Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations

We consider reaction diffusion equations of the prototype form u _t = u _xx + λ u + |u| ^p-1 u on the interval 0 < x < π, with p > 1 and λ > m ². We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u ₁, . . . ,u _m . Since u _k = ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x ₁, . . . ,x _m of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation). Dedicated to Palo Brunovsky on the occasion of his birthday.     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

Lp-Lq Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle

We consider the motion of a viscous fluid filling the whole three-dimensional space exterior to a rotating obstacle with constant angular velocity. We develop the L _p -L _q estimates and the similar estimates in the Lorentz spaces of the Stokes semigroup with rotation effect. We next apply them to the Navier–Stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t → ∞ with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in L _3,∞ (weak-L ₃ space).     

Asymptotic analysis of the mathematical expectation of the total energy of a harmonic oscillator under random pulse perturbation

We study the behavior of the mathematical expectation of the total energy of a harmonic frictionless oscillator as t → ∞ under external Poisson pulse perturbation.     

Stability of Degenerate Stationary Waves for Viscous Gases

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t ^−α/4 as t → ∞, provided that the initial perturbation is in the weighted space L²_a=L²(\mathbb R₊; (1+x)^a dx){L^2_\alpha=L^2({\mathbb R}_+;\,(1+x)^\alpha dx)} . This convergence rate t ^−α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α_*(q), where α_*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L²_a{L^2_\alpha} for α > α^*(q) with another critical value α^*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.     

Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

We perform a mathematical analysis of the steady flow of a viscous liquid, L{\mathcal{L}} , past a three-dimensional elastic body, B{\mathcal{B}} . We assume that L{\mathcal{L}} fills the whole space exterior to B{\mathcal{B}} , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v _∞. As for B{\mathcal{B}} , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of B{\mathcal{B}} as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v _∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.     

A note on the definition of fractional derivatives applied in rheology

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives—Riemann–Liouville (R–L) definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R–L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model (Scott–Blair model). We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R–L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R–L derivatives with lower terminal a →∞ (or, equivalently, the Caputo derivatives with lower terminal a →∞) not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott–Blair model and give an exact solution of this equation under given initial conditions.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Spectral Stability of Small-Amplitude Viscous Shock Waves in Several Space Dimensions

A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues correspond to zeros of an essentially analytic function E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished that relate the small strength e{\epsilon} of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρ ω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

A Beale–Kato–Majda Criterion for Three Dimensional Compressible Viscous Heat-Conductive Flows

We prove a blow-up criterion in terms of the upper bound of (ρ, ρ ⁻¹, θ) for a strong solution to three dimensional compressible viscous heat-conductive flows. The main ingredient of the proof is an a priori estimate for a quantity independently introduced in Haspot (Regularity of weak solutions of the compressible isentropic Navier–Stokes equation, arXiv:1001.1581, 2010) and Sun et al. (J Math Pure Appl 95:36–47, 2011), whose divergence can be viewed as the effective viscous flux.     

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W ^1,p -W ^1,q “duality pairing” with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W ^1,∞-W ^1,1 “duality pairing” which leads to optimal order error estimates in a discrete W ^1,∞-norm.     

On the Well-posedness of the Ideal MHD Equations in the Triebel–Lizorkin Spaces

In this paper, we prove the local well-posedness for the ideal MHD equations in the Triebel–Lizorkin spaces and obtain a blow-up criterion of smooth solutions. Specifically, we fill a gap in a step of the proof of the local well-posedness part for the incompressible Euler equation in Chae (Comm Pure Appl Math 55:654–678 2002).     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

Thermal convection in a porous layer: Effects of anisotropy and surface boundary conditions

The theory describing the onset of convection in a homogeneous porous layer bounded above and below by isothermal surfaces is extended to consider an upper boundary which is partly permeable. The general boundary condition p + λ ∂p/∂n = constant is applied at the top surface and the flow is investigated for various λ in the range 0 ⩽ λ < ∞. Estimates of the magnitude and horizontal distribution of the vertical mass and heat fluxes at the surface, the horizontally-averaged heat flux (Nusselt number) and the fraction of the fluid which recirculates within the layer are found for slightly supercritical conditions. Comparisons are made with the two limiting cases λ → ∞, where the surface is completely impermeable, and λ = 0, where the surface is at constant pressure. Also studied are the effects of anisotropy in permeability, ξ = K _H /K _V , and anisotropy is thermal conductivity, η = k _H /k _V , both parameters being ratios of horizontal to vertical quantities. Quantitative results are given for a wide variety of the parameters λ, ξ and η. In the limit ξ/η → 0 there is no recirculation, all fluid being converted out of the top surface, while in the limit ξ/η → ∞ there is full recirculation.     

A comment on the “linear” law of the wall for fully developed turbulent channel flow

 In fully developed turbulent channel or pipe flows, the validity of a viscous sublayer with a quadratic mean velocity profile strictly requires the gradient of the Reynolds shear stress to be negligible compared to the gradient of the viscous shear stress. Direct numerical simulations suggest that this requirement is satisfied only in the limit y ⁺→0. The use of a quartic, instead of quadratic, profile represents the available numerical and experimental mean velocity data satisfactorily in the region y ⁺<10. Received: 28 July 1997 / Accepted: 6 February 1998     

Resolvent Estimates in L p for the Stokes Operator in Lipschitz Domains

We establish the L ^p resolvent estimates for the Stokes operator in Lipschitz domains in , for . The result implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in L ^p for (3/2) − ε < p < 3 + ε. This gives an affirmative answer to a conjecture of M. Taylor (Progr. Nonlinear Differential Equations Appl., vol. 42, pp. 320–334).     

An Eigenvalue Criterion for Stability of a Steady Navier–Stokes Flow in {\mathbb{R}}^3

We study the resolvent equation associated with a linear operator L{\mathcal{L}} arising from the linearized equation for perturbations of a steady Navier–Stokes flow U^*{\mathbf{U^*}}. We derive estimates which, together with a stability criterion from [33], show that the stability of U^*{\mathbf{U^*}} (in the L²-norm) depends only on the position of the eigenvalues of L{\mathcal{L}}, regardless the presence of the essential spectrum.