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.     

High accuracy time and space transform method for advection–diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection–diffusion equation in the unbounded domain Rⁿ is converted to one on the bounded domain [?1, 1]ⁿ, and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright © 2008 John Wiley & Sons, Ltd.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions

We consider the difference equation

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Contact Transformation Group Classification of Nonlinear Wave Equations

Interest in nonlinear wave equations has been stimulated bynumerous physical applications, such as telecommunication (e.g.nonlinear telegrapher equation), gasdynamics, anisotropic plasticity andnonlinear elasticity, etc. Mathematical models of these phenomena canoften be reduced to particular types of the equation u _tt = f(x, u _x ) u _xx + g(x, u _x ). In this paper, the problem ofclassification of the latter equation with respect to admitted contacttransformation groups is reduced to the investigation of pointtransformation groups of the equivalent system of first-orderquasi-linear equations v _t =a(x, v)w _x , w _t = b(x,v)v _x .     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

A triangular quasi-conforming finite element for transient dynamic analysis

A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation Δu−A ² u=0. The strains in the element are approximated by an exponential function and the string-net function between neighbouring elements is approximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented. The project is supported by the National Natural Science Foundation of China.     

Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form u _t–u=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form u/n _a=g(u(t), u(t–r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.     

On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem:

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      12. Two-phase brine mixtures in the geothermal context and the polymer flood model      Roger Young 《Transport in Porous Media》1993,11(2):179-185 Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.Roman Symbols A,B coefficient matrices (5) - c(x,t) salt concentration (primary dependent variable) - C(p, s, c, q t) wave speed matrix (6) - f source term (5) - g acceleration due to gravity (constant) - h b(p, c) brine specific enthalpy - h v(p) vapour specific enthalpy - j conservation flux (1) - k absolute permeability (constant) - k b(s), kv(s) relative permeabilities of the brine and vapour phases - K conductivity - p(x,t) pressure (primary dependent variable) - q volume flux (Darcy velocity) (3) - s(x,t) brine saturation (primary dependent variable) - t time (primary independent variable) - T=T sat(p) saturation temperature - u b(p, c) brine specific internal energy - u m T rock matrix specific internal energy - u v(p) vapour specific internal energy - U(x, t) shock velocity - x space (primary independent variable) Greek Symbols porosity (constant) - b(p, c) brine dynamic viscosity - v(p) vapour dynamic viscosity - (p, s, c) conservation density (1) - b(p, c) brine density - v(p) vapour density Suffixes b brine - m rock matrix - t total - v vapour - S salt - M mass - E energy      13. Wave Packets, Signaling and Resonances in a Homogeneous Waveguide      Leonid Brevdo 《Journal of Elasticity》1997,49(3):201-237 We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e−iωt + O(e−ɛt ), for t → ∞, where Im ω0 = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.      14. Asymptotic Variational Wave Equations   总被引：1，自引：0，他引：1   Alberto Bressan  Ping Zhang  Yuxi Zheng 《Archive for Rational Mechanics and Analysis》2007,183(1):163-185 We investigate the equation (u t +(f(u)) x ) x =f ′ ′(u) (u x )2/2 where f(u) is a given smooth function. Typically f(u)=u 2/2 or u 3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u tt − c(u) (c(u)u x ) x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.      15. Local energy decay for the wave equation on the exterior of two balls or convex bodies      De-Fu Liu  Nicholas D. Kazarinoff 《Archive for Rational Mechanics and Analysis》1988,103(3):279-288 An algebraic rate of decay of local energy, nonuniform with respect to the initial data, is established for solutions of the Dirichlet and Neumann problems for the scalar wave equation defined on the exterior V3 of two balls or of two convex bodies. That is, for given initial data f(x)=u(x), 0 and g(x)= u t (x, 0), if u solves u tt in V with either u(x, t)=0 or u n (x,t)+(x) u(x,t,)-0 ((x)0) on V, then there exists a constant T 0, depending upon (f, g), such that the local energy (the energy in any compact set) of u at t=T is bounded from above by QE(0)T –1 for TT 0, where E(0) is the total initial energy of u and Q is a positive constant, independent of u, that depends upon V.      16. Numerical studies of the fingering phenomena for the thin film equation      Yibao Li  Hyun Geun Lee  Daeki Yoon  Woonjae Hwang  Suyeon Shin  Youngsoo Ha  Junseok Kim 《国际流体数值方法杂志》2011,67(11):1358-1372 We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is ht + (h2?h3)x = ??·(h3?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h2?h3)x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h3?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.      17. Transient heat transfer in the laminar thermal entry region of a pipe: an analytical solution      James Sucec 《Applied Scientific Research》1986,43(2):115-125 Attention is directed toward the problem of unsteady convective heat transfer to a fluid flowing inside a pipe in a laminar, fully developed fashion when suddenly, an ambient fluid outside the pipe undergoes a step change in temperature. For the fastest portion of the resultant transient, time domain I, an analytical solution of the governing partial differential thermal energy equation is effected via the Laplace transformation. From this solution, response functions are found for the pipe wall temperature, surface heat flux, and fluid bulk mean temperature as a function of non-dimensional time for a range of values of a parameter which characterizes the heat transfer between the ambient and the pipe.Comparison of results is made with a recent finite difference solution in the literature and with the standard quasi-steady type of analysis. It is found that the analytical solution presented herein extends and complements the finite difference solution and that the quasi-steady solution can be severely in error in this part of the transient.Nomenclature â c p R/wcpwb Ratio of thermal energy storage capacity of fluid to wall material - b pipe wall thickness - C n defined by equation (24) - c p , c pw specific heat capacity of fluid and pipe wall, respectively - D n functions defined by equation (23) - erf, erfc error function and complimentary error function, respectively - F t/R 2 Fourier number - g 1–2S - h local surface coefficient of heat transfer between inside of pipe wall and inside flowing fluid - i n erfc n th repeated integral of the error function - k thermal conductivity of the inside fluid - N h(2R)/k Nusselt number - p Laplace transform parameter - q w local, instantaneous surface heat flux at inside of pipe wall - Q w 2Rq w /k(T L –T i ) nondimensional surface heat flux - R pipe inside radius - S UR/k - t time - T local instantaneous fluid temperature - T B , T L , T i bulk mean, ambient, and initial, as well as inlet, temperature, respectively - u, u m local and mass average, fluid velocity, respectively - U overall heat transmission coefficient between ambient fluid outside of pipe and inside pipe wall - X, Y x/R, y/R nondimensional space coordinates along, and radially inward from, the pipe wall, respectively - k/c p thermal diffusivity of inside fluid - , w mass density of inside fluid and wall, respectively - (T(x, y, t)–T i )/(T L –T i ) - w , B wall, bulk mean value of , respectively      18. Contaminant Transport in Fractured Media with Sources in the Porous Domain      Homp  Michelle Reeb  David Logan  J. 《Transport in Porous Media》1997,29(3):341-353 We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu t = u yy – u + f(x,y,t),x,y>0, t>0, u t = –u x + u y – u on y = 0; u(0,0,t) =u0(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.      19. On the First Boundary Value Problem for Quasilinear Parabolic Equations with Two Independent Variables      Alkis S. Tersenov 《Archive for Rational Mechanics and Analysis》2000,152(1):81-92 This paper is concerned with the global solvability of the first initial boundary value problem for the quasilinear parabolic equations with two independent variables: a(t,x,u,uxINF>)uxxm ut=f(t,x,u,uxINF>). We investigate the case when the growth of [(|f(t,x,u,p)|)/(a(t,x,u,p))]{{|f(t,x,u,p)|}\over {a(t,x,u,p)}} with respect to p is faster than p2 when |p|M X. Conditions which guarantee the global classical solvability of the problem are formulated.      20. Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)      José A. Carrillo  Marco Di Francesco  Giuseppe Toscani 《Archive for Rational Mechanics and Analysis》2006,180(1):127-149 We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations ut = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ um at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.

