Global weak solutions to the cometary flow equation with a self-generated electric field

We study the Cauchy problem of a cometary flow equation with a self-generated electric field. This kinetic model originates from the theory of astrophysical plasmas and can be viewed as a perturbation, by a wave-particle collision operator, of the classical Vlasov-Poisson system. By asymptotic methods in kinetic theory, global existence of nonnegative weak solutions to the Cauchy problem in three space variables is established for bounded initial data having finite second order velocity moments.     

On the “Destruction” of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin-Bona-Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin-Bona-Mahony type, we prove the “failure” of a “weakened” solution in finite time.     

Estimates of the Nash–Aronson type for degenerating parabolic equations

We consider second-order parabolic equations describing diffusion with degeneration and diffusion on singular and combined structures. We give a united definition of a solution of the Cauchy problem for such equations by means of semigroup theory in the space L ₂ with a suitable measure. We establish some weight estimates for solutions of Cauchy problems. Estimates of Nash–Aronson type for the fundamental solution follow from them. We plan to apply these estimates to known asymptotic diffusion problems, namely, to the stabilization of solutions and to the “central limit theorem.”     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations

The paper is devoted to studying the distribution of stationary solutions for 3D Navier-Stokes equations perturbed by a random force. Under a non-degeneracy assumption, we show that the support of such a distribution coincides with the entire phase space, and its finite-dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the Lebesgue measure. Similar assertions are true for weak solutions of the Cauchy problem with a regular initial function. The results of this paper were announced in the short note [A. Shirikyan, Controllability of three-dimensional Navier-Stokes equations and applications, in: Sémin. Équ. Dériv. Partielles, 2005-2006, École Polytech., Palaiseau, 2006].     

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall''s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980''s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.     

On the blow up criterion for the 2-D compressible Navier-Stokes equations

Motivated by [10], we prove that the upper bound of the density function j9 controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Weak Dissipative Structure for Compressible Navier-Stokes Equations

This paper concerns the Cauchy problem for compressible Navier-Stokes equations.The weak dissipative structure is explored and a new proof for the classical solutions are shown to exist globally in time if the initial data is sufficiently small.     

Initial Boundary Value Problems of the Camassa–Holm Equation

In this paper we study initial boundary value problems of the Camassa–Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa–Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa–Holm equation on a compact interval possesses no nontrivial global classical solutions.     

On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-?verák’s criterion. We also show that if a weak solution u satisfies $\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $ for some 3 < p < ∞, then the number of singular points is finite.     

Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations

We prove that the smooth solutions to the Cauchy problem for the Navier-Stokes equations with conserved total mass, finite total energy and finite momentum of inertia lose the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported. The cases of isentropic and incompressible fluids are also considered.     

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.     

Convergence estimates of Galerkin-wavelet solutions to a Cauchy problem for a class of periodic pseudodifferential equations

A class of Cauchy problems for interesting complicated periodic pseudodifferential equations is considered. By the Galerkin-wavelet method and with weak solutions one can find sufficient conditions to establish convergence estimates of weak Galerkin-wavelet solutions to a Cauchy problem for this class of equations.

     

Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data n the homogeneous spaces.We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in C(σ,s,p)and L~q([O,T);H~(s,p)by introducing the concept of both admissiblequintuptet and compatible space and establishing estblishing time-space estimates for solutions to the linear parabolic typeequations For the small data,we prove that these results can be extended globally in time. We also study the     

Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the “Double Null Foliation Gauge”

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region much larger than the one provided by the Cauchy–Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the a priori estimates for the Weyl equations, associated with the “Bel-Robinson norms”. In particular, if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a “geometric” way the Einstein equations, then we show how the Cauchy–Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally, we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.     

Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L¹(R²) for positive times is entirely determined by the trace of the vorticity at t=0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa & Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in R² is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions.     

Global classical solution to the three-dimensional isentropic compressible Navier-Stokes equations with general initial data

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in the three space dimensions with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient μ is large enough.     

The point-vortex method for periodic weak solutions of the 2-D Euler equations

Almost-sure convergence of a subsequence of the vorticity to a weak solution is proven for the point-vortex method for 2-D, inviscid, incompressible fluid flow. Here “almost-sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or L^p with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H⁻¹. All these results are also valid for the “vortex-blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point-vortex dynamics are weak solutions of the Euler equations is also discussed.     

Nonlinear instability in Vlasov type equations around rough velocity profiles

In the Vlasov-Poisson equation, every configuration which is homogeneous in space provides a stationary solution. Penrose gave in 1960 a criterion for such a configuration to be linearly unstable. While this criterion makes sense in a measure-valued setting, the existing results concerning nonlinear instability always suppose some regularity with respect to the velocity variable. Here, thanks to a multiphasic reformulation of the problem, we can prove an “almost Lyapounov instability” result for the Vlasov-Poisson equation, and an ill-posedness result for the kinetic Euler equation and the Vlasov-Benney equation (two quasineutral limits of the Vlasov-Poisson equation), both around any unstable measure.