On the Navier�CStokes equations with rotating effect and prescribed outflow velocity

We consider the equations of Navier–Stokes modeling viscous fluid flow past a moving or rotating obstacle in \mathbb R^d{\mathbb R^d} subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use L ^p -techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in \mathbb R^d{\mathbb R^d} is solved by an evolution system on L^p_s(\mathbb R^d){L^p_{\sigma}(\mathbb R^d)} for 1 < p < ∞. For this we use results about time-dependent Ornstein–Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data u₀ ? L^p_s(\mathbb R^d){u_0\in L^p_{\sigma}(\mathbb R^d)}, the existence of a unique mild solution to the full Navier–Stokes system.     

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRⁿ{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRⁿ{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .     

On Kato’s Method for Navier–Stokes Equations

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR³{\mathbb{R}}^{3} and irregular domains in \mathbbRⁿ{\mathbb{R}}^{n}.     

Hydrodynamic Limit of Gradient Exclusion Processes with Conductances

Fix a strictly increasing right continuous with left limits function ${W: \mathbb{R} \to \mathbb{R}}Fix a strictly increasing right continuous with left limits function W: \mathbbR ? \mathbbR{W: \mathbb{R} \to \mathbb{R}} and a smooth function F: [l,r] ? \mathbb R{\Phi : [l,r] \to \mathbb R}, defined on some interval [l, r] of \mathbb R{\mathbb R}, such that 0 < b\leqq F￠\leqq b^-1{0 < b\leqq \Phi'\leqq b^{-1}}. On the diffusive time scale, the evolution of the empirical density of exclusion processes with conductances given by W is described by the unique weak solution of the non-linear differential equation ?_t r = (d/dx)(d/dW) F(r){\partial_t \rho = ({\rm d}/{\rm d}x)({\rm d}/{\rm d}W) \Phi(\rho)}. We also present some properties of the operator (d/dx)(d/dW).     

A New Approach to Equations with Memory

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR⁺ ? \mathbbR⁺{G : \mathbb{R}^+ \to \mathbb{R}^+} such that

