The following list is NOT time-ordered. See the Program schedule, or return to Local Information.
L.K. Grover's quantum search algorithm gives a favorable quadratic speedup in the search of a single object in a large unsorted database. In this talk, we generalize Grover's algorithm to multiobject search in a Hilbert space and operator-theoretic framework along the lines of a recent paper by Farhi and Gutmann ([Phys. Rev. A 57(1998), pp.2403-2405]). The model is an ODE version of the Schrodinger PDE in quantum mechanics. The ideas of amplitude amplification and reducibility to lower dimensional subspaces will be used.The work is based on part of a new manuscript by G. Chen and S.A. Fulling which is available in the speaker's website.
When multiple solutions exist in a nonlinear variational PDE or dynamic system, some are stable and others are unstable. The Morse index and local linking index can be used to measure an instability of a solution. However, Morse index is ineffective in measuring the instability of a degenerate solution, and it is too expensive to compute either the Morse index or the local linking index in addition to numerically compute such unstable solution. We developed a local minimax method to numerically find such unstable solutions in a stable way. Use our local minimax method, we define a local minimax index to measure the instability of a solution which can be degenerate. The local minimax index generalizes the notion of the Morse index and the local linking index in instability analysis and is known even before we numerically compute the solution. Its relations with the Morse index and the local index are established.
Suppose T is a semigroup as in the title. Following recent work by J.R. Dorroh and this writer, T is extended to a semigroup on a space of measures on the metric space X (identifying points of X with Dirac measures on X). A generator is defined for the extended semigroup and it is shown how to recover the extended semigroup, and hence T, from the generator. A complete characterization of such generators is given.
We study the rate of decay to equilibrium of a class of nonlinear degenerate parabolic equations using the entropy dissipation method. We prove the exponential decay of the relative entropy and the entropy dissipation and as a consequence the exponential decay of the L^1 norm. These equations include the porous medium equation, the fast diffusion equation by a suitable change of variables and the linear homogeneous Fokker-Planck equation.
We show that the solution of a Cauchy problem for quasilinear, hyperbolic equations having nonlinear damping, asymptotically approaches the solution of a corresponding nonlinear parabolic equation as time tends to infinity. The hyperbolic equation is derived from a system of evolution equations describing propagation of thermal waves.
We examine the behavior of an incompressible fluid in an infinitely long, porous channel. The Proudman-Johnson model represents the Navier-Stokes equations in this situation, and we consider the influence of the boundary conditions on the lifetime of solutions.
Solitary waves play a distinguished role in the long-term evolution of nonlinear dispersive waves. A discussion will be initiated concerning existence and especially the asymptotic properties and regularity of solitary-wave solutions of a general class on nonlinear wave equations. The results apply equally to the full two-dimensional Euler equations for the propagation of waves on the surface of an ideal fluid.
The equations for rotating MHD confined to a cylindrical shell of infinite extent in the axial direction yield a simplified mathematical model for resistive instability in the Earth's outer core. This model leads to a boundary value problem for a fourth order ordinary differential equation with rational function coefficients. Solutions to this equation are approximated asymptotically by assuming certain parameters of the problem to be large. These approximations have interesting boundary layer structures and are used to approximate the complex frequencies which determine the stability of the ambient terrestrial magnetic field.
We consider the problem of determining a radially-symmetric potential in the three dimensional Schr\"odinger equation from data consisting of eigenvalues associated with spectral sequences from two different angular momentum quantum numbers. In recent years enormous numbers of quite accurate eigenvalues of the sun have been compiled. Our problem represents one of the simplest that model the determination of certain physical parameters in the standard model of the sun from eigenvalue data. This leads to a highly singular eigenvalue problem for which there are no known uniqueness results to the most reasonable conjectures. We are able to give strong evidence for uniqueness in many cases and discuss a computational solution method together with some supporting analysis.
As is well-known, the analysis of the absolutely continuous spectrum of the Schroedinger equation for certain potentials leads to the study of the Lippmann-Schwinger integral equation. For certain potentials, solutions of this equation are known via techniques involving Hilbert-Schmidt operators and the Fredholm Alternative Theorem. Here, we investigate the Lippmann-Schwinger equation for weak solutions also via the Fredholm Alternative Theorem but using estimates of operator norms other than the Hilbert-Schmidt norm, thereby enlarging the class of potentials with which we may analyze such solutions. Some asymptotic estimates of weak solutions are also given for certain cases.
This will be a survey of results on the symmetry of energy minimizing configurations of many-particle systems.
