Transition fronts in inhomogeneous Fisher–KPP reaction–diffusion equations

We use a new method in the study of Fisher–KPP reaction–diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction–diffusion equations in several spatial dimensions. Our method is based on the construction of sub- and super-solutions to the non-linear PDE from solutions of its linearization at zero.     

Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0$u_t+H(x,Du)=0$ in Rn×(0,∞)${\mathbf{R}}^n\times (0,\infty )$, where H(x,p)$H(x,p)$ is continuous on Rn×Rn${\mathbf{R}}^n\times {\mathbf{R}}^n$ and convex in p  . We establish a general convergence result for viscosity solutions u(x,t)$u(x,t)$ of the Cauchy problem as t→∞$t\to \infty$.     

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

The incompressible Navier–Stokes flows in a 2D exterior domain are considered, for which the associated total net force to the boundary may not vanish. The decay properties are shown for the first and second derivatives of the Navier–Stokes flows in L ¹ and weighted spaces, respectively, which improve Theorem 1.2 in Bae and Jin (J Funct Anal 240:508–529, 2006).     

Asymptotic behavior of Cauchy hypersurfaces in constant curvature space–times

We present some new examples of families of cubic hypersurfaces in \(\mathbb {P}^5 (\mathbb {C})\) containing a plane whose associated quadric bundle does not have a rational section.     

Asymptotic Lp stability for transition fronts in Cahn–Hilliard systems

We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn–Hilliard systems on $\mathbb{R}$. Such equations arise naturally in the study of phase separation processes, and systems describe cases in which three or more phases are possible. When a Cahn–Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is not separated from essential spectrum. In cases such as this, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. Our main result asserts that if initial perturbations are small in $L^1\cap L^{\infty }$ then spectral stability—a necessary condition for stability, defined in terms of an appropriate Evans function—implies asymptotic nonlinear stability in $L^p$ for all $1<p?\infty$.     

Asymptotic behaviour of travelling waves for the delayed Fisher–KPP equation

In this work we study the behaviour of travelling wave solutions for the diffusive Hutchinson equation with time delay. Using a phase plane analysis we prove the existence of travelling wave solution for each wave speed c?2$c?2$. We show that for each given and admissible wave speed, such travelling wave solutions converge to a unique maximal wavetrain. As a consequence the existence of a nontrivial maximal wavetrain is equivalent to the existence of travelling wave solution non-converging to the stationary state u=1$u=1$.     

Attractors for the inhomogeneous incompressible Navier–Stokes flows

This paper is devoted to the evolution of Lions’s weak solutions to the inhomogeneous Navier–Stokes equations. After proving that the kinetic energy is eventually bounded, we obtain a weakly compact global attractor that all Lions’s weak solutions approach as time tends to infinity. Furthermore, the existence of attracting sets in strong topology is established for short trajectories satisfying an additional compactness condition on the density.     

Volterra-type Ornstein–Uhlenbeck processes in space and time

We propose a novel class of temporo-spatial Ornstein–Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.     

High-order accurate iterative solution of the Navier–Stokes equations for incompressible flows

An iterative solution scheme for the incompressible Navier–Stokes equations is presented. It is split into inner and outer iteration cycles, such that the momentum and continuity equations are satisfied within prescribed accuracy. The spatial discretization is based on high-order finite differences which makes it well suited for massively parallel computers. This is demonstrated in a scaling test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The linear flows in the space of Krichever–Lax matrices over an algebraic curve

Krichever (Commun Math Phys 229(2):229–269, 2002) invented the space of matrices parametrizing the cotangent bundle of moduli space of stable vector bundles over a compact Riemann surface, which is named as the Hitchin system after the investigation (Hitchin, Duke Math J 54(1):91–114, 1987). We study a necessary and sufficient condition for the linearity of flows on the space of Krichever–Lax matrices in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work by Griffiths (Am J Math 107(6):1445–1484, 1985).      

Quantum mechanics of stationary states of particles in a space–time of classical black holes

Theoretical and Mathematical Physics - We consider interactions of scalar particles, photons, and fermions in Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman gravitational...     

A lattice Boltzmann model for the eddy–stream equations in two-dimensional incompressible flows

A lattice Boltzmann model for two-dimensional incompressible flows with eddy–stream equations is proposed. By using two kinds of distribution functions and employing several higher-order moments of equilibrium distribution functions, the eddy equation and stream function equation with the second-order truncation error are obtained. In the numerical examples, we compared the numerical results of this scheme with those obtained by other classical method. The numerical results agree well with the classical ones.     

Emergent dynamics of Cucker–Smale particles under the effects of random communication and incompressible fluids

We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in ${\mathbb{R}}^d$$(d=2,3)$. Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in ${\mathbb{R}}^2$. In a large coupling regime and periodic spatial domain ${\mathbb{T}}^2:={\mathbb{R}}^2/{\mathbb{Z}}^2$, we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different.     

Planar flows of incompressible heat-conducting shear-thinning fluids—Existence analysis

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called L ^∞-truncation method, used to obtain the strong convergence of the velocity gradient. The important point of the approach consists in the choice of an appropriate form of the balance of energy.