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.     

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      3. Orientational anisotropy for Rouse eigenmodes during creep and recovery process      Hiroshi?WatanabeEmail author  Tadashi?Inoue 《Rheologica Acta》2004,43(6):634-644 The Rouse model is a well established model for nonentangled polymer chains and its dynamic behavior under step strain has been fully analyzed in the literature. However, to the knowledge of the authors, no analysis has been made for the orientational anisotropy for the Rouse eigenmodes during the creep and creep recovery processes. For completeness of the analysis of the Rouse model, this anisotropy is calculated from the Rouse equation of motion. The calculation is simple and straightforward, but the result is intriguing in a sense that respective Rouse eigenmodes do not exhibit the single Voigt-type retardation. Instead, each Rouse eigenmode has a distribution in the retardation time. This behavior, reflecting the interplay among the Rouse eigenmodes of different orders under the constant stress condition, is quite different from the behavior under rate-controlled flow (where each eigenmode exhibits retardation/relaxation associated with a single characteristic time).List of abbreviations and symbols a Average segment size at equilibrium - Ap(t) Normalized orientational anisotropy for the p-th Rouse eigenmode defined by Eq. (14) - p-th Fourier component of the Brownian force (=x, y) - FB(n,t) Brownian force acting on n-th segment at time t - G(t) Relaxation modulus - J(t) Creep compliance - JR(t) Recoverable creep compliance - kB Boltzmann constant - N Segment number per Rouse chain - Qj(t) Orientational anisotropy of chain sections defined by Eq. (21) - r(n,t) Position of n-th segment of the chain at time t - S(n,t) Shear orientation function (S(n,t)=a–2<ux(n,t)uy(n,t)>) - T Absolute temperature - u(n,t) Tangential vector of n-th segment at time t (u = r/n) - V(r(n,t)) Flow velocity of the frictional medium at the position r(n,t) - Xp(t), Yp(t), and Zp(t) x-, y-, and z-components of the amplitudes of p-th Rouse eigenmode at time t - Strain rate being uniform throughout the system - Segmental friction coefficient - 0 Zero-shear viscosity - p Numerical coefficients determined from Eq. (25) - Gaussian spring constant ( = 3kBT/a2) - Number of Rouse chains per unit volume - (t) Shear stress of the system at time t - steady Shear stress in the steadily flowing state - R Longest viscoelastic relaxation time of the Rouse chain      4. Backward Uniqueness for Parabolic Equations   总被引：4，自引：0，他引：4   L.?Escauriaza  G.?Seregin  V.??verákEmail author 《Archive for Rational Mechanics and Analysis》2003,169(2):147-157 It is shown that a function u satisfying |t+u|M(|u|+|u|), |u(x, t)|MeM|x|2 in (n \ (BR) × [0, T] and u(x, 0) = 0 for xn \ BR must vanish identically in n \ BR×[0, T].      5. Symmetry of Mountain Pass Solutions of Some Vector Field Equations      Orlando Lopes  Marcelo Montenegro 《Journal of Dynamics and Differential Equations》2006,18(4):991-999 We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u 1(x),...,u N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].      6. The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws      Heinrich Freistühler  Denis Serre 《Journal of Dynamics and Differential Equations》2001,13(4):745-755 Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u t+f(u) x =u xx on a half-line R+, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u1 induced by L 1(R+). We prove here that v is asymptotically stable with respect to d: if u 0–vL 1, then u(t)–v10 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].      7. Weak Continuity and Lower Semicontinuity Results for Determinants      Irene?FonsecaEmail author  Giovanni?Leoni  Jan?Maly 《Archive for Rational Mechanics and Analysis》2005,178(3):411-448 Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if {un} is bounded in and if u ∈ BV(Ω;ℝN) are such that un→u in L1(Ω;ℝN) and in the sense of measures, then for This result is sharp, and counterexamples are provided in the cases where the regularity of {un} or the type of weak convergence is weakened.      8. Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions      Eduard Feireisl  Frédérique Simondon 《Journal of Dynamics and Differential Equations》2000,12(3):647-673 We show that any global nonnegative and bounded solution to the degenerate parabolic problemut-um+f(u)=0 qquad {\rm on} quad RN,u|{}=0converges to a single stationary state as time goes to infinity. Here m>0, f is a restriction of a real analytic function defined on a sector containing the half-line [0, ), and f(u 1/m ) is a continuously differentiable function of u.      9. A Semilinear Schrödinger Equation in the Presence of a Magnetic Field      Gianni?Arioli  Andrzej?SzulkinEmail author 《Archive for Rational Mechanics and Analysis》2003,170(4):277-295 We consider the semilinear stationary Schrödinger equation in a magnetic field: (–i+A)2 u+V(x)u=g(x,|u|)u in N , where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|2 * –2 and |g(x,|u|)|c(1+|u| p –2), where 2<p<2*). The results are obtained by variational methods. For g critical we use constrained minimization and for subcritical g we employ a minimax-type argument. In the latter case we also study the existence of infinitely many geometrically distinct solutions.      