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 consider the amplitude decay for the linearized equations governing irrotational vortex sheets and water waves with surface tension. Using oscillatory integral estimates, we prove that the magnitude of the amplitude decays faster than t^{-1/3}.

This paper contains two parts. In the first part, we derive a variant of Gagliardo-Nirenberg interpolation inequality involving nonlocal nonlinearity and determine its best (smallest) constant. In the second part, we study two applications of this inequality and its best constant. In the first application, we use this best constant to establish a sharp criterion for the global existence and blow-up of solutions of the inhomogeneous Schr\"{o}dinger equation with harmonic potential and a nonlocal nonlinearity in the critical case $p=2+(2-\alpha)/N$. The result indicates that the existence of blow-up solution not only depends on the mass of the initial data but also on the profile of the initial data. In the second application, we use this best constant to prove that when $2+(2-\alpha)/N < p < (2N-\alpha)/(N-2)$, the solutions exist globally in time for one class of initial data whose norm can be as large as one wants.

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.

This paper is concerned with the linear and nonlinear exponential stability of traveling wave solutions for a system of quasi-linear hyperbolic equations with relaxation. By applying C_0-semigroup theory and detailed spectral analysis, we prove the linear exponential stability of the traveling waves for the semilinear systems, i.e., the Jin-Xin relaxation models, in some exponentially weighted spaces without assuming that the wave strengths are small.

A multiscale, time reversible method for computing the effective behavior of systems of highly oscillatory ordinary differential equations is presented. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the system. The algorithm follows the framework of the heterogeneous multiscale method. The notion of time reversibility in the multiple time-scale setting is discussed. The algorithm requires nontrivial matching between the microscopic state variables and the macroscopic slow ones. Numerical examples show the efficiency of the multiscale method and the advantages of time reversibility.

In linear algebra, both eigenvalue problems and singular value problems for matrices are of fundamental importance. In this paper, we use critical point theory from nonlinear analysis and tools from algebraic topology to study the existence and multiplicity problem of singular values in the higher order tensor setting.

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 present an explicit upper-bound for the life-span of C^3-smooth solutions to the multi-dimensional compressible Euler equations with a certain class of initial data containing compression vacuum states. We also show that the divergence of a fluid velocity will blow up along the particle trajectories issued from the compression vacuum states, which represent a shock formation at the vacuum states.

Recent analytic progress has increased demand for numerical approaches to the Wigner-Fokker-Planck (WFP) equation. We present a Discontinuous Galerkin scheme for the WFP equation with a general potential. Estimates showing convergence and stability of the scheme a re provided. The scheme is adaptable, and may use both polynomial and non-polynomial basis functions.

We prove that smooth solutions of the semigeostrophic equations in the incompressible x-z setting can be derived from the Navier-Stokes equations with the Boussinesq approximation.

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.

A method for photoacoustic tomography is presented that uses circular integrals of the acoustic wave for the reconstrcution of a three-dimensional image. Image reconstruction is a two-step process: in the first step data from a stack of curcular integrating detectors are used to reconstruct the curcular projection of the source distribution. In the second step the inverse circular Radon transform is applied. In this article we establish inversion formulas for the first step, which involves inverse problem for the axially symmetric wave equation. Numerical results are presented that show the validity and robustness of the resulting algorithm.

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.

We consider the derivation of the Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation from the nonlinear isentropic Navier-Stokes and Euler systems. The KZK equation is a mathematical model that describes the nonlinear propagation of a finite-amplitude sound pulse in a thermo-viscous medium. The derivation of the KZK equation has to date been based on the paraxial approximation of small perturbations around a given state of the Navier-Stokes system. However, this method does not guarantee that the solution of the initial Navier-Stokes system can be reconstructed from the solution of the KZK-approximation. We introduce a corrector function in the derivation ansatz that allows one to validate the KZK-approximation. We also give the analysis of other types of derivation. We prove the validation of the KZK-approximation for the non-viscous case, as well as for the viscous nonlinear and linear cases. The results are obtained in Sobolev spaces for functions periodic in one of the space variables and with a mean value of zero. The existence of a unique regular solution of the isentropic Navier-Stokes system in the half space with boundary conditions that are both periodic and mean value zero in time is also obtained.

We develop a discrete network approximation to effective conductivity of high contrast, highly packed particulate composites with nonlinear constituent relations. The key tool is the perforated medium approach, which provides a simple mathematical justification of the discrete network approximation by variational techniques.