Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Global existence for one-dimensional motion of non-isentropic viscous fluids

We study the p-system with viscosity given by v_t ? u_x = 0, u_t + p(v)_x = (k(v)u_x)_x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v₀, u₀(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Beyond quenching for singular reaction-diffusion problems

Let f(u) be twice continuously differentiable on [0, c]) for some constant c such that f(0) > 0,f′ ? 0,f″ ? 0, and lim_u→cf(u) = ∞. Also, let χ(S) be the characteristic function of the set S. This article studies all solutions u with non-negative u_t, in the region where u < c and with continuous u_x for the problem: u_xx – u_t = ? f(u)χ({u < c}), 0 < x < a, 0 < t < ∞, subject to zero initial and first boundary conditions. For any length a larger than the critical length, it is shown that if ∫f(u) du < ∞, then as t tends to infinity, all solutions tend to the unique steady-state profile U(x), which can be computed by a derived formula; furthermore, increasing the length a increases the interval where U(x) ? c by the same amount. For illustration, examples are given.     

Time asymptotics for the polyharmonic wave equation in waveguides

Let Ω denote an unbounded domain in ?ⁿ having the form Ω=?^l×D with bounded cross‐section D??^n?l, and let m∈? be fixed. This article considers solutions u to the scalar wave equation ?u(t,x) +(?Δ)^mu(t,x) = f(x)e^?iωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t→∞ is investigated. Depending on the choice of f ,ω and Ω, two cases occur: Either u shows resonance, which means that ∣u(t,x)∣→∞ as t→∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd.     

Velocity averaging,kinetic formulations,and regularizing effects in quasi‐linear PDEs

We prove in this paper new velocity‐averaging results for second‐order multidimensional equations of the general form ??(?_x, v)f(x, v) = g(x, v) where ??(?_x, v) := a (v) · ?_x ? ? ${}_{\text{x}}^{\text{?}}$ · b (v)?_x. These results quantify the Sobolev regularity of the averages, ∫_v f(x, v)?(v)dv, in terms of the nondegeneracy of the set {v: |??(iξ, v)| ≤ δ} and the mere integrability of the data, (f, g) ∈ (L, L). Velocity averaging is then used to study the regularizing effect in quasi‐linear second‐order equations, ??(?_x, ρ)ρ = S(ρ), which use their underlying kinetic formulations, ??(?_{x, v})χ_ρ = g_S. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection‐diffusion and elliptic equations driven by degenerate, nonisotropic diffusion. © 2007 Wiley Periodicals, Inc.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

Boltzmann diffusive limit beyond the Navier‐Stokes approximation

Given a normalized Maxwellian μ and n ≥ 1, we establish the global‐in‐time validity of a diffusive expansion for a solution F^ε to the rescaled Boltzmann equation (diffusive scaling) inside a periodic box ??³. We assume that in the initial expansion (0.1) at t = 0, the fluid parts of these f_m(0,x,v) have arbitrary divergence‐free velocity fields as well as temperature fields for all 1 ≤ m ≤ n while f₁(0,x,v) has small amplitude in H². For m ≥ 2, these f_m(t,x,v) are determined by a sequence of linear Navier‐Stokes‐Fourier systems iteratively. More importantly, the remainder f(t,x,v) is proven to decay in time uniformly in ε via a unified nonlinear energy method. In particular, our results lead to an error estimate for f₁(t,x,v), the well‐known Navier‐Stokes‐Fourier approximation, and beyond. The collision kernel Q includes hard‐sphere, the cutoff inverse‐power, as well as the Coulomb interactions. © 2005 Wiley Periodicals, Inc.     

Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation: (1) for (t,x) ∈ ?_t × ?. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ∣V(t,x)∣?C(1+t)(1+∣x∣) for a.e. (t,x) ∈ ?_t × ? where C, σ₀>0 are real constants, then for any real number λ that satisfies there exists a real number ρ(f,g,λ)>0 such that the equation has a global solution provided that 0<ρ?ρ(f,g,λ). Copyright © 2004 John Wiley & Sons, Ltd.     