10. Functions of bounded deformation      Roger Temam  Gilbert Strang 《Archive for Rational Mechanics and Analysis》1980,75(1):7-21 We study the space BD(), composed of vector functions u for which all components ij=1/2(u i, j+u j, i) of the deformation tensor are bounded measures. This seems to be the correct space for the displacement field in the problems of perfect plasticity. We prove that the boundary values of every such u are integrable; indeed their trace is in L 1 ()N. We show also that if a distribution u yields ij which are measures, then u must lie in L p() for pN/(N–1).The second author gratefully acknowledges the supprot of the National Science Foundation.      11. Entire Solutions with Merging Fronts to Reaction–Diffusion Equations      Yoshihisa Morita  Hirokazu Ninomiya 《Journal of Dynamics and Differential Equations》2006,18(4):841-861 We deal with a reaction–diffusion equation u t = u xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c 1 t) (c 1 < 0) and ψ2(x + c 2 t) (c 2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c 1 t) and ψ2(x + c 2 t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c 1, we show the existence of an entire solution which behaves as ψ1( − x + c 1 t) in and φ(x + ct) in for t≈ − ∞.      12. Some closure results for inertial manifolds      J. C. Robinson 《Journal of Dynamics and Differential Equations》1997,9(3):373-400 Suppose that the family of evolution equationsdu/dt+Au+f N (u)=0 possesses inertial manifolds of the same dimension for a sequence of nonlinear termsf N withf N f in the C0 norm. Conditions are found to ensure that the limiting equationdu/dt+Au+f(u)=0 also possesses an inertial manifold. There are two cases. The first, where the manifolds for the family have a bounded Lipschitz constant, is straightforward and leads to an interesting result on inertial manifolds for Bubnov-Galerkin approximations. When the Lipschitz constant is unbounded, it is still possible to prove the existence of an exponential attractor of finite Hausdorff dimension for the limiting equation. This more general result is applied to a problem in approximate inertial manifold theory discussed by Sell (1993).For Paul Glendinning, with thanks.      13. Fast Diffusion to Self-Similarity: Complete Spectrum,Long-Time Asymptotics,and Numerology      Jochen?DenzlerEmail author  Robert J.?McCann 《Archive for Rational Mechanics and Analysis》2005,175(3):301-342 The complete spectrum is determined for the operator on the Sobolev space W1,2(Rn) formed by closing the smooth functions of compact support with respect to the norm Here the Barenblatt profile is the stationary attractor of the rescaled diffusion equation in the fast, supercritical regime m the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d2. Formally, the operator H=HessE is the Hessian of this entropy at its minimum , so the spectral gap H:=2–n(1–m) found below suggests the sharp rate of asymptotic convergence: from any centered initial data 0u(0,x)L1(Rn) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)–n(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rn and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of .Dedicated to Elliott H. Lieb on the occasion of his 70th birthday.      14. Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations      Jérôme Le Rousseau  Luc Robbiano 《Archive for Rational Mechanics and Analysis》2010,195(3):953-990 In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, A=-?x02 - ?x ·(c(x) ?x){A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}, where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator ?t - ?x ·(c(x) ?x){{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}} .      15. Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions      R. Koplatadze  G. Kvinikadze 《Nonlinear Oscillations》2009,12(2):184-198 We consider the difference equation D2u(k) + ?l = 1m pl(k)u( tl(k) ) = 0, {\Delta^2}u(k) + \sum\limits_{l = 1}^m {{p_l}(k)u\left( {{\tau_l}(k)} \right) = 0},      16. Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer      P. L. Johnson  R. S. Barlow 《Experiments in fluids》1989,8(3-4):137-144 The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( w /)1/2 - v kinematic viscosity - d p pinhole diameter - l eff spanwise extent of LDV measuring volume viewed by photomultiplier - l + non-dimensional length of measuring volume, l eff u /v - y + non-dimensional coordinate in spanwise direction, y u /v - z + non-dimensional coordinate in spanwise direction, z u /v - U + non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u 21/2 - v RMS normal velocity fluctuation, v 21/2 - Re Reynolds number based on momentum thickness, U 0/v - R uv cross-correlation coefficient, u v/u v - R12(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt2)–1/2      17. On the critical exponent for reaction-diffusion equations   总被引：2，自引：0，他引：2   Peter Meier 《Archive for Rational Mechanics and Analysis》1990,109(1):63-71 In this paper we study the initial-boundary value problem for u t =u+ h(t) u p with homogeneous Dirichlet boundary conditions, where h(t)t q for large t. Let s * sup {s ¦ positive solutions w of u t =u such that . Then for p * 1+(q+1)/s * we show: If p>p *, there are global positive solutions that decay to zero uniformly for t. If 1<p<p *, then all nontrivial solutions blow up in finite time. We determine p * for some conical domains in R 2 and R 3.A similar result is derived for a bounded domain if h(t)e t for large t.      18. Pulsatile flow in a tube with wall injection and suction      F. M. Skalak  C.