On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ ₀=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.     

The existence of solution of a class of two-order quasilinear boundary value problem

Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献   

Eventual C∞-regularity and concavity for flows in one-dimensional porous media

We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation $$\begin{gathered} {\text{ }}u_t = \left( {u^m } \right)_{xx} {\text{ in }}Q = \mathbb{R} \times \left( {{\text{0,}}\infty } \right){\text{,}} \hfill \\ u\left( {x{\text{,0}}} \right) = u_{\text{0}} \left( x \right){\text{ for }}x \in \mathbb{R}{\text{,}} \hfill \\ \end{gathered}$$ with m > 1 and, u ₀a continuous, nonnegative function. It is well known that, across a moving interface x=ζ(t) of the solution u(x, t), the derivatives v tand v _x of the pressure v = (m/(m?1)) u ^m?1 have jump discontinuities. We prove that each moving part of the interface is a C ^∞curve and that v is C ^∞on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t→∞.     

Effects of periodic forcing on delayed bifurcations

This work is concerned with the rigorous analysis of the effects of small periodic forcing (perturbations) on the dynamical systems which present some interesting phenomena known as delayed bifurcations. We study the dynamical behavior of the system (0.1) $$\begin{gathered} \frac{{\partial u}}{{\partial t}} = f\left( {u, I_i + \varepsilon t} \right) + \varepsilon g\left( {u, I_i + \varepsilon t, \varepsilon , t} \right) \hfill \\ \left. {u\left( t \right)} \right|_{t = 0} = u_0 \left( {I_i } \right) + O\left( \varepsilon \right) \hfill \\ \end{gathered} $$ whereu ₀(I) is the solution off(u ₀(I), I)=0 andI(t)=I _i+εt is a slowly varying parameter that moves past a critical pointI_ of the system so that the linear stability aroundu ₀(I) changes from stable to unstable atI_. General results are given with respect to the effects of the perturbation εg(u,I(t),ε, t) to several important types of dynamical systems (0.2) $$\frac{{\partial u}}{{\partial t}} = f\left( {u,I_i + \varepsilon t} \right)$$ which present dynamical patterns that there exist persistent unstable solutions in the dynamical systems (delayed bifurcations) in contrast to bifurcations in the classical sense. It is shown that (1) the delayed bifurcations persist if the frequency ofg(…,…,…,t) ont is a constantΩ which is not a resonant frequency; (2) in case the frequency ofg(…,…,…,t) ont isΩ=Ω(I _i+εt) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending onΩ′(I_)>0 orΩ′(I_)<0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in a codimension one parameter family of periodic perturbations. (1) is a rigorous analysis of the results in [3], (2) is a new and interesting phenomenon, and (3) is a generalization of the results of Diener [8] and Schecter [19].     

Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water

ThePekeris differential operator is defined by $$Au = - c^2 (x_n )\rho (x_n )\nabla \cdot \left( {\frac{1}{{\rho (x_n )}}\nabla u} \right),$$ wherex=(x ₁,x ₂,...x _n )∈R _n ,?=(?/?x ₁, ?/?x ₂,...?/?x _n ), and the functionsc(x _n),σ(x _n) satisfy $$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ and $$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ wherec ₁,c ₂,? ₁,? ₂, andh are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speedc ₁ and density? ₁ which overlays a bottom having sound speedc ₂ and density? ₂. In this paper it is shown that the operatorA, acting on a class of functions u (x) which are defined for x_n≧0 and vanish for x_n=0, defines a selfadjoint operator on the Hilbert space whereR ₊ ⁿ ={x∈R ⁿ :x _n >0} anddx =dx ₁ dx ₂...dx _n denotes Lebesgue measure in R ₊ ⁿ . The spectral family ofA is constructed and the spectrum is shown to be continuous. Moreover an eigenfunction expansion for A is given in terms of a family of improper eigenfunctions. Whenc ₁≧c ₂ each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c₁< c₂, additional eigen-functions arise which can be interpreted as plane waves that are trapped in the layer 0n h by total reflection at the interface x_n=h.     

Desingularization of Vortex Rings and Shallow Water Vortices by a Semilinear Elliptic Problem

Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and shown to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problem $$ \left\{ \begin{aligned} -{\rm div} \left(\frac{\nabla u_{\varepsilon}}{b}\right) & = \frac{1}{\varepsilon^2} b f \left(u_{\varepsilon} - \log \tfrac{1}{\varepsilon} q \right) & & \text{ in } \; \Omega, \\u_\varepsilon & = 0 & & \text{ on } \; \partial \Omega, \end{aligned}\right.$$ for small values of ${\varepsilon > 0}$ .     

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.     

A well posed problem in singular Fickian diffusion

We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L _loc ^∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.     

Degenerate solutions of electron-beam equations

In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in [1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In [2] and in a number of papers referred to in [2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component [3] and to relativistic conical flow [1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left[ {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where q^β denotes orthogonal coordinates with the metric tensor g^ββ (β=1,2,3); A_α is the magnetic potential; A_α = (u_α/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; p_α is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio.     

