We condider the aggregation equation with nonnegative initial data in L^1(R^n) \cap L^\infty(R^n) for n \ge 2. We assume that K is rotationally invariant, nonnegative, decaying at infinity, with at worst a Lipschitz point at the origin. We prove existence, uniqueness, and continuity of solutions. Finite time blow-up (in the L^\infty norm) of solutions is proved when the kernel has precisely a Lipschitz point at the origin.

Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demonstrate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catastrophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations.

With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.

We investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. In the steady-state case, the well-posedness is established under the assuumption of small coefficient in the Beavers-Joseph interface boundary condition. In the time-dependent case, the well-posedness is established via appropriate time discretization of the problem and a novel scaling of the system under isotropic media assumption. Such coupled systems are crucial to the study of subsurface flow problems, in particular, flows in karst aquifers.

A three-mode nonlinear slow-fast system with the fast forcing is studied here as a model for filtering turbulent signals from partial observations. The model describes the interaction of two extremely driven fast modes with a slow mode through catalytic nonlinear coupling. The speical structure of the nonlinear interaction allows for the analytical solution for the first and second order statistics even with fast forcing. These formulas are used for testing the exact Extended Kalman Filter for the slow-fast system with fast forcing. Various practial questions such as the inluence of the strong fast forcing on the slowly varying wave envelope, the role of observation, the frequency and variance of observation, and the model error due to linearization are addressed here.

Squall lines are coherent turbulent travelling waves on scales of order 100 km in the amtosphere that emerge in a few hours from the interaction of strong vertical shear and moist deep convection on scales of order 10 km. They are cononical coherent structures in the tropics and middle lattidudes reflecting upscale conversion of energy from moist buoyant sources to horizontal kinetic energy on larger scales. Here sqauall lines are introduced through high rsolution numerical simlations which reveal a new self-similarity with respect to the shear amplitude. A new muitl-scale model on mesoscales which allows for large vertical shears, appropriate for squall lines, is developed here through systematic multi-scale asymptotics. Mathematial and numerical formulations of the new multi-scale equations are utilized to illustrate both new mathematical and physical phenomena captured by the new models. In partiiculart, non-hydrostatic Taylor-Goldstein equatiosn govern the upscale transports of momentum and temperature from the order 10 km microscales to the order 100 km mesoscales; surprisingly, upright single model convection heating without titls can lead to significant upsale convective momentum transport from the microscales to the mesoscales due to the strong shear. The multi-scale models developed here should be especially useful for dynamics parameterizations of upscale transports as well as for new therory in three-dimensions with a transverse shear component, where comtemporary theoretical understanding in meager.

We propose and analyze an extremely fast, efficient and simple method for solving the problem:
\min { \|u\|_1: Au=f, u \in R^n }.
The motivation was compressive sensing, which now has a vast and exciting history, which seems to have started with Candes, et.al. and Donoho. Our method introduces an improvement called "kicking" of the very efficient method [Darbon and Osher, Yin, Osher, Goldfarb and Darbon], and also applies it to the problem of denoising of undersampled signals. The use of Bregman iteration for denoising of images begin in [Osher, Burger, Goldfarb, Xu and Yin], and led to improved results for total variation based methods. Here we apply it to denoise signals, especially essentially sparse signals, which might even be undersampled.

New linear response formulas for unperturbed chaotic (stochastic) complex dynamical systems with time periodic coefficients are developed here. Such time periodic systems arise naturally in climate change studies due to the seasonal cycle. These response formulas are developed through the mathematical interply between statistical solutions for the time-periodic dynamical systems and the related skew-product system. This interplay is utilized to develop new systematic quasi-Gaussian and adjoint algorithms for calculating the climate response in such time-periodic systems. These new formulas are found in section 4. New linear response formulas are also developed here for general time-dependent statistical ensembles arising in ensemble prediction including the effects of deterministic model errors, initial ensembles, and model noise perturbations simultaneously. An information theoetical perspective is developed in calculating those model perturbations which yield the largest information deficit for the unperturbed system both for climate response and finite ensemble predictions.

We characterize the high intensity limits of minimal free energy states for interacting corpora--that is, for objects with finitely many degrees of freedom, such as articulated rods. These limits are measures supported on zero-level-sets of the interacting potential. We describe a selection mechanism for the limits that is mediated by evanescent entropic contributions.

