Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.     

Similarities Between Rotation and Stratification Effects on Homogeneous Shear Flow

Linear theory is applied to examine rotation and buoyancy effects on homogeneous turbulent shear flows with given vertical velocity shear, S=d/dx ₃. In the rotating shear case (where the rotation vector is perpendicular to the plane of the mean flow, Ω _i =Ωδ _{i 2}), general solutions for the Fourier components of the fluctuating velocity are proposed. These solutions are compared with those proposed in the literature for the Fourier components of the fluctuating velocity and density in the case of a homogeneous stratified shear flow with vertical density gradient, S _ρ=d/dx ₃. It is shown that from the normal mode stability stand point the Bradshaw parameter B=2Ω/S(1+2Ω/S) (in the rotating shear case) and the Richardson number R _i (in the statified shear case) play similar roles in identifying the stability for all the wave components except in the case where Ω·k=0, for which rotation has no effects on the flow. Analysis of the long-time behavior of the non-dimensional spectral density of energy, e ^g , is carried out. In the stable case, e ^g has decaying oscillations or undergoes a power law decay in time. Analytical solutions for the streamwise two-dimensional energy ℰ _ii ¹/2 (i.e. the limit at k ₁=0 of the one-dimensional energy spectra) are proposed. At large time, ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (3R _i +1)/(4R _i ) except at R _i =1 it stays constant in time. Similar behavior for ℰ _ii ¹(t)/ℰ _ii ¹(0) is also observed in the rotating shear case (ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (1+4B)/(4B)). Due to the behavior of the dimensionless spectral density of energy in both flow cases, the turbulent kinetic energy, /2, the production rate, ?, and the rate due to the buoyancy forces, ℬ, are split into two parts, , ?=?₁+?₂, ℬ=ℬ₁+ℬ₂ (in the stratified shear case, both ?₁ and ℬ₁ vanish when R _i >?, while in the rotating shear case one has ℬ=0). It is shown that when rotation is “cyclonic” (i.e. Ω/S>0), part reaches maximum magnitudes at St ≈2, independent of the B value, and the first time to a zero crossing of ?₂ occurs at this particular value. When rotation is “anticyclonic” (i.e. Ω/S<0) one finds St ≈1.6 instead of St ≈2. In the stratified shear case, both ?₂ and ℬ₂ cross zero at Nt=St ≈2, and part reaches maximum magnitudes at this particular value. These results and in particular those for the turbulent kinetic energy are compared with previous direct numerical simulation (DNS) results in homogeneous stratified shear flows. Received 30 July 2001 and accepted 19 February 2002     

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.     

On the perturbational global attractivity of nonautomous delay differential equations

Consider the perturbed nonautonomous linear delay differential equation x(t) = - a(t)x(t-τ) + F(t, x₁, t ⩾ 0 where x₁(s)=x(t+s) for −δ≤s≤0. Suppose that a(t) ∈ C([0,∞), (0,∞)), τ≥0,F:[0, ∞) x C[−δ,0] → R is a continuous functions and F(t,0) ≡ 0. Here C[−δ,0] is the space of continuous functions Φ: [−δ,0] → R with ∥Φ∥<H for the norm | Φ |, where |·| is any norm in R and 0<H≤+∞. Most of the known papers [1–5,7] have been concerned with the local or global asymptotic behavior of the zero solution of Eq. (*) when a(t) is independent of t i. e., a(t) is autonomous. The aim in this paper is to derive the sufficient conditions for the global attractivity of the zero solution of of Eq. (*) When a(t) is nonautomous. Our results, which extend and improve the known results, are even “sharp”. At the same time, the method used in this paper can be applicable to the perturbed nonlinear equation. Project supported by the Natural Science Foundation of Hunan     

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

On the free energy and variational theorem of elastic problem in thermal shock

In some investigations on variational principle for coupled thermoelastic problems, the free energy Φ(e_ij,θ) ,where the state variables are elastic strain e_ij and temperature increment θ, is expressed as Φ(e_ij,θ)=λ/2e_kke_ij=ue_k1e_k1-γe_kkθ-c/2 p θ²/T₀(0.1) This expression is employed only under the condition of |θ|≤T₀(absolute temperature of reference) But the value of temperature increment is great, even greater than T₀ in thermal shock. And the material properties (λ ,μ ,ν ,c , etc.) will not remain constant, they vary with θ. The expression of free energy for this condition.is derived in this paper. Equation (0.1) is its special case.Euler’s equations will be nonlinear while this expression of free energy has been introduced into variational theorem. In order to linearise, the time interval of thermal shock is divided into a number of time elements Δt_k, (Δt_k=t_k-t_k-1,k=1,2…,n), which are so small that the temperature increment θ_k within it is very small, too. Thus, the material properties may be defined by temperature field T_k-1=T(x₁,x₂,x₃,t_k-1) at instant t_k-1 , and the free energy Φ_k expressed by eg. (0.1) may be employed in element Δt_k.Hence the variational theorem will be expressed partly and approximately.     

On the global stability of one nonlinear difference equation

We establish exact sufficient conditions for the global stability of the trivial solution of the difference equation x _n+1 = x _n + f _n (x _n ,...,x _n−k ) n ∈ ℤ, where the nonlinear functions f _n satisfy negative feedback conditions and have sublinear growth. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 525–534, October–December, 2006.     

Trigonometric simplification of a class of conservative nonlinear oscillators

This paper deals with a class of conservative nonlinear oscillators of the form $\ddot x(t)+f(x(t))=0$ , where f(x) is analytic. A transformation of time from t to a new time coordinate τ is defined such that periodic solutions can be expressed in the form x(τ) = A ₀+A ₁ cos 2τ. We refer to this process of trigonometric simplification as trigonometrification. Application is given to the stability of nonlinear normal modes (NNMs) in two-degree-of-freedom systems.     