Turbulent heat transfer in circular tube flows of viscoelastic fluids

The effects of thermal entrance length, polymer degradation and solvent chemistry were found to be critically important in the determination of the drag and heat transfer behavior of viscoelastic fluids in turbulent pipe flow. The minimum heat transfer asymptotic values in the thermally developing and in the fully developed regions were experimentally determined for relatively high concentration solutions of heat transfer resulting in the following correlations: $$\begin{gathered} j_H = 0.13\left( {\frac{x}{d}} \right)^{ - 0.24} \operatorname{Re} _a^{ - 0.45} thermally developing region \hfill \\ x/d< 450 \hfill \\ j_H = 0.03 \operatorname{Re} _a^{ - 0.45} thermally developed region \hfill \\ x/d< 450 \hfill \\ \end{gathered} $$ For dilute polymer solutions the heat transfer is a function ofx/d, the Reynolds number and the polymer concentration. The Reynolds analogy between momentum and heat transfer which has been widely used in the literature for Newtonian fluids is found not to apply in the case of drag-reducing viscoelastic fluids.     

On the symmetry of deformable crystals

We prove existence, uniqueness, and regularity properties for a solution u of the Bellman-Dirichlet equation of dynamic programming: (1) $$\left\{ \begin{gathered} \max {\text{ }}\{ L^i u + f^i = 0{\text{ in }}\Omega \hfill \\ i{\text{ = 1,2 }} \hfill \\ u{\text{ = 0 on }}\partial \Omega , \hfill \\ \end{gathered} \right.$$ where L ¹ and L ² are two second order, uniformly elliptic operators. The method of proof is to rephrase (1) as a variational inequality for the operator K=L ²(L ¹)^?1 in L ²(Ω) and to invoke known existence theorems. For sufficiently nice f ¹ and f ² we prove in addition that u is in H ³(Ω)?C ^2,α(Ω) (for some 0<α<1) and hence is a classical solution of (1).     

Ergodicity of minimal sets in scalar parabolic equations

Skew product semiflowΠ _t :X ×Y → X × Y generated by $$\left\{ \begin{gathered} u_t = u_{xx} + f(y \cdot t,x,u,u_x ), t > 0 x \in (0,1), y \in Y, \hfill \\ D or N boundary conditions \hfill \\ \end{gathered} \right.$$ is considered, whereX is an appropriate subspace ofH ²(0, 1), (Y, ?) is a compact minimal flow. By analyzing the zero crossing number for certain invariant manifolds and the linearized spectrum, it is shown that a minimal setE?X × Y ofΠ, is uniquely ergodic if and only if (Y, ?) is uniquely ergodic andμ(Y ₀)=1, whereμ is the unique ergodic measure of (Y, ?),Y ₀={ity∈Y} Card(E∩P ^?1(y))=1},P:X × Y → Y is the natural projection (it was proved in an authors' earlier paper thatY ₀ is a residual subset ofY). Moreover, if (E, ?) is uniquely ergodic, then it is topologically conjugated to a subflow ofR ¹ ×Y. A consequence of the last result is the following: in the case that (Y, ?) is almost periodic,Π, is expected to have many purely almost automorphic motions which are not ergodic.     

Filtration problems with a piecewise-linear resistance law

The article discusses elementary solutions of problems of nonlinear filtration with a piece-wise-linear resistance law, and analyzes their behavior with a relative increase in the resistance in the region of small velocities, and a transition to the law of filtration with a limiting gradient. The results obtained are applied to a determination of the dimensions of the stagnant zones in stratified strata. The law of filtration with a limiting gradient (0.1) $$\begin{gathered} w = - \frac{k}{\mu }\left( {grad p - G\frac{{grad p}}{{|grad p|}}} \right),|grad p| > G \hfill \\ w = 0,|grad p|< G \hfill \\ \end{gathered}$$ describes motion in some intermediate range of velocities w, but its satisfaction in the region of the smallest velocities, as a rule, remains unverified. It is natural to pose the problem of the degree to which a divergence between the true filtration law and its approximation (0.1) affects the accuracy of calculation of the flow fields, and the significance of a determination of the dimensions of the stagnant zones under such conditions. To answer this problem to some measure, there are considered below several simple exact (elementary) solutions obtained for a more general nonlinear filtration law (0.2) $$\begin{gathered} grad p = - (\mu /k) (w + \lambda )w/w,\lambda = kG/\mu ,w \geqslant w_0 \hfill \\ grad p = - \mu w/k\varepsilon ), w \leqslant w_0 ,\varepsilon = w_0 /w_0 + \lambda \hfill \\ \end{gathered}$$ going over into (0.1) with w₀→ 0. The solutions obtained are applied also to an evaluation of the dimensions of stagnant zones, forming in stratified strata when the effects of the limiting gradient in one of the intercalations are considerable.     

Initial value problems for a class of nonlinear integro-partial differential equations

１ＰｒｏｂｌｅｍｓａｎｄＭａｉｎＲｅｓｕｌｔｓＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｔｕｄｙｔｈｅｎｏｎｌｉｎｅａｒｖｉｂｒａｔｉｏｎｓｏｆｉｎｆｉｎｉｔｅｒｏｄｓｗｉｔｈｖｉｓｃｏｅｌａｓｔｉｃｉｔｙ．Ｔｈｅｃｏｎｓｔｉｔｕｔｉｏｎｌａｗｏｆｔｈｅｒｏｄｓ...     

