The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

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

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.      4. 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.      5. Isolated Singularity for the Stationary Navier—Stokes System      H. J. Choe 《Journal of Mathematical Fluid Mechanics》2000,2(2):151-184 In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|-(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L2n/(n - 1)(BR) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? Lb(BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.      6. A Locally Quadratic Glimm Functional and Sharp Convergence Rate of the Glimm Scheme for Nonlinear Hyperbolic Systems      Fabio Ancona  Andrea Marson 《Archive for Rational Mechanics and Analysis》2010,196(2):455-487 Consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one space dimension ut  +  A(u)ux = 0,        u(0, x) = [`(u)](x),                 (1)u_{t} \, + \, A(u)u_{x} = 0, \qquad u(0, x) = {\bar u}(x), \quad \quad \quad \quad (1)      7. On the fundamental linear fractional order differential equation      Abdelfatah Charef  Hassene Nezzari 《Nonlinear dynamics》2011,65(3):335-348 This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t0s)2m + 2z(t0s)m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t0)2m\fracd2mx(t)dt2m + 2z(t0)m\fracdmx(t)dtm + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.      8. 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)}} .      9. On a Volume‐Constrained Variational Problem      Luigi Ambrosio  Irene Fonseca  Paolo Marcellini  Luc Tartar 《Archive for Rational Mechanics and Analysis》1999,149(1):23-47 . Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = òW W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where LN({u = zi}) = ai, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which ziz_i are extremal points of a compact, convex set in \Bbb Rd\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|2W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.      10. On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases      David Ruiz 《Archive for Rational Mechanics and Analysis》2010,198(1):349-368 This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem: - Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      11. Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result      M. M. Cavalcanti  V. N. Domingos Cavalcanti  R. Fukuoka  J. A. Soriano 《Archive for Rational Mechanics and Analysis》2010,197(3):925-964 Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      12. Hydrodynamic Limit of Gradient Exclusion Processes with Conductances      Tertuliano Franco  Claudio Landim 《Archive for Rational Mechanics and Analysis》2010,195(2):409-439 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).      13. 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,      14. 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≈ − ∞.      15. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      16. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      17. 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.      18. An Example of Finite-time Singularities in the 3d Euler Equations      Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410 Let be the exterior of the closed unit ball. Consider the self-similar Euler system Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution , vanishing at infinity, precisely The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v.      19. Entropy Solutions for Nonlinear Degenerate Problems   总被引：9，自引：0，他引：9   José Carrillo 《Archive for Rational Mechanics and Analysis》1999,147(4):269-361 We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.      20. 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.      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

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.     

Isolated Singularity for the Stationary Navier—Stokes System

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in B_R \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|^{-(n - 1)/2}) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L^{2n/(n - 1)}(B_R) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? L^b(B_R) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.     

A Locally Quadratic Glimm Functional and Sharp Convergence Rate of the Glimm Scheme for Nonlinear Hyperbolic Systems

Consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one space dimension

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.     

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)}} .     

On a Volume‐Constrained Variational Problem

. Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = ò_W W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where L^N({u = z_i}) = a_i, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which z_iz_i are extremal points of a compact, convex set in \Bbb R^d\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|²W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.     

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

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

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      11. Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result      M. M. Cavalcanti  V. N. Domingos Cavalcanti  R. Fukuoka  J. A. Soriano 《Archive for Rational Mechanics and Analysis》2010,197(3):925-964 Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      12. Hydrodynamic Limit of Gradient Exclusion Processes with Conductances      Tertuliano Franco  Claudio Landim 《Archive for Rational Mechanics and Analysis》2010,195(2):409-439 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).      13. 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,      14. 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≈ − ∞.      15. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      16. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      17. 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.      18. An Example of Finite-time Singularities in the 3d Euler Equations      Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410 Let be the exterior of the closed unit ball. Consider the self-similar Euler system Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution , vanishing at infinity, precisely The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v.      19. Entropy Solutions for Nonlinear Degenerate Problems   总被引：9，自引：0，他引：9   José Carrillo 《Archive for Rational Mechanics and Analysis》1999,147(4):269-361 We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.      20. 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.      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

Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result

Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ^∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by

l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      12. Hydrodynamic Limit of Gradient Exclusion Processes with Conductances      Tertuliano Franco  Claudio Landim 《Archive for Rational Mechanics and Analysis》2010,195(2):409-439 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).      13. 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,      14. 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≈ − ∞.      15. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      16. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      17. 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.      18. An Example of Finite-time Singularities in the 3d Euler Equations      Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410 Let be the exterior of the closed unit ball. Consider the self-similar Euler system Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution , vanishing at infinity, precisely The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v.      19. Entropy Solutions for Nonlinear Degenerate Problems   总被引：9，自引：0，他引：9   José Carrillo 《Archive for Rational Mechanics and Analysis》1999,147(4):269-361 We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.      20. 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.      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

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).     

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,      14. 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≈ − ∞.      15. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      16. Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth      Giovany M. Figueiredo  Marcelo F. Furtado 《Journal of Dynamics and Differential Equations》2012,24(1):13-28 We consider the quasilinear problem -ep\textdiv(|?u|p-2?u) + V(z)up-1 = f(u) + up*-1, u ? W1,p(\mathbbRN), -\varepsilon^p\text{div}(|\nabla u|^{p-2}\nabla u) + V(z)u^{p-1} = f(u) + u^{p^*-1},\,u \in W^{1,p}\left(\mathbb{R}^N\right),      17. 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.      18. An Example of Finite-time Singularities in the 3d Euler Equations      Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410 Let be the exterior of the closed unit ball. Consider the self-similar Euler system Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution , vanishing at infinity, precisely The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L 2 − norm of curl v.      19. Entropy Solutions for Nonlinear Degenerate Problems   总被引：9，自引：0，他引：9   José Carrillo 《Archive for Rational Mechanics and Analysis》1999,147(4):269-361 We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.      20. 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.

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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≈ − ∞.     

Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent

In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l₁ u = m₁ u³+buv²,     x ? W,-Dv +l₂ v = m₂ v³+bvu²,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.     

Positive Solutions for a Quasilinear Schr?dinger Equation with Critical Growth

We consider the quasilinear problem

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.     

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

Entropy Solutions for Nonlinear Degenerate Problems

We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)_t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.     

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.