Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations, if they exist, they blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then we show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.     

Global stability of large solutions to the 3D Navier-Stokes equations

We prove the stability of mildly decaying global strong solutions to the Navier-Stokes equations in three space dimensions. Combined with previous results on the global existence of large solutions with various symmetries, this gives the first global existence theorem for large solutions with approximately symmetric initial data. The stability of unforced 2D flow under 3D perturbations is also obtained.     

Initial-boundary value problems for a class of nonlinear thermoelastic plate equations

This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions, it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations, by means of the Galerkin method. Moreover, it also proves the existence of strong and classical solutions.     

Global Well–Posedness of the 3<Emphasis Type="Italic">D</Emphasis> Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion

The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.     

Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals

The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution.     

Random solutions to a system of fractional differential equations via the Hadamard fractional derivative

In this work, we use a new random fixed point theorem in vector metric spaces due to Sinacer et al. [M.L. Sinacer et al., Random Oper. Stoch. Equ. 24, 93 (2016)] to prove the existence of solutions and the compactness of solution sets of a random system of fractional differential equations via the Hadamard-type derivative. The existence, modification and stochastically continuity of an M²-solution are also proved.     

A class of blowup and global analytical solutions of the viscoelastic Burgersʼ equations

In this Letter, by employing the perturbational method, we obtain a class of analytical self-similar solutions of the viscoelastic Burgers? equations. These solutions are of polynomial-type whose forms, remarkably, coincide with that given by Yuen for the other physical models, such as the compressible Euler or Navier–Stokes equations and two-component Camassa–Holm equations. Furthermore, we classify the initial conditions into several groups and then discuss the properties on blowup and global existence of the corresponding solutions, which may be readily seen from the phase diagram.     

Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the Euler equations in two space dimensions, in the case where the initial rarefaction waves are large. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar, and we characterize how large the rarefaction waves must be. We use the methods of hodograph transformation, characteristic decomposition, and phase space analysis. We resolve binary interactions of simple waves in the process.     

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE PERIODIC HUNTER–SAXTON EQUATION WITH WEAK DISSIPATION

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.     

Global Well-Posedness for the 2D Micro-Macro Models in the Bounded Domain

In this paper, we establish new a priori estimates for the coupled 2D Navier-Stokes equations and Fokker-Planck equation. As its applications, we prove the global existence of smooth solutions for the coupled 2D micro-macro models for polymeric fluids in the bounded domain.     

Asymptotic property of solutions to the random cauchy problem for wave equations

Based on the existence and uniqueness theorem for random wave equations, we consider asymptotic behaviors of solutions to the random initial value problem. We describe conditions for the equipartition of stochastic energy (or SE for short) by making use of the random spectral theory and, according to Goldstein's semigroup method, we prove the asymptotically equipartitioned SE theorem and the so-called virial theorem of classical mechanics, and also study the probabilistic characterization of the conditions for equipartition. In addition, we show at last the equipartition of SE from a finite time onwards.     

Numerical simulation of non-Abelian particle-field dynamics

Numerical 1D-3V solutions of the Wong-Yang-Mills equations with anisotropic particle momentum distributions are presented. They confirm the existence of plasma instabilities for weak initial fields and of their saturation at a level where the particle motion is affected, similar to Abelian plasmas. The isotropization of the particle momenta by strong random fields is shown explicitly, as well as their nearly exponential distribution up to a typical hard scale, which arises from scattering off field fluctuations. By variation of the lattice spacing we show that the effects described here are independent of the UV field modes near the end of the Brioullin zone.     

Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data

We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at a certain time under some circumstance. The central feature is the strengthening of waves as they move radially inward. A longstanding open, fundamental problem is whether concentration could be formed at the origin. In this paper, we develop a method of vanishing viscosity and related estimate techniques for viscosity approximate solutions, and establish the convergence of the approximate solutions to a global finite-energy entropy solution of the isentropic Euler equations with spherical symmetry and large initial data. This indicates that concentration is not formed in the vanishing viscosity limit, even though the density may blow up at a certain time. To achieve this, we first construct global smooth solutions of appropriate initial-boundary value problems for the Euler equations with designed viscosity terms, approximate pressure function, and boundary conditions, and then we establish the strong convergence of the viscosity approximate solutions to a finite-energy entropy solution of the Euler equations.     

Dynamical Stability of Non-Constant Equilibria for the Compressible Navier–Stokes Equations in Eulerian Coordinates

In this paper we establish global existence and uniqueness of strong solutions to the non-isothermal compressible Navier–Stokes equations in bounded domains. The initial data have to be near equilibria that may be non-constant due to considering large external forces. We are able to show exponential stability of equilibria in the phase space and, above all, to study the problem in Eulerian coordinates. The latter seems to be a novelty, since in works by other authors, global strong L _p -solutions have been investigated only in Lagrangian coordinates; Eulerian coordinates are even declared as impossible to deal with. The proof is based on a careful derivation and study of the associated linear problem.     

Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications

In this paper we study blow-up phenomena in general coupled nonlinear Schrödinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose-Einstein condensates and to degenerate boson-fermion mixtures.     

Global Solutions to Gross–Neveu Equation

We prove global existence of solutions to Gross–Neveu equations. Given a local solution, we obtain a uniform L ^∞ bound of the solution by applying local form of charge conservation.     

Decay estimates for Schrödinger equations

We prove global existence and optimal decay estimates for classical solutions with small initial data for nonlinear nonlocal Schrödinger equations. The Laplacian in the Schrödinger equation can be replaced by an operator corresponding to a non-degenerate quadratic form of arbitrary signature. In particular, the Davey-Stewartson system is included in the the class of equations we discuss.Partially supported by NSF grant DMS-860-2031. Sloan Research Fellow     

Viscous Multipolar Fluids-Physical Background and Mathematical Theory-

We introduce the theory of multipolar fluids in which constitutive laws depend linearly not only on the first spatial gradients of velocity as in classical Navier-Stokes theory of newtonian fluids but also on its higher order spatial gradients up to the order 2k − 1, k = 2, 3,… Such fluids are called k-polar fluids. A thermodynamic theory of the constitutive equations satisfying the second law of thermodynamics and the principle of material frame indifference is developed. Special thermodynamic processes as isothermal, barotropic, adiabatic and general heat-conductive motion for compressible multipolar fluids are studied. It is well known that there does not exist adequate existence theory for compressible newtonian fluids. We given a consistent theory for compressible multipolar fluids in two or three dimensions, i. e. we prove the global in time existence of weak solutions for the initial boundary value problems in bounded domains for the systems of partial differential equations describing isothermal, barotropic, adiabatic and general compressible motion. Under some assumptions on the regularity of the initial data and external forces, we prove existence of strong solutions, uniqueness and regularity. Some other properties as e. g. cavitation of density are discussed. We put stress on the lowest possible polarity of the fluid. In the isothermal case we consider the polarity k ≧ 2 and in barotropic and heat-conductive gas the polarity k ≧ 3.     

Global existence and asymptotic behavior of solutions for the Maxwell-Schrödinger equations in three space dimensions

In this paper we study the global existence and asymptotic behavior of solutions for the Maxwell-Schrödinger equations under the Coulomb gauge condition in three space dimensions with the final states given att=+. This leads to the construction of the modified wave operator for certain scattered data. It is also shown that for the initial data in the range of the modified wave operator, the initial value problem of the Maxwell-Schrödinger equations has the global solutions in time.Dedicated to Professor R. Iino on his 70th birthday     

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique solution for the fractal Navier-Stokes equation with initial data in \mathbb W^1,p{{\mathbb W}^{1,p}} is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with time dependent and discontinuous drifts are proved.