A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

In this paper we study the fully nonlinear free boundary problem $$\left\{\begin{array}{ll}F(D^{2}u) = 1 & {\rm almost \, everywhere \, in}\, B_{1} \cap \Omega\\ |D^{2} u| \leqq K & {\rm almost \, everywhere \, in} \, B_{1} \setminus \Omega,\end{array}\right.$$ where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W ^2,n solutions are locally C ^1,1 inside B ₁. Under the extra condition that ${\Omega \supset \{D{u} \neq 0 \}}$ and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary ${\partial \Omega \cap B_1}$ .     

Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l ³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is any portion of?ω withlength γ ₀>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ ₀, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ ₀) is contained in a generatrix ofS. We show that, if the applied body force density isO(? ²) with respect to?, the fieldu(?)=(u _i (?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH ¹(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts _?1 ¹ u dx ₃, which belongs to the spaceV _F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η _i ) ∈V _F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified.     

Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation

Li and Qiao studied the bifurcations and exact traveling wave solutions for the generalized two-component Camassa–Holm equation $$\begin{aligned} \left\{ \begin{array}{l} m_{t}+\sigma um_{x}-Au_{x}+2m \sigma u_{x}+3(1-\sigma )uu_{x}\\ \quad +\rho \rho _{x}=0, \\ \rho _{t} +(\rho u)_{x}=0, \end{array} \right. \end{aligned}$$ \(m=u-u_{xx}, A>0\) . They showed that there exist solitary wave solutions, cusp wave solutions, and periodic wave solutions for the equation, and their analysis focused on the bifurcations when \(\sigma >0\) . In this paper, we first complement the bifurcations when \(\sigma <0\) by following the same procedure as that of Li, and then show the existence and implicit expressions of several new types of bounded wave solutions, including solitary waves, periodic waves, compacton-like waves, and kink-like waves. In addition, the numerical simulations of the bounded wave solutions are given to show the correctness of our results.     

Improved theoretical model for wavy filmwise condensation

This paper presents a numerical solution for wavy laminar film-wise condensation on vertical walls. Integral method is achieved based on the recently developed simple wave equations. Solutions are obtained for ranges of dimensionless groups as follows: $$1.5 \leqslant \left( {Pr = \frac{{^{\mu C} p}}{k}} \right) \leqslant 6.0$$ $$10 \leqslant \left( {G = \frac{{^h fg}}{{^{C_p \Delta T} }}} \right) \leqslant 400$$ $$100 \leqslant \left( {S = \left( {\frac{{\sigma ^2 \rho }}{{g_\rho \mu ^4 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} } \right) \leqslant 400$$ $$1000 \leqslant \left( {L = \frac{{{\rm H}_t }}{{^\delta cr}}} \right) \leqslant 10000$$ . Such ranges cover the expected situations in industrial applications. It is found that the Reynolds number (Re=h_LΔTH_t/h_fg) is a linear function of L on the log-log plane. It is also relatively insensitive to small variations of Pr at high values of this number. At situations where G less than 200 the Re appears to be dependent on S. Agreement with experimental observation is improved over that obtained from previous analytical theories.     

Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities

On the basis of some very plausible assumptions about the response of physical systems to stimuli, such as Boltzmann's superposition principle and the causality principle, Spence showed that the following characteristics obtain for the modulus and compliance functions: (i) They are analytic in the lower half of the complex frequency plane, (ii) they are limited if the frequency tends to infinity, and (iii) the real and imaginary parts are even and odd functions, respectively, of the frequencyω. It can generally be demonstrated that the real and imaginary parts of every function satisfying these three requirements and (iv) without singularities on the real frequency axis, are interrelated by Kramers-Kronig transforms. Similar relations hold between the logarithm of the modulus and the argument of the function. Under certain conditions the Kramers-Kronig relations may be approximated by rather simple equations. For linear viscoelastic materials, for instance, the following approximate relations were obtained for the components of the complex dynamic shear modulus,G ^* (iω) = G′(ω) + iG″(ω) = G _d (ω) expiδ(ω): $$\begin{gathered} G'' (\omega ) \simeq \frac{\pi }{2}\left( {\frac{{dG'(u)}}{{d In u}}} \right)_{u = \omega } , \hfill \\ G' (\omega ) - G'(o) \simeq - \frac{{\omega \pi }}{2}\left( {\frac{{d[G''(u)/u]}}{{d In u}}} \right)_{u = \omega } , \hfill \\ \delta (\omega ) \simeq \frac{\pi }{2}\left( {\frac{{d In G_d (u)}}{{d In u}}} \right)_{u = \omega } . \hfill \\ \end{gathered} $$ The first of these relations was published long ago by Staverman and Schwarzl and is useful over broad frequency ranges, as is the second relation. The last equation is the most general one, and also is better supported by experiment.