Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

Local-in-time regularity results for some flux-limited diffusion equations of porous media type

We prove some regularity results for solutions of some flux limited diffusion equations of porous media type for Lipschitz initial data (or assuming a uniform gradient bound on some power of the data), including the fast diffusion case in which the results are global in time. We also develop the existence and uniqueness theory for solutions of the fast diffusion case, which was not covered in the current literature.     

Large‐time behavior of solutions to the Rosenau equation with damped term

In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity‐loss type, which causes the difficulty in high‐frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for n≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for n≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for n = 1. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Applications of Schochet''s methods to parabolic equations

Methods used by S. Schochet in [32] enable one to find a lower bound for the life span of solutions of hyperbolic PDEs with a small parameter. We prove a similar theorem for such equations where a diffusion term has been added, with the minimal assumption on the Sobolev regularity of the initial data ( in the d-dimensional torus). When the data is smooth and under a “small divisor” assumption on the perturbation, the first term of an asymptotic expansion of the solution is computed. Those results are then applied to prove global existence theorems, for arbitrary initial data, in the case of the primitive system of the quasigeostrophic equations, followed by the rotating fluid equations. We finally prove a more precise existence theorem for the latter, using anisotropic Sobolev and Besov spaces.     

LOCAL THEORY IN CRITICAL SPACES FOR COMPRESSIBLE VISCOUS AND HEAT-CONDUCTIVE GASES

We are concerned with local existence and uniqueness of solutions for a general model of viscous and heat-conductive gases with low regularity assumptions on the initial data (the velocity and the temperature may be discontinuous). Local well-posedness is showed to hold in spaces which are critical with respect to the scaling of the equations, provided that the initial density is close enough to a positive constant. When initial data are a trifle more regular, local well-posedness holds for any initial density bounded away from zero. This former result lies on new estimates for linear heat equations with a non constant diffusion coefficient.     

Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh–Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.     

Asymptotic profile of solutions for some wave equations with very strong structural damping

We consider the Cauchy problem in R ⁿ for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by using a simple method introduced in by the first author. The obtained results will include regularity loss type estimates, which are essentially new in this kind of equation.     

The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

It is proved herein that certain smooth, global solutions of a class of quasi-linear, dissipative wave equations have precisely the same leading order, long-time, asymptotic behavior as the solutions with the same initial data of the corresponding linearized equations. The solutions of the nonlinear equations are shown to be asymptotically self-similar with explicitly determined profiles. The equations considered have homogeneous nonlinearities and homogeneous dispersive and dissipative symbols. By relating these degrees of homogeneous to the leading order asymptotic behavior of the Fourier transform of the initial data near k= 0, different classes of long-time asymptotic behavior are characterized. These results cover the case where dissipation is not asymptotically negligible in comparison with dispersion, and where nonlinear effect are asymptotically negligible in comparison with linear effect, i.e., dissipation and dispersion. They always hold for solutions with "small" initial data. In most circumstances however a new a priori bound on certain negative homogeneous Sobolev norms of solutions is obtained, which implies that any solution, even one which is initially "large" will eventually satisfy the smallness condition, and hence will have the above described asymptotic behavior     

Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy

This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one if the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. It is shown that the critical Fujita exponent depends on the degeneracy of the equation and the asymptotic behavior of the diffusion coefficient at infinity, and it may be equal to one or infinity. Furthermore, the critical case is proved to belong to the blowing-up case.     

Gevrey regularity for a class of dissipative equations with applications to decay

In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via “caloric extension”. We apply our result to the Navier–Stokes equations, the surface quasi-geostrophic equations, the Kuramoto–Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier–Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation.     

Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations

We establish one-to-one transformations and self-maps between nonlinear diffusion equations in nonhomogeneous media, where the density function is given by a power. We use these transformations to deduce new interesting self-similar, radially symmetric solutions of the equations. In particular, Barenblatt, dipole and focusing Aronson-Graveleau type solutions are deduced, and some equations with singular potentials are studied. The new solutions are example of interesting or unexpected mathematical features of these equations, providing also natural candidates for the asymptotic behavior.     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study C ^∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006.     

Optimal control for multi-phase fluid Stokes problems

Optimal control for a system consistent of the viscosity dependent Stokes equations coupled with a transport equation for the viscosity is studied. Motivated by a lack of sufficient regularity of the adjoint equations, artificial diffusion is introduced to the transport equation. The asymptotic behavior of the regularized system is investigated. Optimality conditions for the regularized optimal control problems are obtained and again the asymptotic behavior is analyzed. The lack of uniqueness of solutions to the underlying system is another source of difficulties for the problem under investigation.     

Propagation dynamics of a time periodic diffusion equation with degenerate nonlinearity

This paper is on study of traveling wave solutions and asymptotic spreading of a class of time periodic diffusion equations with degenerate nonlinearity. The asymptotic behavior of traveling wave solutions is investigated by using auxiliary equations and a limit process. In addition, the monotonicity and uniqueness, up to translation, of traveling wave solution with critical speed are determined by sliding method. Finally, combining super and sub-solutions and the stability of steady states, some sufficient conditions on asymptotic spreading are given, which indicates that the success or failure of asymptotic spreading are dependent on the degeneracy of nonlinearity as well as the size of compact support of initial value.     

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

We consider partial differential equations of drift‐diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure‐valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0⁺. The time evolution of these singular masses is also explicitly described and, as a by‐product, uniqueness of this measure solution is obtained.     

Well‐posedness of a parabolic free boundary problem driven by diffusion and surface tension

Well‐posedness and regularity results are shown for a class of free boundary problems consisting of diffusion on a free domain where the boundary movement depends on its mean curvature of the boundary and the diffusion on the boundary, and initial conditions are radially symmetric. Short‐time existence and uniqueness of solutions in a suitable Sobolev space are shown using a fixed‐point argument. Higher regularity is a posteriori. Finally, it is shown that solutions exist globally in time and converge to equilibrium if the boundary movement depends on the mean curvature of the boundary and diffusion in a specific way. A mathematical model describing the swelling of a cell due to osmosis is treated as an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence of weak‐type solutions for models of monotone type in the theory of inelastic deformations

This article introduces the notion of weak‐type solutions for systems of equations from the theory of inelastic deformations, assuming that the considered model is of monotone type (for the definition see [Lecture Notes in Mathematics, 1998, vol. 1682]). For the boundary data associated with the initial‐boundary value problem and satisfying the safe‐load condition the existence of global in time weak‐type solutions is proved assuming that the monotone model is rate‐independent or of gradient type. Moreover, for models possessing an additional regularity property (see Section 5) the existence of global solutions in the sense of measures, defined by Temam in Archives for Rational Mechanics and Analysis, 95 : 137, is obtained, too. Copyright © 2002 John Wiley & Sons, Ltd.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.