-Y. Wang 《Applied Scientific Research》1977,33(3-4):269-307 This paper studies similarity solutions for pulsatile flow in a tube with wall injection and suction. The Navier-Stokes equations are reduced to a system of three ordinary differential equations. Two of the equations represent the effects of suction and injection on the steady flow while the third represents the effects of suction and injection on pulsatile flow. Since the equations for steady flow have been studied previously, the analysis centers on the third equation. This equation is solved numerically and by the method of matched asymptotic expansions. The exact numerical solutions compare well with the asymptotic solutions.The effects of suction and injection on pulsatile flow are the following: a) Small values of suction can cause a resonance-like effect for low frequency pulsatile flow. b) The annular effect still occurs but for large injection or suction the frequency at which this effect becomes dominant depends on the cross-flow Reynolds number. c) The maximum shear stress at the wall is decreased by injection, but may be increased or decreased by suction.Nomenclature a radius of the tube - a 0 2 i 2 - A0, B0, C0, D0, E0 constant coefficients appearing in the expression for pressure - b a non-dimensionalized length - b 0 2 i 2 2 - b k complex coefficients of a power series - B - C 1, C 2, D complex constants - d - D 1,2 - f() F(a 1/2)/aV - f 0,f 1,... functions of order one used in asymptotic expansions of f() - F(r) rv r - g() - G(r) a steady component of velocity in axial direction - h() 4/C0 a 2 H(a 1/2) - h 0,h 1,h 2,...;l 0,l 1,l 2,... functions of order one used in asymptotic expansions for h() in outer regions - H(r) complex valued function giving unsteady component of velocity - H 0, H 1, H 2, ... K 0, K 1, K 2, ...; L 0, L 1, L 2, ... functions of order one used in asymptotic expansions for h() in inner regions - i - J 0, J 1, Y 0, Y 1 Bessel functions of first and second kind - k - K Rk/2b 2 - O order symbol - p pressure - p 1(z, t) arbitrary function related to pressure - r radial coordinate - r 0 (1+16 4 4)1/4 - R Va/, the crossflow Reynolds number - t time - u() G(r)/V - v r radial velocity - v z axial velocity - V constant velocity at which fluid is injected or extracted - z axial coordinate - 2 a 2/4 - 4.196 - small parameter; =–2/R (Sect. 4); =–R/2 (Sect. 5); =2/R(Sect. 6) - r 2/a 2 - * 0.262 - Arctan (4 2 2) - , inner variables - kinematic viscosity - b - * zero of g() - density - (r, t) arbitrary function related to axial velocity - frequency      19. The effect of Hall currents on MHD flow and heat transfer between two parallel porous plates      A. Bharali  A. K. Borkakati 《Applied Scientific Research》1982,39(2):155-165 The effect of Hall currents on magneto hydrodynamic (MHD) flow of an incompressible viscous electrically conducting fluid between two non-conducting porous plates in the presence of a strong uniform magnetic field is studied. The flow is generated by a small uniform suction at the plates. Solutions are obtained for suction Reynolds number R1, considering two cases for the imposed magnetic field, viz. (i) when the magnetic field is perpendicular to the plates (parallel to y-axis), and (ii) when the magnetic field is parallel to the plates and perpendicular to the primary flow direction (parallel to z-axis). The effect of the Hall currents on the flow as well as on the heat transfer is studied. It is observed that in the absence of Hall currents, the change of the direction of the applied magnetic field does not affect the primary flow.Nomenclature B total magnetic induction vector - V velocity vector - E electric field vector - J current density vector - U 0 suction velocity - T temperature of the fluid at any point - B 0 imposed magnetic field - u x-component of fluid velocity - v y-component of fluid velocity - w z-component of fluid velocity - density of the fluid - kinematic viscosity of the fluid - c p specific heat at constant pressure - p fluid pressure - electrical conductivity of the fluid - K coefficient of thermal conductivity - e magnetic permeability - n e number density of electrons - e electric charge - dimensionless distance (=y/h) - f(), g(), Q(), () dimensionless functions defined in (14) - R suction Reynolds number (=U 0 h/) - M Hartmann number (=B 0 h(/)1/2) - m Hall parameter (=B 0/en e) - Pr Prandtl number of the fluid (=c p/K) - s dimensionless quantity defined as s=(T 1–T 0)/[vU 0/(hc p)]      20. Strong Traces for Solutions to Scalar Conservation Laws with General Flux   总被引：1，自引：0，他引：1   Young-Sam Kwon  Alexis Vasseur 《Archive for Rational Mechanics and Analysis》2007,185(3):495-513 In this paper we consider bounded weak solutions u of scalar conservation laws, not necessarily of class BV, defined in a subset . We define a strong notion of trace at the boundary of reached by L 1 convergence for a large class of functionals of u, G(u). The functionals G depend on the flux function of the conservation law and on the boundary of . The result holds for a general flux function and a general subset.