A stochastic model for representing the missing variability in gloabl climate models due to unresolved features of organized topical convection is presented here. We use a Markov chain lattice model to represent small scale convective elements which inreract with each other and with the large scale environmental variables through convective available potential energy (CAPE) and middle troposphere dryness. Each lattice site is either occupied by a cloud of a certain type (congestus, deep or stratiform) or it is a clear sky site. The lattice sites are assumed to be independent from each other so that a coarse-grained stochastic birth-death system, which can be evolved with a very low computational overhead, is obtained for the cloud area fractions alone. The stochastic multicloud model is then coupled to a simple tropical climate model consisting of a system of ode's, mimicking the dynamics over a single GCM grid box. Physical intuition and observations are employed here to constrain the design of the models. Numerical simulations showcasing some of the dynamical features of the coupled model are presented below.

A methodology is developed to assign, from an observed sample, a joint-probability distribution to a set of continuous variables. The algorithm proposed performs this assignment by mapping the original variables onto a jointly-Gaussian set. The map is built iteratively, ascending the log-likelihood of the observations, through a series of steps that move towards normality the marginal distributions along a random set of orthogonal directions.

We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with perfectly conducting boundary. In the scheme, velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently developed formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray project operators. We prove stability of the time-difference scheme, and deduce a local-time well-posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.

We present a time-dependent semiclassical transport model for coherent pure-state scattering with quantum barriers. The model is based on a complex-valued Liouville equation, with interface conditions at quantum barriers computed from the steady-state Schrodinger equation. By retaining the phase information at the barrier, this coherent model adequately describes quantum scattering and interference at quantum barriers, with a computational cost comparable to that of the classical mechanics. We construct both Eulerian and Lagrangian numerical methods for this model, and validate it using several numerical examples, including multiple quantum barriers.

The Navier-Stokes-Voigt (NSV) model of viscoelastic incompressible fluid has been recently proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we investigate its statistical properties by employing phenomenological heuristic arguments, in combination with Sabra shell model simulations of the analogue of the NSV model. For large values of the regularizing parameter, compared to the Komogorov length scale, simulations exhibit multiscaling inertial range, and the dissipation range displaying low intermittency. These facts provide evidence that the NSV regularization may reduce the stiffness of direct numerical simulations of turbulent flows, with a small impact on the energy containing scales.

An idealized framework to study the impacts of phase transitions on atmospheric dynamics is described. Condensation of water vapor releases a significant amount of latent heat, which directly affects the atmospheric temperature and density. Here, phase transitions are treated by asusming that air parcels are in local thermodynamic equilibrium, which implies that condensed water can only be present when the air parcel is saturated. This reduces the number of variables necessary to describe the thermodynamic state of moist air to three. It also introduces a discontinuity in the partial derivatives of the equation of state. A simplified version of the equation of state is obtained by a separate linearization for saturated and unsaturated parcels. When this equation of state is implemented in a Boussinesq system, the buoyancy can be expressed as a piecewise linear function of two prognostic thermodynamic variables, D and M, and height z. Numerical experiments on the nonlinear evolution of the convection and the impact of laten heat release on the buoyant flux are presented.

We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system, and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type system.

In this note we revisit the homogenization theory of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic environments. We present a new simple proof for the homogenization in probability. The argument uses some a prior bounds (uniform modulus of continuity) on the solution and the convexity and coercivity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula of the solution as was the case with all previously known proofs. We also introduce a new formula for the effective Hamiltonian for Hamilton-Jacobi and "viscous" Hamilton-Jacobi equations.

We investigate noise-induced transitions in non-gradient systems when complex invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number regimes. It is found that before the homoclinic explosion bufurcation, the only transition state is the saddle point, and the transition is similar to that in gradient systems. However, when the chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a relative attractor, plays the role of the most probable exit set in the transition process. This example demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved in the transition process in smooth dynamical systems.

Recently the author developed a numerical method for the multidimensional moment-constrained maximum entropy problem, which is practically capable of solving maximum entropy problems in the two-dimensional domain with moment constraints of order up to 8, in the three-dimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. In this work, the author assembles together key algorithms and observations from his previous works as well as other literature in an attempt to present a comprehensive exposition of the current methods and results for the multidimensional maximum entropy moment problem.