Two-phase brine mixtures in the geothermal context and the polymer flood model

Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.Roman Symbols A,B coefficient matrices (5) - c(x,t) salt concentration (primary dependent variable) - C(p, s, c, q _t) wave speed matrix (6) - f source term (5) - g acceleration due to gravity (constant) - h _b(p, c) brine specific enthalpy - h _v(p) vapour specific enthalpy - j conservation flux (1) - k absolute permeability (constant) - k _b(s), k_v(s) relative permeabilities of the brine and vapour phases - K conductivity - p(x,t) pressure (primary dependent variable) - q volume flux (Darcy velocity) (3) - s(x,t) brine saturation (primary dependent variable) - t time (primary independent variable) - T=T _sat(p) saturation temperature - u _b(p, c) brine specific internal energy - u _m T rock matrix specific internal energy - u _v(p) vapour specific internal energy - U(x, t) shock velocity - x space (primary independent variable) Greek Symbols porosity (constant) - _b(p, c) brine dynamic viscosity - _v(p) vapour dynamic viscosity - (p, s, c) conservation density (1) - _b(p, c) brine density - _v(p) vapour density Suffixes b brine - m rock matrix - t total - v vapour - S salt - M mass - E energy     

Wave Packets, Signaling and Resonances in a Homogeneous Waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e^−iωt + O(e^−ɛt ), for t → ∞, where Im ω₀ = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Asymptotic Variational Wave Equations

We investigate the equation (u _t +(f(u)) _x ) _x =f ^{′ ′}(u) (u _x )²/2 where f(u) is a given smooth function. Typically f(u)=u ²/2 or u ³/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _tt − c(u) (c(u)u _x ) _x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.     

