hyperbolic systems of balance laws with relaxation in the absence of conserved quantities Eduard Feireisl, Academy of Sciences of the Czech Republic Title: Weak and strong solutions to problems arising in fluid mechanics Abstract: We discuss solvability of certain problems arising in the dynamics of inviscid fluids in the framework of weak and strong solutions. In particular, the following topics will be addressed: (1) The property of weak-strong uniqueness (2) Uniqueness/multiplicity of weak solutions for given data (3) Admissibility criteria, relative energy functionals Mikhail Feldman, University of Wisconsin at Madison Title: Shock reflection and von Neumann conjectures: existence and properties of solutions Abstract: We discuss free boundary problems for elliptic equations, motivated by shock reflection problem for compressible fluid flow. When a shock hits a convex wedge, shock reflection-diffraction phenomena occur, and various reflection patterns are observed. We will discuss von Neumann conjectures on transition between regular and Mach reflections, and recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type.

9 We also discuss properties of solutions, including their regularity and convexity of the free boundary. The talk is based on the joint works with Gui-Qiang Chen, as well as Myoungjean Bae and Wei Xiang. Feimin Huang, Chinese Academy of Sciences Title: Sonic-subsonic limit of approximate solutions to multidimensional steady Euler equations Abstract: A compactness framework is established for approximate solutions to sonicsubsonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework applies for both non-isentropic and rotational flows. One of our main observations in this paper is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional sonic-subsonic full Euler flows through infinitely long nozzles. Alexandru Ionescu, Princeton University Title: On the regularity of certain water wave models in 2D Abstract: I will discuss some recent work on the local and the global regularity of several water wave models in 2 dimensions. The results concern the pure gravity model, the capillary waves equation, and the two-fluid model. Gabriela Jaramillo, Title: Pacemakers in a large array of oscillators University of Minnesota Abstract: We look at a one dimensional array of oscillators with nonlocal coupling and analyze the pacemaker effects of an algebraically localized heterogeneity. We assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be modeled by a continuous nonlocal evolution equation. Thinking of the heterogeneity as a perturbation, we show that steady solutions exist and that the heterogeneity behaves as a wave source. These results are obtained using a series of isomorphisms which connect the nonlocal problem to the viscous Eikonal equation and then using the Fredholm properties of the Laplace operator in Kondratiev spaces to obtain solutions. Fucai Li, Nanjing University, China Title: Low Mach number limit to the compressible magnetohydrodynamic equations Abstract: In this talk we report our recent results on the low Mach number limit to the compressible magnetohydrodynamic equations. Siran Li, University of Oxford Title: A generalised div-curl lemma and its application to isometric immersions of higher dimensional Riemannian manifolds 9

10 10 Abstract: In this talk we sketch a proof for a generalised version of the Div-Curl Lemma. As a central tool in the weak convergence methods of nonlinear PDEs, the classical Div- Curl Lemma guarantees the convergence in distribution of inner products of two weakly convergent sequences of vector fields in R 3, one with compactly confined divergence and the other with compactly confined curl. We generalise this result to suitable differential forms on arbitrary n-dimensional Riemannian manifolds, based on the Hodge Theory for the Laplace-Beltrami operator. Notably, our proof is global and geometric in nature. Moreover, we apply our result to the analysis of the Gauss-Ricci-Codazzi Equations for n- dimensional manifolds, which has implications to the isometric immersion problem. This project is carried out under the supervision of, and in collaboration with, Prof. Gui-Qiang G. Chen. Peter Miller, University of Michigan Title: Semiclassical initial-boundary value problems for the defocusing nonlinear Schrödinger equation Abstract: The so-called unified transform method is a variant of the inverse-scattering transform that is tailored for mixed initial-boundary value problems for integrable partial differential equations possessing Lax pairs. It is based on the use of both equations of the Lax pair to deduce spectral transforms of the initial and boundary data, and it leads to a Riemann-Hilbert problem of inverse scattering whose solution encodes that of the initialboundary value problem. The main difficulty with the method is that computation of the spectral transforms requires knowledge of more boundary conditions than are needed to make the problem well-posed. Rather than resort to the global relation, an identity satisfied by the spectral transforms of consistent boundary data in the transform domain but one that is not easy to handle in practice, we propose an explicit approximation to the nonlinear Dirichlet-to-Neumann mapping for the defocusing nonlinear Schrödinger equation that is expected to be meaningful in the semiclassical limit and enables the explicit elimination of the unknown boundary values. We use this approximation to generate an approximate solution of the mixed initial-boundary value problem, and we study it near the initial time and near the boundary in the semiclassical limit. In particular, we prove the existence of a vacuum domain, an unbounded region of the (x, t)-plane in which the solution is small given homogeneous initial data. We analyze the solution in the vacuum domain using the steepest descent method, a variant of the steepest descent method for Riemann-Hilbert problems that works even when jump matrices fail to be analytic. This is joint work with Zhenyun Qin (Fudan University). Arnd Scheel, University of Minnesota Title: Pinning and unpinning in nonlocal equations Abstract: I ll present recent work on pulses and fronts in systems with nonlocal coupling. I ll first discuss pinning and unpinning of fronts in bistable nonlocal Allen-Cahn-type systems. Near the Maxwell point, that is, when potential energies of the asymptotic equilibria are close, interfaces are often discontinuous and cannot propagate: they are pinned. We study these pinning regions and asymptotics for speeds near the pinning region theoretically and numerically. Our main result shows that speeds obey an unusual but universal s µ 3/2 asymptotic which is different from conventional µ 1/2 asymptotics in discrete systems. We also give some motivation and speculation how speed asymptotics