The geometric non-linear Schroedinger equation arises as in a simplification of the Landau-Lifschitz equations for a macroscopic ferro-magnetic continuum. The continuum is taken to be $R^n$, where $ n=1,2,3$ and the equation is written \[ \frac{\partial}{\partial t}U = U\times\Delta U \] for $U = (u_1,u_2,u_3)$ satisfying $u_1^2 + u_2^2 + u_3^2 = 1$. For the domain $R^1$, this equation is equivalent to the integrable (focusing) non-linear Schroedinger equation. There is also an interpretation as the equation governing the tangent of a curve in $R^3$ propagating in the direction of the binormal with speed equal to the curvature. The equation itself can be thought of as a non-linear Schroedinger equation governing the flow of a map from $R^n$ to $S^2$, and the target manifold $S^2$ can be replaced by any Kaehler manifold, yielding a family of non-linear Schroedinger equations whose behavior depends both on the dimension of the independent variable and on the geometry of the target. This equation belongs to the sequence of equations based on harmonic maps, including the harmonic map equation, the heat flow for the harmonic map equation and wave maps. We discuss the well-posedness of this equation, and some of the basic estimates which are used in proving long-time existence for non-linear Schroedinger equations.
We discuss the recent study (a joint work with S. Canic and B. Keyfitz) of a transonic regular reflection problem for the unsteady transonic small disturbance equation, using a free boundary problem approach. Our method applies to self-similar shock reflection when the incident shock angle is large enough to permit a regular reflection configuration with a subsonic state behind the reflected shock. For the small-disturbance approximation in weak shock reflection, this corresponds to relatively large wedge angles. One contribution of our study is the development of an asymptotic formula for the reflected shock, far from the reflection point, and for the subsonic state far downstream. These asymptotic series are valid for the small-disturbance approximation, for any incident shock angles.The main focus in this talk is an existence result for the nonuniform subsonic flow behind the reflected shock. The flow velocity satisfies a quasilinear elliptic equation which is coupled to the Rankine-Hugoniot equations for the reflected shock, forming a free boundary problem on part of the boundary. We prove that the flow we have constructed solves the original problem in a domain of small finite size around the reflection point.
The concept of multimodel coupling raises many interesting physical, mathematical and computational issues. The domain decomposition-based implementation has been recently developed under the reservoir simulator framework IPARS at Center for Subsurface Modeling in TICAM, UT,In the talk we discuss the coupling of single phase flow equations with two phase flow model. The former describes the flow of water in an aquifer and the latter describes the oil and water flow in a reservoir connected to the aquifer by a (fixed) interface. The coupling across interface preserves mass and momentum. From mathematical point of view the coupling expresses the matching or approximate matching between the single phase flow equations and the (degenerate) limit of two phase flow equations in case the oil phase is missing.
Consider positive solutions to nonlinear elliptic equation : $\Delta u+\beta(u)=0$ in the upper half space $H=\{(x', x_n) \ \ | \ \ x_n>0\}$ with Dirichlet boundary condition $u=0$. Under certain assumptions on $\beta(s)$ we show that bounded solution is unique and is a function of $x_n$ alone. Here we present a new approach to obtain such classification results. This method also allows us to obtain similar results on the problem with Neumann boundary conditions.
We present in this talk the recent results on the zero-dissipation limit of the Navier-Stokes equations in the spatial domain of a unit disk.
In previous joint work of A. Castro, J. Cossio and J. M. Neuberger it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of $-\Delta$. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the $k$-th and $k+1$-st eigenvalues there still exists a solution which changes sign at most $k$ times. In particular, when $k=1$ the sign-changing {\it exactly-once } solution persists although one-sign solutions no longer ex$ This is joint work with Pavel Drabek (University of West Bohemia) and John M. Neuberger (Northern Arizona University).
We establish the existence, uniqueness and stability of almost periodic traveling wave solutions for nonlocal evolution equations with bistable nonlinearity.
This talk is about the regularity of the scalar and the vector potentials in Hodge_Weyl decomposition for smooth vector fields on a bounded region in $\mathbb{R}^3$ with smooth boundary. These results will be used to describe some simple equivalent norms for vector-valued Sobolev spaces. These equivalent norms primarily involve Sobolev norms of curl(u) and dvi(u).
The KdV equation which has been derived as a model for surface water waves is thought to constitute an infinite-dimensional integrable system. We consider three similar equations that arise in the modeling of internal waves. We focus on properties of solitary-wave solutions, showing numerical simulations which suggest that two of these equations do not form integrable systems.
We give sufficient conditions to show that the sign-changing solution given by the minmax principle developed by A. Castro, J. Cossio, and J. M. Neuberger in Rocky Mountain J. Math., Vol. 27, 4 (1997), is non-radial. This minmax principle was motivated by the interest of proving the existence of sign-changing solutions to equations such as $\Delta u + f(u) = 0$ in $\Omega$, $u = 0$ on $\partial \Omega$. Here $\Omega$ is a smooth bounded region in $R^n$, $f$ is a function of class $C^1$ satisfying $f(0) = 0$, $f$ is superlinear, subcritical, and $f'(u) > f(u)/u$ for $u \not = 0$. This is joint work with A. Castro and C. Mejia.
Existence, uniqueness and regularity theory will be developed for a general initial-boundary-value problem for a system of partial differential equations which describes diffusion in a deformable composite porous medium. The system will be resolved as a linear implicit evolution equation in Hilbert space. This is joint work with R.E. Showalter.