We derive a multi-scale model of moist tropical dynamics which is valid on horizontal synoptic scales, zonal planetary scales, synoptic and intraseasonal time scales. The intraseasonal Multi-Scale Moist Dynamics (IMMD) framework builds on the IPESD framework of (Majda and Klein 2003). It generalizes the latter by allowing for strong zonal winds (the Trade Winds) and the pressure and stratification variations that they generate. The framework consists of three pieces. The first, called TH, are planetary scale climatology modulation equations which govern the Trade Winds and Hadley Circulation. Self-consistency of the asymptotic theory requires that the meridional component of the Hadley Circulation is an order of magnitude weaker than the zonal component. The second piece, S, is a linear system of equations which govern aynoptic scale velocity, temperature and pressure fluctuations forced by synoptic scale heating fluctuations. Unlike the IPESD theory, these fluctuations are advected by part of the planetary scale climatology from TH. Since the meridional component of TH is an order of magnitude weaker than the zonal component, the synoptic scale fluctuations are only advected by the latter. The third, P, govern the planetary scale anomalies which, like IPESD, are driven both by planetary scale mean heating and by upscale flux from the synoptic scales. These planetary scale anomalies are advected, both, by the zonal component of the Trade Winds and by the meridional component of the Hadley Circulation and, furthermore, respond to an in-scale flux from the mean climatology. We also present an asymptotic analysis of the equations of bulk cloud thermodynamics in order to lay out a self-contained path for incorporating synoptic scale cloud models into the IMMD framework. This framework has potentially important implications for the development of models describing the Madden-Jullian Oscillation (MJO) since the MJO manifests itself as planetary scale anomalies from a mean climatology which it modulates on intraseasonal time scales.

We study the impact of stochastic mechanisms on a coupled hybrid system consisting of a general advection diffusion reaction partial differential equation and a spatially distributed stochastic lattice noise model. The stochastic dyamics include both spin-flip and spin-exchange type inter-particle interactions. Furthermore, we consider a new, asymmetric, single exclusion process, studied elsewhere in the context of traffic flow modeling, with an one-sided interaction potential which imposes advective trends on the stochastic. This is our look-ahead stochastic mechanics which is responsible for rich nonlinear behavior in solutions.

Our appraoch relies heavily on first deriving approximate differential mesoscopic equations. These approximations become exact either in the long range, Kac interaction partial differential equation case, or, given sufficient time separation conditions, between the partial differential equation and the stochastic model giving rise to a stochastic averaging partial differential equation. Although these approximations can in some cases be crude, they can still give a first indication, via linearized stability analysis, of the interesting regimes for the stochastic model.

Motivated by this linearized stability analysis we choose particular regimes where interacting nonlinear stochastic waves are responsible for phenomena, such as random switching, convective instability and metastability, all driven by stochasticity. Numerical kinetic Monte Carlo simulations of the coarse grained hybrid system are implemented to assist in producing solutions and understanding their behavior.

We investigate the nonlinear dynamics of inertia-gravity (IG) wave modes in three-dimensional (3D) rotating stratified fluids. Starting from the rotating Boussinesq equations, we derive a reduced partial differential equation system, the GGG model, consisting of only wave-mode interactions. We note that this subsystem conserves energy and is not restricted to resonant wave-mode interactions. In principle, comparing this model to the full rotating Boussinesq system allows us to gauge the importance of wave-vortical-wave vs. wave-wavewave interactions in determining the transfer and distribution of wave-mode energy. As in many atmosphere-ocean phenomena we work in a skewed aspect ratio domain H/L (H and L are the vertical and horizontal lengths) with Fr = Ro < 1 such that Bu = 1, where Fr, Ro and Bu are the Froude, Rossby and Burger numbers, respectively. Our focus is on the equilibration of wave-mode energy and its spectral scaling under the influence of random large-scale (kf ) forcing. We present results from two sets of parameters: (i) Fr = Ro \approx 0.05, H/L =1/5, and (ii) Fr = Ro \approx 0.1, H/L =1/3. As anticipated from prior work, when forcing is applied to all modes with equal weight, with Fr = Ro \approx 0.05 and H/L =1/5, the wave-mode energy of the full system equilibrates and its spectrum scales as a power-law that lies between k^{-1} and k^{-5/3} for kf < k < kd, where kd is the dissipation scale. For the same parameters, when forcing is restricted to only wave modes, the wave-mode energy fails to equilibrate in both the full system as well as the GGG subsystem at the resolutions we can achieve. This clearly demonstrates the importance of the vortical mode (by facilitating wave-vortical-wave interactions) in determining the wave-mode energy in the rotating Boussinesq system. Proceeding to the second set of simulations, i.e. for the larger Fr = Ro \approx 0.1 in a less skewed aspect ratio domain with H/L =1/3, we observe that the energy of the GGG subsystem equilibrates and is resolution independent. Further, the full system with forcing restricted to wave modes also equilibrates and both yield identical power-law scaling of wave-mode energy spectra. Thus it is clear that the wave-wave-wave interactions play a role in the overall dynamics at moderate Ro, Fr and aspect ratios. From a practical standpoint these results highlight the difficulty in properly resolving wave-mode interactions when simulating realistic geophysical phenomena.