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Orientational anisotropy for Rouse eigenmodes during creep and recovery process

The Rouse model is a well established model for nonentangled polymer chains and its dynamic behavior under step strain has been fully analyzed in the literature. However, to the knowledge of the authors, no analysis has been made for the orientational anisotropy for the Rouse eigenmodes during the creep and creep recovery processes. For completeness of the analysis of the Rouse model, this anisotropy is calculated from the Rouse equation of motion. The calculation is simple and straightforward, but the result is intriguing in a sense that respective Rouse eigenmodes do not exhibit the single Voigt-type retardation. Instead, each Rouse eigenmode has a distribution in the retardation time. This behavior, reflecting the interplay among the Rouse eigenmodes of different orders under the constant stress condition, is quite different from the behavior under rate-controlled flow (where each eigenmode exhibits retardation/relaxation associated with a single characteristic time).List of abbreviations and symbols a Average segment size at equilibrium - A_p(t) Normalized orientational anisotropy for the p-th Rouse eigenmode defined by Eq. (14) - p-th Fourier component of the Brownian force (=x, y) - F_B(n,t) Brownian force acting on n-th segment at time t - G(t) Relaxation modulus - J(t) Creep compliance - J_R(t) Recoverable creep compliance - k_B Boltzmann constant - N Segment number per Rouse chain - Q_j(t) Orientational anisotropy of chain sections defined by Eq. (21) - r(n,t) Position of n-th segment of the chain at time t - S(n,t) Shear orientation function (S(n,t)=a^–2<u_x(n,t)u_y(n,t)>) - T Absolute temperature - u(n,t) Tangential vector of n-th segment at time t (u = r/n) - V(r(n,t)) Flow velocity of the frictional medium at the position r(n,t) - X_p(t), Y_p(t), and Z_p(t) x-, y-, and z-components of the amplitudes of p-th Rouse eigenmode at time t - Strain rate being uniform throughout the system - Segmental friction coefficient - ₀ Zero-shear viscosity - _p Numerical coefficients determined from Eq. (25) - Gaussian spring constant ( = 3k_BT/a²) - Number of Rouse chains per unit volume - (t) Shear stress of the system at time t - _steady Shear stress in the steadily flowing state - _R Longest viscoelastic relaxation time of the Rouse chain     

Backward Uniqueness for Parabolic Equations

It is shown that a function u satisfying |_t+u|M(|u|+|u|), |u(x, t)|Me^M|x|2 in (ⁿ \ (B_R) × [0, T] and u(x, 0) = 0 for xⁿ \ B_R must vanish identically in ⁿ \ B_R×[0, T].     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem [2].     

Weak Continuity and Lower Semicontinuity Results for Determinants

Weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper. In particular, it is shown that if {u_n} is bounded in and if u ∈ BV(Ω;ℝ^N) are such that u_n→u in L¹(Ω;ℝ^N) and in the sense of measures, then for This result is sharp, and counterexamples are provided in the cases where the regularity of {u_n} or the type of weak convergence is weakened.     

Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions

We show that any global nonnegative and bounded solution to the degenerate parabolic problemu_t-u^m+f(u)=0 qquad {\rm on} quad R^N,u|{}=0converges to a single stationary state as time goes to infinity. Here m>0, f is a restriction of a real analytic function defined on a sector containing the half-line [0, ), and f(u ^1/m ) is a continuously differentiable function of u.     

A Semilinear Schrödinger Equation in the Presence of a Magnetic Field

We consider the semilinear stationary Schrödinger equation in a magnetic field: (–i+A)² u+V(x)u=g(x,|u|)u in ^N , where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|^{2 * –2} and |g(x,|u|)|c(1+|u| ^{p –2}), where 2<p<2^*). The results are obtained by variational methods. For g critical we use constrained minimization and for subcritical g we employ a minimax-type argument. In the latter case we also study the existence of infinitely many geometrically distinct solutions.     

Functions of bounded deformation

We study the space BD(), composed of vector functions u for which all components _ij=1/2(u _{i, j}+u _{j, i}) of the deformation tensor are bounded measures. This seems to be the correct space for the displacement field in the problems of perfect plasticity. We prove that the boundary values of every such u are integrable; indeed their trace is in L ¹ ()^N. We show also that if a distribution u yields _ij which are measures, then u must lie in L ^p() for pN/(N–1).The second author gratefully acknowledges the supprot of the National Science Foundation.     

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

We deal with a reaction–diffusion equation u _t = u _xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ₁(x + c ₁ t) (c ₁ < 0) and ψ₂(x + c ₂ t) (c ₂ > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ₁(x + c ₁ t) and ψ₂(x + c ₂ t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c ₁, we show the existence of an entire solution which behaves as ψ₁( − x + c ₁ t) in and φ(x + ct) in for t≈ − ∞.     

Some closure results for inertial manifolds

Suppose that the family of evolution equationsdu/dt+Au+f _N (u)=0 possesses inertial manifolds of the same dimension for a sequence of nonlinear termsf _N withf _N f in the C⁰ norm. Conditions are found to ensure that the limiting equationdu/dt+Au+f(u)=0 also possesses an inertial manifold. There are two cases. The first, where the manifolds for the family have a bounded Lipschitz constant, is straightforward and leads to an interesting result on inertial manifolds for Bubnov-Galerkin approximations. When the Lipschitz constant is unbounded, it is still possible to prove the existence of an exponential attractor of finite Hausdorff dimension for the limiting equation. This more general result is applied to a problem in approximate inertial manifold theory discussed by Sell (1993).For Paul Glendinning, with thanks.     

Fast Diffusion to Self-Similarity: Complete Spectrum,Long-Time Asymptotics,and Numerology

The complete spectrum is determined for the operator on the Sobolev space W^1,2(Rⁿ) formed by closing the smooth functions of compact support with respect to the norm Here the Barenblatt profile is the stationary attractor of the rescaled diffusion equation in the fast, supercritical regime m the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d₂. Formally, the operator H=HessE is the Hessian of this entropy at its minimum , so the spectral gap H:=2–n(1–m) found below suggests the sharp rate of asymptotic convergence: from any centered initial data 0u(0,x)L¹(Rⁿ) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)^–n(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rⁿ and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of .Dedicated to Elliott H. Lieb on the occasion of his 70th birthday.     

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, A=-?_x0² - ?_x ·(c(x) ?_x){A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}, where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator ?_t - ?_x ·(c(x) ?_x){{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}} .     

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

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

On the critical exponent for reaction-diffusion equations

In this paper we study the initial-boundary value problem for u _t =u+ h(t) u ^p with homogeneous Dirichlet boundary conditions, where h(t)t ^q for large t. Let s ^* sup {s ¦ positive solutions w of u _t =u such that . Then for p ^* 1+(q+1)/s ^* we show: If p>p ^*, there are global positive solutions that decay to zero uniformly for t. If 1<p<p ^*, then all nontrivial solutions blow up in finite time. We determine p ^* for some conical domains in R ² and R ³.A similar result is derived for a bounded domain if h(t)e ^t for large t.     

