Unified approach to KdV modulations

We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial‐value problem for the zero‐dispersion KdV as the steepest descent for the scalar Riemann‐Hilbert problem [6] and on the method of generating differentials for the KdV‐Whitham hierarchy [9]. By assuming the hyperbolicity of the zero‐dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)‐plane. The resulting system effectively solves the zero‐dispersion KdV with an arbitrary initial datum. © 2001 John Wiley & Sons, Inc.     

Local well‐posedness and blow‐up criteria of solutions for a rod equation

In this paper we consider a new rod equation derived recently by Dai [Acta Mech. 127 No. 1–4, 193–207 (1998)] for a compressible hyperelastic material. We establish local well‐posedness for regular initial data and explore various sufficient conditions of the initial data which guarantee the blow‐up in finite time both for periodic and non‐periodic case. Moreover, the blow‐up time and blow‐up rate are given explicitly. Some interesting examples are given also. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

This study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in one‐ and two‐dimensional cases are provided to show the effectiveness of the proposed method.     

Wave Breaking and Measure of Momentum Support for an Integrable Camassa‐Holm System with Two Components

We discuss a new integrable two‐component Camassa‐Holm equation which describes the motion of fluid. This paper is concerned with the wave breaking mechanism for the Cauchy problem with periodic condition where two special classes of initial data are involved. Moreover, the estimate of momentum support is also shown.     

Stability and error analysis for a spectral Galerkin method for the Navier‐Stokes equations with H2 or H1 initial data

In this article we consider a spectral Galerkin method with a semi‐implicit Euler scheme for the two‐dimensional Navier‐Stokes equations with H² or H¹ initial data. The H²‐stability analysis of this spectral Galerkin method shows that for the smooth initial data the semi‐implicit Euler scheme admits a large time step. The L²‐error analysis of the spectral Galerkin method shows that for the smoother initial data the numerical solution u exhibits faster convergence on the time interval [0, 1] and retains the same convergence rate on the time interval [1, ∞). © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2

We analyze the two‐dimensional parabolic‐elliptic Patlak‐Keller‐Segel model in the whole Euclidean space ?². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local‐in‐time existence for any mass of “free‐energy solutions,” namely weak solutions with some free‐energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free‐energy solutions with initial data as before for the critical mass 8π/χ. Actually, we prove that solutions blow up as a delta Dirac at the center of mass when t → ∞ when their second moment is kept constant at any time. Furthermore, all moments larger than 2 blowup as t → ∞ if initially bounded. © 2007 Wiley Periodicals, Inc.     

A Crank‐Nicolson and ADI Galerkin method with quadrature for hyperbolic problems

We propose, analyze, and implement fully discrete two‐time level Crank‐Nicolson methods with quadrature for solving second‐order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265–1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564–576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577–598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Combination of standard Galerkin and subspace methods for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H¹‐stability and L²‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Local Regularity for the Modified SQG Patch Equation

We study the patch dynamics on the whole plane and on the half‐plane for a family of active scalars called modified surface quasi‐geostrophic (SQG) equations. These involve a parameter α that appears in the power of the kernel in their Biot‐Savart laws and describes the degree of regularity of the equation. The values α =0 and α =½ correspond to the two‐dimensional Euler and SQG equations, respectively. We establish here local‐in‐time regularity for these models, for all α ? (0,½) on the whole plane and for all small α > 0 on the half‐plane. We use the latter result in [16], where we show existence of regular initial data on the half‐plane that lead to a finite‐time singularity.© 2016 Wiley Periodicals, Inc.     

Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

We study a class of partial differential equations (PDEs) in the family of the so‐called Euler–Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses nonsmooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particle system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when nonhomogeneous “constant stream” boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic‐like to dispersive with a nontrivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two‐particle dynamics under the resulting lack of Lipschitz continuity. Curiously, we find that for the conically shaped kernels the motion of the related two‐dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two‐dimensional system to the one‐dimensional completely integrable Shallow‐Water equation [1], while retaining the full dependence on two spatial dimensions for the single channel solutions. Finally, by comparing with an operator‐splitting pseudospectral method we illustrate the performance of the particle algorithms with respect to their Eulerian counterpart for this class of nonsmooth kernels.     

Functional large deviations for burgers particle systems

We consider Burgers particle systems, i.e., one‐dimensional systems of sticky particles with discrete white‐noise‐type initial data (not necessarily Gaussian), and describe functional large deviations for the state of the systems at any given time. For specific functionals such as maximal particle mass, particle speed, rarefaction interval, momentum, and energy, the research was initiated by Avellaneda and E [1, 2] and pursued further by Ryan [14]. Our results extend those of Ryan and contain many other examples. © 2006 Wiley Periodicals, Inc.     

A uniformly optimal‐order estimate for finite volume method for advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for finite volume method for two‐dimensional time‐dependent advection–diffusion equations on a uniform space‐time partition of the domain. The generic constants in the estimates depend only on certain norms of the true solution but not on the scaling parameter. These estimates, combined with a priori stability estimates of the governing partial differential equations with full regularity, yield a uniform estimate of the finite volume method, in which the generic constants depend only on the Sobolev norms of the initial and right side data but not on the scaling parameter. We use the interpolation of spaces and stability estimates to derive a uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right‐hand side data. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 17‐43, 2014     

Supersonic flow onto a solid wedge

We consider the problem of two‐dimensional supersonic flow onto a solid wedge, or equivalently in a concave corner formed by two solid walls. For mild corners, there are two possible steady state solutions, one with a strong and one with a weak shock emanating from the corner. The weak shock is observed in supersonic flights. A longstanding natural conjecture is that the strong shock is unstable in some sense. We resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed. More precisely, we prove that for upstream state as initial data in the entire domain, the time‐dependent solution is self‐similar, with a weak shock at the tip of the wedge. We construct analytic solutions for self‐similar potential flow, both isothermal and isentropic with arbitrary γ ≥ 1. In the process of constructing the self‐similar solution, we develop a large number of theoretical tools for these elliptic regions. These tools allow us to establish large‐data results rather than a small perturbation. We show that the wave pattern persists as long as the weak shock is supersonic‐supersonic; when this is no longer true, numerics show a physical change of behavior. In addition, we obtain rather detailed information about the elliptic region, including analyticity as well as bounds for velocity components and shock tangents. © 2007 Wiley Periodicals, Inc.     