Local energy decay for the wave equation on the exterior of two balls or convex bodies

An algebraic rate of decay of local energy, nonuniform with respect to the initial data, is established for solutions of the Dirichlet and Neumann problems for the scalar wave equation defined on the exterior V³ of two balls or of two convex bodies. That is, for given initial data f(x)=u(x), 0 and g(x)= u _t (x, 0), if u solves u _tt in V with either u(x, t)=0 or u _n (x,t)+(x) u(x,t,)-0 ((x)0) on V, then there exists a constant T ₀, depending upon (f, g), such that the local energy (the energy in any compact set) of u at t=T is bounded from above by QE(0)T ^–1 for TT ₀, where E(0) is the total initial energy of u and Q is a positive constant, independent of u, that depends upon V.     

Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²?h³)_x = ??·(h³?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²?h³)_x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h³?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.     

Transient heat transfer in the laminar thermal entry region of a pipe: an analytical solution

Attention is directed toward the problem of unsteady convective heat transfer to a fluid flowing inside a pipe in a laminar, fully developed fashion when suddenly, an ambient fluid outside the pipe undergoes a step change in temperature. For the fastest portion of the resultant transient, time domain I, an analytical solution of the governing partial differential thermal energy equation is effected via the Laplace transformation. From this solution, response functions are found for the pipe wall temperature, surface heat flux, and fluid bulk mean temperature as a function of non-dimensional time for a range of values of a parameter which characterizes the heat transfer between the ambient and the pipe.Comparison of results is made with a recent finite difference solution in the literature and with the standard quasi-steady type of analysis. It is found that the analytical solution presented herein extends and complements the finite difference solution and that the quasi-steady solution can be severely in error in this part of the transient.Nomenclature â c _p R/_wc_pwb Ratio of thermal energy storage capacity of fluid to wall material - b pipe wall thickness - C _n defined by equation (24) - c _p , c _pw specific heat capacity of fluid and pipe wall, respectively - D _n functions defined by equation (23) - erf, erfc error function and complimentary error function, respectively - F t/R ² Fourier number - g 1–2S - h local surface coefficient of heat transfer between inside of pipe wall and inside flowing fluid - i ⁿ erfc n ^th repeated integral of the error function - k thermal conductivity of the inside fluid - N h(2R)/k Nusselt number - p Laplace transform parameter - q _w local, instantaneous surface heat flux at inside of pipe wall - Q _w 2Rq _w /k(T _L –T _i ) nondimensional surface heat flux - R pipe inside radius - S UR/k - t time - T local instantaneous fluid temperature - T _B , T _L , T _i bulk mean, ambient, and initial, as well as inlet, temperature, respectively - u, u _m local and mass average, fluid velocity, respectively - U overall heat transmission coefficient between ambient fluid outside of pipe and inside pipe wall - X, Y x/R, y/R nondimensional space coordinates along, and radially inward from, the pipe wall, respectively - k/c _p thermal diffusivity of inside fluid - , _w mass density of inside fluid and wall, respectively - (T(x, y, t)–T _i )/(T _L –T _i ) - _w , _B wall, bulk mean value of , respectively     

Contaminant Transport in Fractured Media with Sources in the Porous Domain

We study contaminant flow with sources in a fractured porous mediumconsisting of a single fracture bounded by a porous matrix. In the fracturewe assume convection, decay, surface adsorption to the interface, and lossto the porous matrix; in the porous matrix we include diffusion, decay,adsorption, and contaminant sources. The model leads to a nonhomogeneous,linear parabolic equation in a quarter-space with a differential equationfor an oblique boundary condition. Ultimately, we study the problemu _t = u _yy – u + f(x,y,t),x,y>0, t>0, u _t = –u _x + u _y – u on y = 0; u(0,0,t) =u₀(t), t>0,with zero initial data. Using Laplace transforms we obtain the Green'sfunction for the problem, and we determine how contaminant sources in theporous media are propagated in time.     

On the First Boundary Value Problem for Quasilinear Parabolic Equations with Two Independent Variables

This paper is concerned with the global solvability of the first initial boundary value problem for the quasilinear parabolic equations with two independent variables: a(t,x,u,u_xINF>)u_xxm u_t=f(t,x,u,u_xINF>). We investigate the case when the growth of [(|f(t,x,u,p)|)/(a(t,x,u,p))]{{|f(t,x,u,p)|}\over {a(t,x,u,p)}} with respect to p is faster than p² when |p|M X. Conditions which guarantee the global classical solvability of the problem are formulated.     

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L¹₊(R) for solutions of general nonlinear diffusion equations u_t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d₂. In the particular case of ϕ(u) ~ u^m at first order when u ~ 0, we also obtain an optimal rate of convergence in L¹ towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.