We study the diffusion limit of the Vlasov-Poisson-Fokker-Planck system. Here, we generalize the local in time results and the two dimensional results of Poulaud-Soler and Goudon to the case of several space dimensions. Renormalization techniques, the method of moments and a velocity averaging lemma are used to prove the convergence of free energy solutions (renormalized solutions) to the Valsov-Poisson-Fokker-Planck system towards a global weak solution of the Drift-Diffusion-Poisson model.

Synchronous and asynchronous dynamics in all-to-all coupled networks of identical excitatory, current-based, integrate-and-fire (I\&F) neurons with delta-impulse coupling currents and Poisson spike-train external drive are studied. Repeating synchronous total firing events, during which all the neurons fire simultaneously, are observed using numerical simulations and found to be the attracting state of the network for a large range of parameters. Mechanisms leading to such events are then described in two regimes of external drive: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed computationally in networks with random transmission delays. Neuronal voltage probability density functions (PDFs) and gain curves---graphs depicting the dependence of the network firing rate on the external drive strength---are analyzed using the steady solutions of the self-consistency problem for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic solutions for the gain curves are obtained in several physiologically relevant limits. The absence of chaotic dynamics is proved for the type of network under investigation by demonstrating convergence in time of its trajectories.

We consider application of the stochastic mode-reduction strategy to a particular class of coupled models where a part of self-interactions of the slow variables is given by a rotationally invariant gradient system. The stochastic mode-reduction strategy is utilized to derive stochastic reduced models which yield a simple description of the phenomena resulting from breaking the original rotational symmetry. It is demonstrated that the direction of the symmetry breaking can be predicted a-priori without any knowledge of the statistical behavior of the fast modes.

We examine stochastic coarse-graining strategies for two biomolecular systems. First, we compute the large-scale transport properties of the basic flashing ratchet mathematical model for (Brownian) molecular motors and consider in this light whether the underlying continuous-space, continous-time Markowian model can be coarse-grained as a discrete-state, continuous-time Markovian random walk model. Through careful computation of associated statistical signatures of Markovianity, we find that such a discrete coarse-graining is an excellent approximation over much but not all of the parameter regime. In particular, for the parameter values associated with the fastest transport by the flashing ratchet, the discretized model displays non-Markovian features such as waiting times between jumps which are not exponentially distributed. We provide a theoretical framework for understanding the conditions under which Markovianity is to be expected in the discretized model and two mechanisms by which the flashing ratchet model coarse-grains to a non-Markovian discretized model. Next we turn to a basic question of how the dynamics of water molecules near the surface of a solute can be represented by a simple drift-diffusion stochastic model. This quesiton is of most interest for the purpose of accelerating molecular dynamics simulations of proteins, but for simplicity, we here examine the simple case where the solute is a C_{60} buckyball, which has a homogeneous, roughly isotropic form. We compare the mathematical drift-diffusion framework with a statistical quantification of water dynamics near a solute discussed in the biophysical literature. A key concern is the choice of time interval on which to sample the molecular dynamics data to generate estimators for the drift and diffusivity. We use a simple mathematial toy model to establish insight and a strategy, but find for the actual molecular dynamics data, that the sampling times which produce the most faithful drift coefficient do not overlap, so that scarifice of quality in one or the other parameter appears necessary.