On the Global Stability of One Nonlinear Difference Equation

We give exact sufficient conditions for the global stability of the zero solution of the difference equation x _{n + 1} = qx _n + f _n (x _n , ..., x _{n – k} ), n , where the nonlinear functions f _n satisfy the conditions of negative feedback and sublinear growth.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 487–494, October–December, 2004.     

Uniqueness of positive radial solutions of △u+g(r)u+h(r)u p =0 in Rn

Positive radial solutions of a semilinear elliptic equation u+g(r)u+h(r)u ^p =0, where r=|x|, xR ⁿ , and p>1, are studied in balls with zero Dirichlet boundary condition. By means of a generalized Pohoaev identity which includes a real parameter, the uniqueness of the solution is established under quite general assumptions on g(r) and h(r). This result applies to Matukuma's equation and the scalar field equation and is shown to be sharp for these equations.     

The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

We show that, in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable. The crucial point here is that the boundary is not flat. In fact (see Beir?o da Veiga et al. in J Math Anal Appl 377:216–227, 2011) if  W = \mathbb R³₊,{\,\Omega = \mathbb R^3_+,} convergence holds in C([0,T]; W^k,p(\mathbb R³₊)){C([0,T]; W^{k,p}(\mathbb R^3_+))}, for arbitrarily large k and p. For this reason, the negative answer given here was not expected.     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Uniform Local Existence for Inhomogeneous Rotating Fluid Equations

We investigate the equations of anisotropic incompressible viscous fluids in , rotating around an inhomogeneous vector B(t, x ₁, x ₂). We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniform local existence result with respect to the Rossby number in the same functional spaces under the additional assumption that B = B(t, x ₁) or B = B(t, x ₂). We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.     

On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[w;α]=0 where F:X×R^k→Y,X,Y are Banach spaces and XY. We focus attention on problems whose bifurcation equations are of the form

f_i(α₁,α₂;λ,μ)=(a_iμ+b_iλ)α_i+p_iα_i³+q_iα_i∑_j=1,j≠i^kα_j+1²+α_ih_i(λ,μ;α₁,α₂,…α_k) i=1,2,…k

which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.     

Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

We consider degenerate reaction diffusion equations of the form u _t ?=???u ^m ?+?f(x, u), where f(x, u) ~ a(x)u ^p with 1??? p m. We assume that a(x)?>?0 at least in some part of the spatial domain, so that ${u \equiv 0}$ is an unstable stationary solution. We prove that the unstable manifold of the solution ${u \equiv 0 }$ has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as ${t\to -\infty}$ while its support shrinks to an arbitrarily chosen point x* in the region where a(x)?>?0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.     

Bounded solutions of some nonlinear elliptic equations in cylindrical domains

The existence of a (unique) solution of the second-order semilinear elliptic equation $$\sum\limits_{i,j = 0}^n {a_{ij} (x)u_{x_i x_j } + f(\nabla u,u,x) = 0}$$ withx=(x ₀,x ₁,?,x _n )?(s ₀, ∞)× Ω′, for a bounded domainΩ′, together with the additional conditions $$\begin{array}{*{20}c} {u(x) = 0for(x_1 ,x_2 ,...,x_n ) \in \partial \Omega '} \\ {u(x) = \varphi (x_1 ,x_2 ,...,x_n )forx_0 = s_0 } \\ {|u(x)|globallybounded} \\ \end{array}$$ is shown to be a well-posed problem under some sign and growth restrictions off and its partial derivatives. It can be seen as an initial value problem, with initial value?, in the spaceC ₀ ⁰ $(\overline {\Omega '} )$ and satisfying the strong order-preserving property. In the case thata _ij andf do not depend onx ₀ or are periodic inx ₀, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions onf are given under which all the solutions tend to zero asx ₀ tends to infinity. Proofs are strongly based on maximum and comparison techniques.     

Some new classes of inverse coefficient problems in non-linear mechanics and computational material science

Three classes of inverse coefficient problems arising in engineering mechanics and computational material science are considered. Mathematical models of all considered problems are proposed within the J₂-deformation theory of plasticity. The first class is related to the determination of unknown elastoplastic properties of a beam from a limited number of torsional experiments. The inverse problem here consists of identifying the unknown coefficient g(ξ²) (plasticity function) in the non-linear differential equation of torsional creep −(g(|∇u|²)u_x1)_x1−(g(|∇u|²)u_x2)_x2=2?, x∈Ω⊂R², from the torque (or torsional rigidity) T(?), given experimentally. The second class of inverse problems is related to the identification of elastoplastic properties of a 3D body from spherical indentation tests. In this case one needs to determine unknown Lame coefficients in the system of PDEs of non-linear elasticity, from the measured spherical indentation loading curve P=P(α), obtained during the quasi-static indentation test. In the third model an inverse problem of identifying the unknown coefficient g(ξ²(u)) in the non-linear bending equation is analyzed. The boundary measured data here is assumed to be the deflections w_i[τ_k]?w(λ_i;τ_k), measured during the quasi-static bending process, given by the parameter τ_k, , at some points , of a plate. An existence of weak solutions of all direct problems are derived in appropriate Sobolev spaces, by using monotone potential operator theory. Then monotone iteration schemes for all the linearized direct problems are proposed. Strong convergence of solutions of the linearized problems, as well as rates of convergence is proved. Based on obtained continuity property of the direct problem solution with respect to coefficients, and compactness of the set of admissible coefficients, an existence of quasi-solutions of all considered inverse problems is proved. Some numerical results, useful from the points of view of engineering mechanics and computational material science, are demonstrated.     

Third-order systems: Periodicity conditions

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation

x+f(t,x,x^′,x^″)=0