On Dual Configurational Forces

The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) and the dual configuration force (dual energy–momentum tensor) ,

On the normal forms of Hamiltonian systems

We propose a novel method to analyze the dynamics of Hamiltonian systems with a periodically modulated Hamiltonian. The method is based on a special parametric form of the canonical transformation , using Poincaré generating function Ψ (t,x,y). As a result, stability problem of a periodic solution is reduced to finding a minimum of the Poincaré function. The proposed method can be used to find normal forms of Hamiltonians. It should be emphasized that we apply the modified concept of Zhuravlev [Introduction to Theoretical Mechanics. Nauka Fizmatlit, Moscow (1997); Prikladnaya Matematika i Mekhanika 66(3), (2002) in Russian] to define an invariant normal form, which does not require any partition to either autonomous – non-autonomous, or resonance – non-resonance cases, but it is treated in the frame of one approach. In order to find the corresponding normal form asymptotics, a system of equations is derived analogous to Zhuravlev's chain of equations. Instead of the generator method and guiding Hamiltonian, a parametrized guiding function is used. It enables a direct (without the transformation to an autonomous system as in Zhuravlev's method) computation of the chain of equations for non-autonomous Hamiltonians. For autonomous systems, the methods of computation of normal forms coincide in the first and second approximations. Using this method we will present solutions of the following problems: nonlinear Duffing oscillator; oscillation of a swinging spring; dynamics of solid particles in the acoustic wave of viscous liquid, and other problems.     

Generalized Homoclinic Solutions of a Coupled Schr?dinger System Under a Small Perturbation

This paper is devoted to study a coupled Schr?dinger system with a small perturbation $$\begin{array}{ll}u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R} \end{array}$$ where β is a constant and ε is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schr?dinger-KdV system.     

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

Consider a homogeneous, isotropic, hyperelastic body occupying the region ${A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}}$ 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 $${\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)$$ that maps each sphere ${S_R\subset\,A}$ , of radius R > 0, centred at the origin into another such sphere ${S_r={\bf u}^{\rm rad}(S_R)\subset\,A^*}$ that encloses the same volume as u(S _R ). Since the volumes enclosed by the surfaces u(S _R ) and u ^rad (S _R ) are equal, the classical isoperimetric inequality implies that ${{{\rm Area}( {\bf u}^{\rm rad} (S_R))\leqq {\rm Area}({\bf u} (S_R))}}$ . The equality of the enclosed volumes together with this reduction in surface area is then shown to give rise to a reduction in total energy for many of the constitutive relations used in nonlinear elasticity. These results are also extended to classes of Sobolev deformations and applied to prove that the radially symmetric solutions to these boundary-value problems are local or global energy minimisers in various classes of (possibly nonsymmetric) deformations of a thick spherical shell.     