Two well-known variational principles for geophysical flows are combined into a single minimax principle that characterizes distinguished steady solutions of the rotating shallow water (RSW) equations. On the one hand, in the limit of small Rossby number $\epsilon$, in which the dynamics becomes quasi-geostrophic and closes terms of the potential vorticity field $Q$, steady coherent states are characterized as minimizers of (generalized) enstrophy $\A$ at a given value of total energy $\H$. On the other hand, for small amplitude motions at finite $\epsilon$, balanced states resulting from geostrophic adjustment are characterized as minimizers of the total energy $\H$ subject to a given potential vorticity $Q$. Moreover, the organization into a coherent state through potential vorticity mixing occurs on a slow time scale relative to the fast time scale of adjustment through inertia-gravity wave radiation. These two complementary principles suggest a variational characterization of steady balanced states for the RSW equations at finite $\epsilon$. Namely, the functional, $\A + \theta \H$, where $\theta <0$ is a parameter, is first maximized over all RSW fields with given $Q$, and then minimized over all $Q$. Any such minimax critical point of $\A+\theta \H$ is an exact steady solution of the RSW equations, which represents a physically relevant equilibrium state at finite Rossby number. This minimax principle is implemented numerically for zonal shear flows, and branches of solutions are computed to first-order in $\epsilon$. The results quantify the breakdown of quasi-geostrophy and the asymmetry between cyclonic and anticyclonic structures. In addition, the $O(\epsilon)$-correction is computed for a model of the zonally-averaged winds in Jupiter's weather layer.

In 1953 G.I. Taylor showed theoretically and experimentally that a passive tracer diffusing in the presence of laminar pipe flow would experience an enhanced diffusion in the longitudinal direction beyond the bare molecular diffusivity, \kappa, in the amount a^2 U^2/193 \kappa, where a is the pipe radius and U is the maximum fluid velocity. This behavior is predicted to arise after a transient timescale a^2/\kappa, the diffusive timescale for the tracer to cross the pipe. Typically \kappa is very small, so provided a farily long time has passed, this is a very large diffusive boost. Before this timescale, the evoluation is expected to be anomalous, meaning the scalar variance does not grow linearly in time. A few attempts to compute this anomalous growth have been made in the literature for different special cases with different approximations. Here we derive an exact approach which provides the scalar variance evolution valid for all times for channel and pipe flow for the case of vanishing Neumann boundary conditions. We show how this formula limits to the Taylor regime, and study rigorously the anomalous regime for a range of initial data. We find that the anomalous timescales and exponents depend strongly upon the form of the data. For initial data whose transverse variation is a delta function on the centerline, the anomalous regime emerges after a timescale, (a^4/\kappa U^2)^{1/2}, with variance growing as t^\alpha, with \alpha=4. In contrast, for the case of uniform data (independent of the transverse variable), the anomalous timescale is \kappa/U^2, with exponent, alpha=2, and this result is generalized for generic shear flows given that the initial condition is not a transverse Dirac delta. Further, these exact formulae show explicitly what features the short time approximations which ignore physical boundaries are able to capture.

Many important equations in science and engineering contain rapidly varying operators that cannot practically be resolved sufficiently for accurate solutions. in some cases it is possible to obtain approximate solutions by replacing the rapidly varying operator with an appropriately averaged operator. In this paper we use formal asymptotic techniques to recover a formula for the averaged form of a second order, non-divergence structure, linear elliptic operator. For several special cases the averaged operator is obtained analytically. For genuinely multi-dimensional cases, the averaged operator is also obtained numerically, using a finite difference method, which also has a probabilistic interpretation.

The problem of estimating the eddy diffusivity from Lagrangian observations in the presence of measurement error is studied in this paper. We consider a class of incompressible velocity fields for which it can be rigorously proved that the small scale dynamics can be parametrized in terms of an eddy diffusivity tensor. We show, by means of analysis and numerical experiments, that subsampling of the data is necessary for the accurate estimation of the eddy diffusivity. The optimal sampling rate depends on the detailed properties of the velocity field. Furthermore, we show that averaging over the data only marginally reduces the bias of the estimator due to the multiscale structure of the problem, but that it does significantly reduce the effect of observation error.

A new space semi-discretization for the dynamic Signorini problem, based on a modification of the mass term, has been recently proposed. We prove the convergence of the space semi-discrete solutions to a solution of the continuous problem in the case of a visco-elastic material.

A method is presented to solve two-phase problems involving a material quantity on an interface. The interface can be advected, stretched and change topology, and material can be absorbed to or desorbed from it. The method is based on the use of a diffuse interface framework, which allows a simple implementation using standard finite-difference or finite-element techniques. Here, finite-difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Interfacial flow with soluble surfactants is used an as example of the application of the method, and several test cases are presented demonstrating its accuracy and convergence.

In this paper, we obtain many traveling wave solutions for some nonlinear partial differential equations. The modified tanh-coth method with the symbolic computation is implemented for constructing multiple traveling wave solutions for the two dimensional coupled Burgers, ZK-MEW and one dimensional Ostrovsky equations. The results reveal that the implemented technique is very effective and convenient for solving nonlinear partial differential equations arising in mathematical physics.