$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      6. Boundary Regularity in Variational Problems      Jan Kristensen  Giuseppe Mingione 《Archive for Rational Mechanics and Analysis》2010,198(2):369-455 We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRn ? \mathbbRN{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      7. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks      Anna L. Mazzucato  Victor Nistor 《Archive for Rational Mechanics and Analysis》2010,195(1):25-73 We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.      8. Harmonic Maps and Ideal Fluid Flows      A.?Aleman  A.?ConstantinEmail author 《Archive for Rational Mechanics and Analysis》2012,204(2):479-513 Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in \mathbb Cn{\mathbb {C^n}} with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.      9. A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical L p Framework      Frédéric Charve  Raphaël Danchin 《Archive for Rational Mechanics and Analysis》2010,198(1):233-271 The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in the whole space \mathbbRd{\mathbb{R}^d} with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L p -type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows.      10. Billiards and Two-Dimensional Problems of Optimal Resistance      Alexander Plakhov 《Archive for Rational Mechanics and Analysis》2009,194(2):349-381 A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR2{B \subset \mathbb{R}^2} we assign a measure ν B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR2 \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.      11. On Spherically Symmetric Motions Of a Viscous Compressible Barotropic and Selfgravitating Gas      Bernard Ducomet  ?��rka Ne?asov��  Alexis Vasseur 《Journal of Mathematical Fluid Mechanics》2011,13(2):191-211 We consider the Cauchy problem for the equations of spherically symmetric motions in \mathbb R3{\mathbb {R}^3}, of a selfgravitating barotropic gas, with possibly non monotone pressure law, in two different situations: in the first one we suppose that the viscosities μ(ρ), and λ(ρ) are density-dependent and satisfy the Bresch–Desjardins condition, in the second one we consider constant densities. In the two cases, we prove that the problem admits a global weak solution, provided that the polytropic index γ satisfy γ > 1.      12. Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold      Matteo Bonforte  Gabriele Grillo  Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2010,196(2):631-680 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 ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb Rd{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 Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb Rd{\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.      13. On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows      Mahesh Nerurkar 《Journal of Dynamics and Differential Equations》2011,23(3):451-473 Consider the class of C r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracpnqn| ￡ C exp (-qr+1+kn)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracPnqn{\frac{P_n}{q_n}} is some sequence of irreducible fractions.      14. Heteroclinic cycles arising in generic unfoldings of nilpotent singularities      Pablo G. Barrientos  Santiago Ibáñez  J. Ángel Rodríguez 《Journal of Dynamics and Differential Equations》2011,23(4):999-1028 In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR4{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR3{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.      15. The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations      Jiayun Lin  Jian Zhai 《Journal of Mathematical Fluid Mechanics》2011,13(4):573-591 In this paper, we consider v(t) = u(t) − e tΔ u 0, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u0 ? L2(\mathbb Rn)?Ln(\mathbb Rn){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L 2 norm of D β v(t) decays like t-\frac |b|-1 2-\frac n4{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u 1(t) such that the L 2 norm of D β (v(t) − u 1(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.      16. Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians      Naoyuki Ichihara  Hitoshi Ishii 《Archive for Rational Mechanics and Analysis》2009,194(2):383-419 We investigate the long-time behavior of viscosity solutions of Hamilton–Jacobi equations in \mathbbRn{\mathbb{R}^n} with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.      17. Uniqueness and Nonuniqueness of Viscosity Solutions to Aronsson’s Equation      Robert Jensen  Changyou Wang  Yifeng Yu 《Archive for Rational Mechanics and Analysis》2008,190(2):347-370 For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation: For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and that of the Hamilton–Jacobi equation . Supported by NSF DMS 0601162. Supported by NSF DMS 0601403.      18. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      19. On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity I: Incompressible Materials      Jeyabal Sivaloganathan  Scott J. Spector 《Archive for Rational Mechanics and Analysis》2010,196(2):363-394 Let ${A=\{{\bf x} \in \mathbb{R}^n : a < |{\bf x}| < b\}, n \geqq 2, a > 0}Consider a homogeneous, isotropic, hyperelastic body occupying the region A = {x ? \mathbbRn :  a <  |x |  < b }{A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}} in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation urad(x)=\fracr(R)Rx,     R=|x|,                        (0.1){\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)      20. Study of pulsating flow in close-coupled catalyst manifolds using phase-locked hot-wire anemometry      T.?PersoonsEmail author  E.?Van den Bulck  S.?Fausto 《Experiments in fluids》2004,36(2):217-232       设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      7. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks      Anna L. Mazzucato  Victor Nistor 《Archive for Rational Mechanics and Analysis》2010,195(1):25-73 We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.      8. Harmonic Maps and Ideal Fluid Flows      A.?Aleman  A.?ConstantinEmail author 《Archive for Rational Mechanics and Analysis》2012,204(2):479-513 Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in \mathbb Cn{\mathbb {C^n}} with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.      9. A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical L p Framework      Frédéric Charve  Raphaël Danchin 《Archive for Rational Mechanics and Analysis》2010,198(1):233-271 The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in the whole space \mathbbRd{\mathbb{R}^d} with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L p -type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows.      10. Billiards and Two-Dimensional Problems of Optimal Resistance      Alexander Plakhov 《Archive for Rational Mechanics and Analysis》2009,194(2):349-381 A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR2{B \subset \mathbb{R}^2} we assign a measure ν B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR2 \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.      11. On Spherically Symmetric Motions Of a Viscous Compressible Barotropic and Selfgravitating Gas      Bernard Ducomet  ?��rka Ne?asov��  Alexis Vasseur 《Journal of Mathematical Fluid Mechanics》2011,13(2):191-211 We consider the Cauchy problem for the equations of spherically symmetric motions in \mathbb R3{\mathbb {R}^3}, of a selfgravitating barotropic gas, with possibly non monotone pressure law, in two different situations: in the first one we suppose that the viscosities μ(ρ), and λ(ρ) are density-dependent and satisfy the Bresch–Desjardins condition, in the second one we consider constant densities. In the two cases, we prove that the problem admits a global weak solution, provided that the polytropic index γ satisfy γ > 1.      12. Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold      Matteo Bonforte  Gabriele Grillo  Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2010,196(2):631-680 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 ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb Rd{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 Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb Rd{\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.      13. On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows      Mahesh Nerurkar 《Journal of Dynamics and Differential Equations》2011,23(3):451-473 Consider the class of C r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracpnqn| ￡ C exp (-qr+1+kn)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracPnqn{\frac{P_n}{q_n}} is some sequence of irreducible fractions.      14. Heteroclinic cycles arising in generic unfoldings of nilpotent singularities      Pablo G. Barrientos  Santiago Ibáñez  J. Ángel Rodríguez 《Journal of Dynamics and Differential Equations》2011,23(4):999-1028 In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR4{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR3{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.      15. The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations      Jiayun Lin  Jian Zhai 《Journal of Mathematical Fluid Mechanics》2011,13(4):573-591 In this paper, we consider v(t) = u(t) − e tΔ u 0, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u0 ? L2(\mathbb Rn)?Ln(\mathbb Rn){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L 2 norm of D β v(t) decays like t-\frac |b|-1 2-\frac n4{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u 1(t) such that the L 2 norm of D β (v(t) − u 1(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.      16. Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians      Naoyuki Ichihara  Hitoshi Ishii 《Archive for Rational Mechanics and Analysis》2009,194(2):383-419 We investigate the long-time behavior of viscosity solutions of Hamilton–Jacobi equations in \mathbbRn{\mathbb{R}^n} with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.      17. Uniqueness and Nonuniqueness of Viscosity Solutions to Aronsson’s Equation      Robert Jensen  Changyou Wang  Yifeng Yu 《Archive for Rational Mechanics and Analysis》2008,190(2):347-370 For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation: For H = H(p, x) depending on x, we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation and that of the Hamilton–Jacobi equation . Supported by NSF DMS 0601162. Supported by NSF DMS 0601403.      18. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      19. On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity I: Incompressible Materials      Jeyabal Sivaloganathan  Scott J. Spector 《Archive for Rational Mechanics and Analysis》2010,196(2):363-394 Let ${A=\{{\bf x} \in \mathbb{R}^n : a < |{\bf x}| < b\}, n \geqq 2, a > 0}Consider a homogeneous, isotropic, hyperelastic body occupying the region A = {x ? \mathbbRn :  a <  |x |  < b }{A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}} in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation urad(x)=\fracr(R)Rx,     R=|x|,                        (0.1){\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)      20. Study of pulsating flow in close-coupled catalyst manifolds using phase-locked hot-wire anemometry      T.?PersoonsEmail author  E.?Van den Bulck  S.?Fausto 《Experiments in fluids》2004,36(2):217-232       设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