On Σ1-definable Functions Provably Total in I ∏

We prove the following theorem: Let φ(x) be a formula in the language of the theory PA^? of discretely ordered commutative rings with unit of the form ?yφ′(x,y) with φ′ and let ∈ Δ₀ and let fφ: ? → ? such that f_φ(x) = y iff φ′(x,y) & (?z < y) φ′(x,z). If I ∏ ∈(?x ≥ 0), φ then there exists a natural number K such that I ∏ ? ?y?x(x > y ? ?φ(x) < x^K). Here I ∏₁^? denotes the theory PA^? plus the scheme of induction for formulas φ(x) of the form ?yφ′(x,y) (with φ′) with φ′ ∈ Δ₀.     

Tangential Convergence of Temperatures and Harmonic Functions in Besov and in Triebel-Lizorkin Spaces

We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R₊^N+1 of a distribution in the Besov space B^α_p,q(R^N) or in the Triebel-Lizorkin space F^α_p,q(R^N). In particular, we prove that when Ω= {|_y|^{N/ (N-αp)} < t < 1} the operator M is bounded from F (R^N) into L^p (R^N). The admissible regions for the spaces B (R^N) with p < q are more complicated.     

The ground state eigenvalue of Hill's equation with white noise potential

The distribution of the ground state eigenvalue λ₀(Q) of Hill's operator Q = −d²/dx² + q(x) on the circle of perimeter 1 is expressed in two different ways in case the potential q is standard white noise. Let WN be the associated white noise measure, and let CBM be the measure for circular Brownian motion p(x), 0 ≤ x < 1, formed from the standard Brownian motion b(x), 0 ≤ x ≤ 1, starting at b(0) = a, by conditioning so that b(1) = a, and distributing this common level over the line according to the measure da. The connection is based upon the Ricatti correspondence q(x) = λ + p′ (x) + p²(x). The two versions of the distribution are (1) in which $\overline{p}$ is the mean value ∫ pdx, and (2) the left‐hand side of (2) being the density for (1) and CBM₀ the conditional circular Brownian measure on $\overline{p}$ = 0. (1) and (2) are related by the divergence theorem in function space as suggested by the recognition of the Jacobian factor the outward‐pointing normal component of the vector field v(x) = ∂Δ(λ)/∂q(x), 0 ≤ x < 1, Δ being the Hill's discriminant for Q. The Ricatti correspondence prompts the idea that (1) and (2) are instances of the Cameron‐Martin formula, but it is not so: The latter has to do with the initial value problem for Ricatti, but it is the periodic problem that figures here, so the proof must be done by hand, by finite‐dimensional approximation. The adaptation of 1 and 2 to potentials of Ornstein‐Uhlenbeck type is reported without details. © 1999 John Wiley & Sons, Inc.     

Two‐dimensional chemotaxis models with fractional diffusion

In this paper we study the Cauchy problem for the fractional diffusion equation u_t + (?Δ)^α/2u=?·(u?(Δ^?1u)), generalizing the Keller–Segel model of chemotaxis, for the initial data u₀ in critical Besov spaces ?(?²) with r∈[1, ∞], where 1<α<2. Making use of some estimates of the linear dissipative equation in the frame of mixed time–space spaces, the Chemin ‘mono‐norm method,’ Fourier localization technique and the Littlewood–Paley theory, we obtain a local well‐posedness result. We also consider analogous ‘doubly parabolic’ models. Copyright © 2008 John Wiley & Sons, Ltd.     

An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u₁ + au_x1 ? vδu = 0 in ?ⁿ × ?⁺ with initial data u⁰; the Support of u⁰ is contained in ?(x₁ < 0) and a: ?ⁿ → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?⁺, we propose the artificial boundary condition: u₁ + au_x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v²) for small values of the viscosity v.     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

Copyright © 2010 John Wiley & Sons, Ltd.     

Strong asymptotics of orthogonal polynomials with respect to exponential weights

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e^−Q(x) dx on the real line, where Q(x) = Σ q_k x^k, q_2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [22, 23]. We employ the steepest‐descent‐type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel‐Rotach‐type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. © 1999 John Wiley & Sons, Inc.