A constrained string method is developed to solve the saddle-point problem with constraints. Based on the intrinsic description of the string method, Lagrange multipliers are employed for treating the constraints. Various mathematical properties are established such as the conservation of the constaints and the energy dissipation law. We also investigate time discretization schemes for the constrained string method and discuss possible alternative ways to enforce the constraints. Some numerical examples are proposed as illustration.

We extend previous work on injectivity in chemical reaction networks to general interaction networks. Matrix-theoretic conditions for injectivity of these systems are presented. A particular signed, directed, labelled, bipartite multifraph, terms the "DSR graph", is shown o be a useful representation of an interaction network when discussing questions on injectivity. A graph-theoretic condition, developed previously in the context of chemical reaction networks, is shown to be sufficient to gaurantee injectivity for a large class of systems. The graph-theoretic condition is simple to state and often easy to check. Examples are presented to illustrate the wide applicability of the theory developed.

We introduce and discuss kinetic models for wealth distribution in a simple market economy, which are able to reproduce the salient features of the wealth distribution by including taxes to each trading process and redistributing the collected money among the population according to a given criterion. Our analysis gives a theoretical basis to some recent researches that analyzed discrete simplified models for the exploitation of finite resources by interacting agents, where each agent receives a random fraction of the available resources. It is shown that in general the redistribution is able to modify the Pareto index, and that this modification can be quantified in terms of the redistribution operator.

We study a two-dimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended Onsager-Maier-Saupe free energy may be decomposed into Landau-de Gennes-type and relative entropy-type contributions. We then prove that in the high concentration limit the states of the system display characteristic vortex-like patterns and derive asymptotic expansion for the free energy of the system.

In this paper, we study the existence and long-time behavior of global strong solutions to a system describing the mixture of two viscous incompressible Newtonian fluids of the same density. The system consists of a coupling of Navier-Stokes and Cahn-Hilliard equations. We first show the global existence of strong solutions in several cases. Then we prove that the global strong solution of our system will converge to a steady state as time goes to infinity. Besides, we also provide an estimate on the convergence rate.

This paper investigates the connection between discrete and continuous models describing prion prolification. The scaling parameters are interpreted on biological grounds and we establish rigorous convergence statements. We also discuss, based on the asymptotic analysis, relevant boudary conditions that can be used to complete the continuous model.

We prove that a family of solutions to a Cauchy problem for a two dimensional scalar conservation law with a discontinuous smoothed flux and the vanishing viscosity is strongly L^1_{loc}-precompact under a new genuine nonlinearity condition, weaker than in previous works on the subject.

We consider the scalar wave equation in a bounded convex domain of R^2 and R^3. The boundary condition is of Dirichlet or Neumann type and the initial conditions have a compact support in the considered domain. We construct a family of approximate high frequency solutions by a Gaussian beams summation. We gave a rigorous justification of the asymptotics in the sense of energy and show that the error can be reduced to any arbitrary power of epsilon, the high frequency parameter.

This paper is concerned with the time-asymptotic behavior of solutions to the three-dimensional magnetohydrodynamics (MHD) for viscous, compressible, isentropic fluids. By exploiting some L^p-L^q estimates of solutions for the heat equation and the linearized Navier-Stokes system, the optimal decay estimates of the solution in L^p with 2\le q\le 6 and its first order derivative in L^2 are obtained when the H^3-norm of initial perturbation around a constant state is sufficiently small and its L^p-norm with any given 1\le p <6/5 is bounded. As a byproduct, the global existence theorem is also proved.

Assuming that initial velocity has finite energy and initial vorticity is bounded in the plane, we show that the unique solutions of the Navier-Stokes equations converge to the unique solution of the Euler equations in the L^\infty-norm uniformly over finite time as viscosity approaches zero. We also establish a rate of convergence.

An evolution equation, arising in the study of the Dynamical System Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving ill-posed problems with monotone nonlinear operators F. Local and global existence of the unique solution to this evolution equation are proved, apparently for the first time, under the only assumption that F'(u) exists and is continuous with respect to u. The earlier published results required more smoothness of F. The Dynamical Systems method (DSM) for solving equations F(u)=0 with monotone Frechet differentiable operator F is justified under the above assumption apparently for the first time.

We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order (n=2) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation (n=1) corresponds to another famous model from imaging processing, namely Perona and Malik's anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically obsereved.