Harmonic Maps and Ideal Fluid Flows

Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in \mathbb Cⁿ{\mathbb {C^n}} with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.     

A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical L p Framework

The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in the whole space \mathbbR^d{\mathbb{R}^d} with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L ^p -type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows.     

Billiards and Two-Dimensional Problems of Optimal Resistance

A body moves in a medium composed of noninteracting point particles; the interaction of the particles with the body is completely elastic. The problem is: find the body’s shape that minimizes or maximizes resistance of the medium to its motion. This is the general setting of the optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary B ì \mathbbR²{B \subset \mathbb{R}^2} we assign a measure ν _B on ∂(conv B)×[ − π/2, π/2] generated by the billiard in \mathbbR² \B{\mathbb{R}^2 \setminus B} and characterize the set of measures {ν _B }. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge–Kantorovich problems.     

On Spherically Symmetric Motions Of a Viscous Compressible Barotropic and Selfgravitating Gas

We consider the Cauchy problem for the equations of spherically symmetric motions in \mathbb R³{\mathbb {R}^3}, of a selfgravitating barotropic gas, with possibly non monotone pressure law, in two different situations: in the first one we suppose that the viscosities μ(ρ), and λ(ρ) are density-dependent and satisfy the Bresch–Desjardins condition, in the second one we consider constant densities. In the two cases, we prove that the problem admits a global weak solution, provided that the polytropic index γ satisfy γ > 1.     

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.     

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR⁴{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR³{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians

We investigate the long-time behavior of viscosity solutions of Hamilton–Jacobi equations in \mathbbRⁿ{\mathbb{R}^n} with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.     

Uniqueness and Nonuniqueness of Viscosity Solutions to Aronsson’s Equation

For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:

Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R²{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity I: Incompressible Materials

Let ${A=\{{\bf x} \in \mathbb{R}^n : a < |{\bf x}| < b\}, n \geqq 2, a > 0}Consider a homogeneous, isotropic, hyperelastic body occupying the region A = {x ? \mathbbRⁿ :  a <  |x |  < b }{A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}} in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation

urad(x)=\fracr(R)Rx,     R=|x|,                        (0.1){\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)      20. Study of pulsating flow in close-coupled catalyst manifolds using phase-locked hot-wire anemometry      T.?PersoonsEmail author  E.?Van den Bulck  S.?Fausto 《Experiments in fluids》2004,36(2):217-232