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

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)      7. Stabilization in a two-dimensional chemotaxis-Navier–Stokes system      Michael Winkler 《Archive for Rational Mechanics and Analysis》2014,211(2):455-487 This paper deals with an initial-boundary value problem for the system $$\left\{ \begin{array}{llll} n_t + u\cdot\nabla n &=& \Delta n -\nabla \cdot (n\chi(c)\nabla c), \quad\quad & x\in\Omega, \, t > 0,\\ c_t + u\cdot\nabla c &=& \Delta c-nf(c), \quad\quad & x\in\Omega, \, t > 0,\\ u_t + \kappa (u\cdot \nabla) u &=& \Delta u + \nabla P + n \nabla\phi, \qquad & x\in\Omega, \, t > 0,\\ \nabla \cdot u &=& 0, \qquad & x\in\Omega, \, t > 0,\end{array} \right.$$ which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains ${\Omega \subset \mathbb{R}^2}$ and under appropriate assumptions on the parameter functions χ, f and ?, for each ${\kappa\in\mathbb{R}}$ and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium ${(\overline{n_0},0,0)}$ , where ${\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}$ , in the sense that as t→∞, $$n(\cdot,t) \to \overline{n_0}, \qquad c(\cdot,t) \to 0 \qquad \text{and}\qquad u(\cdot,t) \to 0$$ hold with respect to the norm in ${L^\infty(\Omega)}$ .      8. Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures      Filip Rindler 《Archive for Rational Mechanics and Analysis》2011,202(1):63-113 We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form $ {ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). $ \begin{array}{ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). \end{array}      9. A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux      Boris Andreianov  Kenneth Hvistendahl Karlsen  Nils Henrik Risebro 《Archive for Rational Mechanics and Analysis》2011,201(1):27-86 We propose a general framework for the study of L 1 contractive semigroups of solutions to conservation laws with discontinuous flux: $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \right.\quad\quad\quad (\rm CL) $ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x < 0,\\ f^r(u), & x > 0, \end{array} \right.\quad\quad\quad (\rm CL)      10. On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations      Dongho Chae 《Journal of Mathematical Fluid Mechanics》2010,12(2):171-180 Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z0=(x0, t0)z_{0}=(x_{0}, t_{0}), and let Qz0,r = Bx0,r ×(t0 -r2, t0)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where Lγ, αx,t denotes the Serrin type of class, then z0 is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.      11. Mappings of Least Dirichlet Energy and their Hopf Differentials      Tadeusz Iwaniec  Jani Onninen 《Archive for Rational Mechanics and Analysis》2013,209(2):401-453       12. Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations      Hideo Kozono  Erika Ushikoshi 《Archive for Rational Mechanics and Analysis》2013,208(3):1005-1055 For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.      13. A well posed problem in singular Fickian diffusion      Ana Rodríguez  Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》1990,110(2):141-163 We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL1(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L loc ∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.      14. On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations      Albrecht Bertram  Thomas Böhlke  Miroslav Šilhavý 《Journal of Elasticity》2007,86(3):235-243 The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form where is a function satisfying , and α 1, α 2, α 3 are the singular values of the deformation gradient . Two general situations are determined under which is not rank 1 convex: (a) if (simultaneously) the Hessian of W at α is positive definite, , and f is strictly monotonic, and/or (b) if f is a Seth strain measure corresponding to any . No hypotheses about the range of f are necessary.       15. Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy      Charles Morris  Ali Taheri 《Journal of Elasticity》2018,133(2):201-222 In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\): $$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$ where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.       16. Maximal and Minimal Spreading Speeds for Reaction Diffusion Equations in Nonperiodic Slowly Varying Media      Jimmy Garnier  Thomas Giletti  Gregoire Nadin 《Journal of Dynamics and Differential Equations》2012,24(3):521-538 This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is ${f(x,u) = \mu_0 (\phi (x)) u(1-u)}$ , where??? 0 is a 1-periodic function and ${\phi}$ is a ${\mathcal{C}^1}$ increasing function that satisfies ${\lim_{x \to+\infty}\phi (x) = +\infty}$ and ${\lim_{x \to +\infty}\phi' (x) =0}$ . Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for u for large times, depending on the speed of the convergence of ${\phi}$ at infinity. If ${\phi}$ grows sufficiently slowly, then we prove that the spreading speed of u oscillates between two distinct values. If ${\phi}$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation w ??, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.      17. The existence of solution of a class of two-order quasilinear boundary value problem      何清  冀春慈 《应用数学和力学(英文版)》1992,13(10):983-986 Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems: but it restricts O相似文献    18. Generalized Flows Satisfying Spatial Boundary Conditions      B. Buffoni 《Journal of Mathematical Fluid Mechanics》2012,14(3):501-528 In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.      19. On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term      I. Peral  J. L. Vazquez 《Archive for Rational Mechanics and Analysis》1995,129(3):201-224 We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ 0=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.      20. Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise      Viorel Barbu  Michael Röckner 《Archive for Rational Mechanics and Analysis》2013,209(3):797-834 In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.      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