11 may depend in a universal fashion on kernel regularity properties. As time allows, I ll explore some of the techniques involved in the study of such traveling wave problems. We summarize some of the classical geometric singular perturbation techniques that are used when the problem can be reduced to a local traveling-wave equation, and explain how those techniques translate into methods for nonlocal problems. This is joint work with Gregory Faye (EHESS, Paris). Marshall Slemrod, University of Wisconsin at Madison Title: Friedrichs positive symmetric systems and isometric embedding the case of higher space dimensions Abstract: In our recent paper the authors (Chen, Clelland, Slemrod, Wang, Yang) have shown that an extremely delicate argument allows the application of the theory of strongly positive symmetric systems to prove local existence of an isometric embedding of a three dimensional Riemannian manifold into six dimensional Euclidean space. This raises the question as what can be done in the more general case of embedding an n dimensional Riemannian manifold into n(n + 1)/2 dimensional Euclidean space, i.e. a Euclidean space of critical Janet dimension where the underlying PDEs form a determined system. In this talk I ll address recent results of the authors and conclude that there exist closed subsystems of the relevant symmetric quasi-linear PDEs (the equations of Gauss-Codazzi- Ricci) for which their linearizations possess weak solutions. The proof again uses Friedrichs idea but it is perhaps a bit unusual in that uses ideas from linear programming which seems a technique not in the usual PDE bag of tricks. Monica Torres, Purdue University Title: Characterizations of measures in the dual of BV Abstract: For a bounded open set Ω with Lipschitz boundary, we characterize the measures in the dual space BV 0 (Ω). We make precise the definition of BV 0 (Ω), which is the space of functions of bounded variation with zero trace on the boundary of Ω. This result extends, to bounded domains, a previous characterization of the signed measures in R n that belong to BV (R n ) obtained by the authors and a characterization of the positive measures in BV (R n ) obtained by Meyers and Ziemer. We also discuss the space BV n (Rn ), defined as the space of all functions u in L n n 1 n 1 (R n ) such that Du is a finite vector-valued measure. We show that BV (R n ) and BV n (Rn ) are isometrically isomorphic. As a consequence of our characterizations, an old issue raised by Meyers and Ziemer n 1 is resolved by constructing a locally integrable function f such that f belongs to BV (R n ) but f does not. Moreover, we show that the measures in BV n (Rn ) coincide with the n 1 measures in Ẇ 1,1 (R n ), the dual of the homogeneous Sobolev space Ẇ 1,1 (R n ), and that the measures in BV 0 (Ω) coincide with the measures in W 1,1 0 (Ω). Finally, the class of finite measures in BV (Ω) is also characterized. We remark that the space Ẇ 1,1 (R n ) is used in image processing to model the noise of an image. Moreover, the full characterization of the space BV is still an open problem in geometric measure theory. Konstantina Trivisa, University of Maryland Title: On a nonlinear model for tumor growth with drug application 11