Pulsatile flow in a tube with wall injection and suction

This paper studies similarity solutions for pulsatile flow in a tube with wall injection and suction. The Navier-Stokes equations are reduced to a system of three ordinary differential equations. Two of the equations represent the effects of suction and injection on the steady flow while the third represents the effects of suction and injection on pulsatile flow. Since the equations for steady flow have been studied previously, the analysis centers on the third equation. This equation is solved numerically and by the method of matched asymptotic expansions. The exact numerical solutions compare well with the asymptotic solutions.The effects of suction and injection on pulsatile flow are the following: a) Small values of suction can cause a resonance-like effect for low frequency pulsatile flow. b) The annular effect still occurs but for large injection or suction the frequency at which this effect becomes dominant depends on the cross-flow Reynolds number. c) The maximum shear stress at the wall is decreased by injection, but may be increased or decreased by suction.Nomenclature a radius of the tube - a ₀ ² i ² - A₀, B₀, C₀, D₀, E₀ constant coefficients appearing in the expression for pressure - b a non-dimensionalized length - b ₀ ² i ^{2 2} - b _k complex coefficients of a power series - B - C ₁, C ₂, D complex constants - d - D _1,2 - f() F(a ^1/2)/aV - f ₀,f ₁,... functions of order one used in asymptotic expansions of f() - F(r) rv _r - g() - G(r) a steady component of velocity in axial direction - h() 4/C₀ a ² H(a ^1/2) - h ₀,h ₁,h ₂,...;l ₀,l ₁,l ₂,... functions of order one used in asymptotic expansions for h() in outer regions - H(r) complex valued function giving unsteady component of velocity - H ₀, H ₁, H ₂, ... K ₀, K ₁, K ₂, ...; L ₀, L ₁, L ₂, ... functions of order one used in asymptotic expansions for h() in inner regions - i - J ₀, J ₁, Y ₀, Y ₁ Bessel functions of first and second kind - k - K Rk/2b ² - O order symbol - p pressure - p ₁(z, t) arbitrary function related to pressure - r radial coordinate - r ₀ (1+16 ^{4 4})^1/4 - R Va/, the crossflow Reynolds number - t time - u() G(r)/V - v _r radial velocity - v _z axial velocity - V constant velocity at which fluid is injected or extracted - z axial coordinate - ² a ²/4 - 4.196 - small parameter; =–2/R (Sect. 4); =–R/2 (Sect. 5); =2/R(Sect. 6) - r ²/a ² - * 0.262 - Arctan (4 ^{2 2}) - , inner variables - kinematic viscosity - b - * zero of g() - density - (r, t) arbitrary function related to axial velocity - frequency     

The effect of Hall currents on MHD flow and heat transfer between two parallel porous plates

The effect of Hall currents on magneto hydrodynamic (MHD) flow of an incompressible viscous electrically conducting fluid between two non-conducting porous plates in the presence of a strong uniform magnetic field is studied. The flow is generated by a small uniform suction at the plates. Solutions are obtained for suction Reynolds number R1, considering two cases for the imposed magnetic field, viz. (i) when the magnetic field is perpendicular to the plates (parallel to y-axis), and (ii) when the magnetic field is parallel to the plates and perpendicular to the primary flow direction (parallel to z-axis). The effect of the Hall currents on the flow as well as on the heat transfer is studied. It is observed that in the absence of Hall currents, the change of the direction of the applied magnetic field does not affect the primary flow.Nomenclature B total magnetic induction vector - V velocity vector - E electric field vector - J current density vector - U ₀ suction velocity - T temperature of the fluid at any point - B ₀ imposed magnetic field - u x-component of fluid velocity - v y-component of fluid velocity - w z-component of fluid velocity - density of the fluid - kinematic viscosity of the fluid - c _p specific heat at constant pressure - p fluid pressure - electrical conductivity of the fluid - K coefficient of thermal conductivity - _e magnetic permeability - n _e number density of electrons - e electric charge - dimensionless distance (=y/h) - f(), g(), Q(), () dimensionless functions defined in (14) - R suction Reynolds number (=U ₀ h/) - M Hartmann number (=B ₀ h(/)^1/2) - m Hall parameter (=B ₀/en _e) - Pr Prandtl number of the fluid (=c _p/K) - s dimensionless quantity defined as s=(T ₁–T ₀)/[vU ₀/(hc _p)]     

Strong Traces for Solutions to Scalar Conservation Laws with General Flux

In this paper we consider bounded weak solutions u of scalar conservation laws, not necessarily of class BV, defined in a subset . We define a strong notion of trace at the boundary of reached by L ¹ convergence for a large class of functionals of u, G(u). The functionals G depend on the flux function of the conservation law and on the boundary of . The result holds for a general flux function and a general subset.