Stabilization in a two-dimensional chemotaxis-Navier–Stokes system

This paper deals with an initial-boundary value problem for the system $$\left\{ \begin{array}{llll} n_t + u\cdot\nabla n &=& \Delta n -\nabla \cdot (n\chi(c)\nabla c), \quad\quad & x\in\Omega, \, t > 0,\\ c_t + u\cdot\nabla c &=& \Delta c-nf(c), \quad\quad & x\in\Omega, \, t > 0,\\ u_t + \kappa (u\cdot \nabla) u &=& \Delta u + \nabla P + n \nabla\phi, \qquad & x\in\Omega, \, t > 0,\\ \nabla \cdot u &=& 0, \qquad & x\in\Omega, \, t > 0,\end{array} \right.$$ which has been proposed as a model for the spatio-temporal evolution of populations of swimming aerobic bacteria. It is known that in bounded convex domains ${\Omega \subset \mathbb{R}^2}$ and under appropriate assumptions on the parameter functions χ, f and ?, for each ${\kappa\in\mathbb{R}}$ and all sufficiently smooth initial data this problem possesses a unique global-in-time classical solution. The present work asserts that this solution stabilizes to the spatially uniform equilibrium ${(\overline{n_0},0,0)}$ , where ${\overline{n_0}:=\frac{1}{|\Omega|} \int_\Omega n(x,0)\,{\rm d}x}$ , in the sense that as t→∞, $$n(\cdot,t) \to \overline{n_0}, \qquad c(\cdot,t) \to 0 \qquad \text{and}\qquad u(\cdot,t) \to 0$$ hold with respect to the norm in ${L^\infty(\Omega)}$ .     

Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form

A Theory of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

We propose a general framework for the study of L ¹ contractive semigroups of solutions to conservation laws with discontinuous flux:

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

A well posed problem in singular Fickian diffusion

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

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation \(u=(u_{1}, \ldots, u_{N})\):

$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$

where \({\mathbf {X}}\) is a finite, open, symmetric \(N\)-annulus (with \(N \ge2\)), \(\mathscr{P}=\mathscr{P}(x)\) is an unknown hydrostatic pressure field and \(\varphi\) is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when \(N=3\), the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when \(N=2\), the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions \(N \ge4\) and discuss a number of closely related issues.

     

Maximal and Minimal Spreading Speeds for Reaction Diffusion Equations in Nonperiodic Slowly Varying Media

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is ${f(x,u) = \mu_0 (\phi (x)) u(1-u)}$ , where??? ₀ is a 1-periodic function and ${\phi}$ is a ${\mathcal{C}^1}$ increasing function that satisfies ${\lim_{x \to+\infty}\phi (x) = +\infty}$ and ${\lim_{x \to +\infty}\phi' (x) =0}$ . Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for u for large times, depending on the speed of the convergence of ${\phi}$ at infinity. If ${\phi}$ grows sufficiently slowly, then we prove that the spreading speed of u oscillates between two distinct values. If ${\phi}$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation w _??, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.     

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

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

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in ${\mathbb{R}^2}$ or ${\mathbb{R}^3}$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at ${x\in D}$ at time t and ${p(t,x)\in\mathbb{R}}$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l?≥ 1 is prescribed). The third equation in the system says that ${\kappa\in\mathbb{R}^l}$ is convected by the flow and the second one that ${\phi}$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)?=?x on D (and thus l?=?2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411–452, 1999) in his Eulerian–Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross ${\partial D}$ and that carry each “particle” at time t?=?0 at a prescribed location at time t?=?T?>?0, that is, κ(T, x) is prescribed in D for all ${x\in D}$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary ${\partial D}$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through ${\partial D}$ of particles labelled by each value of κ at each point of ${\partial D}$ . One of the main novelties is the introduction of a prescribed “generalized” Bernoulli’s function ${H:\mathbb{R}^l\rightarrow \mathbb{R}}$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with ${\phi,p,\kappa}$ periodic in time of prescribed period T?>?0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of “Lamb’s surfaces” and “isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier’s formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions ${\phi}$ and κ are given in terms of the stream function ψ.     

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

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

Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.