Well‐posedness for the density‐dependent incompressible flow of liquid crystals

In this paper, we are concerned with the Cauchy problem for the density‐dependent incompressible flow of liquid crystals in thewhole space (N ≥ 2).We prove the localwell‐posedness for large initial velocity field and director field of the system in critical Besov spaces if the initial density is close to a positive constant. We show also the global well‐posedness for this system under a smallness assumption on initial data. In particular, this result allows us to work in Besov space with negative regularity indices, where the initial velocity becomes small in the presence of the strong oscillations. Copyright © 2014 John Wiley & Sons, Ltd.     

Spatial behaviour of solutions of the dual‐phase‐lag heat equation

In this paper we study the spatial behaviour of solutions of some problems for the dual‐phase‐lag heat equation on a semi‐infinite cylinder. The theory of dual‐phase‐lag heat conduction leads to a hyperbolic partial differential equation with a third derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary‐value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial‐time lines. A class of non‐standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T are assumed proportional to their initial values. Copyright © 2004 John Wiley & Sons, Ltd.     

Stokes‐Fourier and acoustic limits for the Boltzmann equation: Convergence proofs

We establish a Stokes‐Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension two or more. Appropriately scaled families of DiPerna‐Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes‐Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L² initial data of the Stokes‐Fourier equations. Both the motion and heat equations are both recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna‐Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna‐Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L² initial data. © 2001 John Wiley & Sons, Inc.     

Relative energy approach to a diffuse interface model of a compressible two‐phase flow

We propose a simple model for a two‐phase flow with a diffuse interface. The model couples the compressible Navier‐Stokes system governing the evolution of the fluid density and the velocity field with the Allen‐Cahn equation for the order parameter. We show that the model is thermodynamically consistent, in particular, a variant of the relative energy inequality holds. As a corollary, we show the weak‐strong uniqueness principle, meaning any weak solution coincides with the strong solution emanating from the same initial data on the life span of the latter. Such a result plays a crucial role in the analysis of the associated numerical schemes. Finally, we perform the low Mach number limit obtaining the standard incompressible model.     

Modeling Sediment Deposition Patterns Arising From Suddenly Released Fixed-Volume Turbulent Suspensions

Models presented in several recent papers [1–3] dealing with particle transport by, and deposition from, bottom gravity currents produced by the sudden release of dilute, well‐mixed fixed‐volume suspensions have been relatively successful in duplicating the experimentally observed long‐time, distal, areal density of the deposit on a rigid horizontal bottom. These models, however, fail in their ability to capture the experimentally observed proximal pattern of the areal density with its pronounced dip in the region initially occupied by the well‐mixed suspension and its equally pronounced local maximum at roughly the one‐third point of the total reach of the deposit. The central feature of the models employed in [1–3] is that the particles are always assumed to be vertically well‐mixed by fluid turbulence and to settle out through the bottom viscous sublayer with the Stokes settling velocity for a fluid at rest with no re‐entrainment of particles from the floor of the tank. Because this process is assumed from the outset in the models of [1–3], the numerical simulations for a fixed‐volume release will not take into account the actual experimental conditions that prevail at the time of release of a well‐mixed fixed‐volume suspension. That is, owing to the vigorous stirring that produces the well‐mixed suspension, the release volume will initially possess greater turbulent energy than does an unstirred release volume, which may only acquire turbulent energy as a result of its motion after release through various instability mechanisms. The eddy motion in the imposed fluid turbulence reduces the particle settling rates from the values that would be observed in an unstirred release volume possessing zero initial turbulent energy. We here develop a model for particle bearing gravity flows initiated by the sudden release of a fixed‐volume suspension that takes into account the initial turbulent energy of mixing in the release volume by means of a modified settling velocity that, over a time scale characteristic of turbulent energy decay, approaches the full Stokes settling velocity. Thereafter, in the flow regime, we assume that the turbulence persists and, in accord with current understanding concerning the mechanics of dense underflows, that this turbulence is most intense in the wall region at the bottom of the flow and relatively coarse and on the verge of collapse (see [22]) at the top of the flow where the density contrast is compositionally maintained. We capture this behavior by specifying a “shape function” that is based upon experimental observations and provides for vertical structure in the volume fraction of particles present in the flow. The assumption of vertically well‐mixed particle suspensions employed in [1–5] corresponds to a constant shape function equal to unity. Combining these two refinements concerning the settling velocity and vertical structure of the volume fraction of particles into the conservation law for particles and coupling this with the fluid equations for a two‐layer system, we find that our results for areal density of deposits from sudden releases of fixed‐volume suspensions are in excellent qualitative agreement with the experimentally determined areal densities of deposit as reported in [1, 3, 6]. In particular, our model does what none of the other models do in that it captures and explains the proximal depression in the areal density of deposit.     

Global well‐posedness for the 2D dissipative quasi‐geostrophic equations in modulation spaces

We prove the existence of global weak solution of the two‐dimensional dissipative quasi‐geostrophic equations with small initial data in and local well‐posedness with the large initial data in the same space. Our proof is based on constructing a commutator related to the problem, as well as its estimate. Copyright © 2017 John Wiley & Sons, Ltd.