We will discuss L^\infty estimate for u_t and L^p estimate for D^2 u for the Parabolic Monge-Ampere equation -u_t\det D^2u=f which arises in some geometric problems such as deformation of surfaces. Parabolic sections are introduced and their basic geometric properties are established. We use parabolic sections as a tool to obtain L^p estimate of D^2u.
We study the general forced pendulum equation in the presence of friction, $$u'' + a(t)u' + b(t) \sin u = f(t)$$ with $a,b\in C([0,T])$ and $f\in L^2(0,T)$. $T$-periodic solutions may be obtained as zeroes of a $2\pi$-periodic continuous real function. Furthermore, the existence of infinitely many solutions is proved under appropiate conditions on $a,b$ and $f$.
In order to calculate the magnetic properties of a molecule in presence of an external magnetic field, we find a gauge that minimizes locally the paramagnetic contribution to the energy functional.
We derive a system of first order hyperbolic conservation laws which describe the motion of a hyperelastic material with polyconvex stored energy. We consider such materials with variable as well as constant temperature. We show that these equations have a convex extension. Consequently they are symmetric-hyperbolic and the Cauchy problem for this system is well posed for small smooth initial data.
We will present a mathematical model describing the critical situation of a boiling fluid, when the fluid is detaching from the heating surface. Then we discuss known mathematical results and open problems.
Long time dynamics of solutions to a strongly coupled system of parabolic equations modelling the competition in bio-reactors with chemotaxis will be studied. If the parameters of the system are periodic, sufficient conditions for positive periodic solutions will be derived in terms of parabolic eigenvalue problems.
In the 1870's Boussinesq derived a system of equations to model surface water waves, under the assumption that the waves being modelled are long and weakly nonlinear. Under the additional assumption that the waves being modeled are unidirectional, he showed that his system reduced to an equation which was rediscovered twenty years later by Korteweg and de Vries, and which today bears their names. We prove that, for a certain regularized version of the Boussinesq system, unidirectional long weakly nonlinear solutions exist over long time scales, and do indeed resemble solutions of the Korteweg-de Vries equation. This is joint work with Abdulrahman Al-Azman, Jerry Bona, and Min Chen.
In numerical simulation of incompressible fluid flows, vorticity boundary values are usually needed to decouple the equations for vorticity and stream-function. We derive an explicit formula for it for Stokes equations in circular regions. Numerical experiments based on it are presented.
New variational formulation to compute propagation constants is proposed. Based on it, vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property. Convergence analysis is conducted in the framework of collectively compact operators. Reported theoretical results apply to a wide class of vector finite elements including two families of Nedelec and their generalization, the hp-edge elements. Numerical experiments fully support theoretical estimates for convergence rates. This is joint work with Leszek Demkowicz.
We consider the compressible Navier-Stokes equations for a p'th power gas law. We obtain new a priori bounds on the density for the Dirichlet initial-boundary value problem for large data. This work is part of a broader program aimed at understanding the temporal asymptotics for materials which may change phase; e.g. Van-der-Waals fluid.
Long waves $y=h\eta(x)$ of speed $c$ on the interface $y=0$ between a thin lower layer of heavy fluid (density $\rho_1$) and a light upper fluid (density $\rho_2<<\rho_1$) extending to $y=+\infty$ satisfy % \begin{equation} -2\left[\frac{c}{c_0}-1\right]\eta(x) +\frac{3}{2}\bigg[\eta(x)\bigg]^2 +\frac{1}{3}h^2\eta''(x) +\frac{\rho_2}{\rho_1}h{\cal H}\eta'(x) =\frac{3}{2}\overline{\eta^2} \end{equation} where $\cal H$ is a Hilbert transform, $c_0=\sqrt{(1-\rho_2/\rho_1)gh}$ is the linear wave speed, and an overbar denotes an average. Comparison is made for periodic waves between numerical results from the above equation and numerical solutions of the Laplace equation subject to the full nonlinear free-surface conditions, and also with analytic solutions of the Korteweg-de Vries and Benjamin-Ono equations which are special cases corresponding to omission of one of the dispersion terms involving $\cal H$ and $\eta''$ respectively.
We consider the standard finite element methods based on a highly nonuniform anisotropic meshes, which does not satisfy the "quasi-uniformity" (which is required by the classic finite element analysis). Through detailed analysis for the derivatives of the analytic solution, we obtain the optimal L^2 uniform convergence for the first time, which improves the results of Apel and Lube (1998) and solves some open problems posed by Roos (1998). Numerical results are also provided.
A steady-state hydrodynamic model for quantum fluids is analyzed. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects are incorporated. The phenomena that admit positivity of the solutions are studied. The cases: dispersive or non-dispersive, viscous or non-viscous are thoroughly analyzed with respect to positivity and existence or non-existence of solutions, in one space dimension. It is proven that in the dispersive, non-viscous model, a classical positive solution only exists for ``small'' (positive) particle current densities and no weak solution can exist for ``large'' current densities, whereas the dispersive, viscous problem admits a classical positive solution for all current densities. These results are valid under appropriate assumptions on the pressure function. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. The results are extended to general third-order equations in any space dimension depending on the boundary conditions. This is joint work with Ansgar Juengel.