12 12 Abstract: This work deals with the investigation of the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and extra-cellular cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Brinkman s equation. Tumor is viewed in this setting as a growing continuum such as both the domain occupied by the tumor as well as its boundary evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data. Alexis Vasseur, University of Texas at Austin Title: Global weak solutions to the inviscid 3D quasi-geostrophic equation Abstract: We will show the existence of global weak solutions to the inviscid threedimensional quasi-geostrophic system of equations. This system of equations models the evolution of the temperature in the atmosphere. It is widely used in geophysics and meteorology. It involves a coupling between a transport equation in the whole domain, and an other one on the boundary of the domain, at the surface of the earth. This is a joint work with Marjolaine Puel. Samuel Walsh, University of Missouri Title: Instability of traveling water waves with compact vorticity Abstract: In this talk, we present some recent results on the stability of traveling waves with a point vortex. These are steady (weak) solutions of the 2-d incompressible free boundary Euler equations whose vorticity is a Dirac δ-measure. We show that, unlike most traveling rotational waves, this system admits a (non-canonical) Hamiltonian formulation: they can be realized as the constrained minimizers of an energy subject to fixed momentum. Using this fact, we are able to prove that small-amplitude and slow moving traveling waves with a point vortex are (orbitally) unstable. Our method involves a nontrivial generalization of the seminal results of Grillakis Shatah Strauss. This is joint work with E. Wahlén. Mile Wheeler, New York University Title: The slope of steady water waves with vorticity Abstract: We consider the angle between the free surface of a steady two-dimensional water wave and the horizontal. In the absence of vorticity, McLeod proved that this angle exceeds 30 degrees for some waves, while Amick proved that it is always bounded above by degrees. With adverse vorticity, on the other hand, there is numerical evidence that steady waves can become much steeper and even overturn. We prove an upper bound of 45 degrees for a large class of waves with favorable vorticity, in particular for constant vorticity of the appropriate sign. We also prove that any overturning wave must have a pressure sink. This is joint work with Walter Strauss (Brown University).

13 Deane Yang, Title: The logarithmic Minkowski problem NYU Polytechnic School of Engineering Abstract: The logarithmic Minkowski problem, like the classical Minkowski problem, is a fundamental question in the affine geometry of convex bodies. For symmetric convex bodies, it asks what are the necessary and sufficient conditions for a measure in (n 1)- dimensional projective space to be the cone-volume measure of the unit ball in an n- dimensional normed space. Solving this problem is equivalent to establishing existence of a solution to a Monge-Ampere equation. This talk outlines a complete solution to the symmetric logarithmic Minkowski problem and will present related open problems. Hao Ying, Title: Shock formation at the sonic Line Ohio State University Abstract: We study a problem of self-similar unsteady transonic small disturbance equation which contains a shock forming at the sonic line. This is a mixed elliptic-hyperbolic type problem where the elliptic region and hyperbolic region are separated by a shock and a sonic line. It can be reformulated as a free boundary problem for nonlinear degenerate elliptic PDE of second order. The interesting feature of this problem is that at the shock formation point, the elliptic PDE becomes degenerate and at the same time the oblique derivative condition we pose on the free boundary becomes tangential. We establish a result on existence of the solution to this configuration, and we present an analysis to understand the solution structure of this problem. Cheng Yu, University of Texas at Austin Title: Existence of global weak solutions for the compressible Navier-Stokes equations with degenerate viscosity Abstract: In this work, we prove the existence of global weak solutions for compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation. The main contribution of this paper is to derive the Mellet-Vasseur type inequality for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any γ > 1 in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions. This is a joint work with Alexis Vasseur. Fang Yu, Pennsylvania State University Title: Stability of vortex sheets for three dimensional compressible steady Euler flows Abstract: In this talk, we discuss the nonlinear structural stability of contact discontinuities in three dimensional compressible isentropic steady Euler equations. First, we obtain a necessary and sufficient condition for the linear stability of supersonic planar contact discontinuities and a priori estimates of solutions to the linearized problem. Moreover, using the calculus of paradifferential operators and taking advantage of vorticities of velocity fields, we get higher order estimates of solutions to the linearized problem of a non-planar contact discontinuity. As there is a loss of regularity in the estimates of solutions to the linearized problem, we adapt the Nash-Moser-Hörmander iteration scheme to obtain the 13

14 14 nonlinear stability of supersonic contact discontinuities in three dimensional compressible isentropic steady Euler equations. This is a joint work with Ya-Guang Wang from Shanghai Jiao Tong University. Chongchun Zeng, Georgia Institute of Technology Title: Wind-driven water waves and instabilities of the Euler equation Abstract: In this talk, we discuss the mathematical theory of wind-generated water waves in the framework of the interface between two incompressible inviscid fluids under the influence of gravity. This entails the careful study of the stability of the shear flow solutions to the interface problem of the two-phase Euler equation. Based on a rigorous derivation of the linearized equations about an arbitrary steady solution, we obtained rigorously the linear instability criterion of Miles due to the presence of the critical layer in the steady shear flows. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air sea interface). This is a joint work with Bühler, Shatah, and Walsh. Yuxi Zheng, Pennsylvania State University Title: No blow-up to a variational wave equation Abstract: We establish the global existence of smooth solutions to the Cauchy problem for a system of variational wave equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and bend coefficients. Joint work with Jingchi Huang.