On the number of singular points of weak solutions to the Navier‐Stokes equations

We consider a suitable weak solution to the three‐dimensional Navier‐Stokes equations in the space‐time cylinder Ω × ]0, T[. Let Σ be the set of singular points for this solution and Σ (t) ≡ {(x, t) ∈ Σ}. For a given open subset ω ? Ω and for a given moment of time t ∈]0, T[, we obtain an upper bound for the number of points of the set Σ(t) ? ω. © 2001 John Wiley & Sons, Inc.     

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2

We analyze the two‐dimensional parabolic‐elliptic Patlak‐Keller‐Segel model in the whole Euclidean space ?². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local‐in‐time existence for any mass of “free‐energy solutions,” namely weak solutions with some free‐energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free‐energy solutions with initial data as before for the critical mass 8π/χ. Actually, we prove that solutions blow up as a delta Dirac at the center of mass when t → ∞ when their second moment is kept constant at any time. Furthermore, all moments larger than 2 blowup as t → ∞ if initially bounded. © 2007 Wiley Periodicals, Inc.     

Nonlinear degenerate parabolic equations with time dependent singular coefficients

We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^NWe are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^N$ (resp. a Carnot Carathéodory metric ball in $\mathbf {R}^{2N+1})$ with smooth boundary and the time dependent singular potential function V(x, t) ∈ L¹_loc(Ω × (0, T)), $\alpha , \beta \in \mathbf {R}$, 1 < p < N, p ? 1 + α + β > 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in Lp

The equation $\text{Lu = ?;(x, u)}\text{on}\text{B × (0, ∞), B}$ bounded, smooth domain in $\text{R}$ⁿ with nonlinear boundary conditions $\text{?u}\text{?v}\text{= g(x, u)}\text{on}\text{?B × (0, ∞)}$ is studied, L being the uniformly parabolic operator with time independent coefficients. Under suitable conditions on the nonlinearities (that do not involve monotonicity) global existence, uniqueness, compactness of the orbits and certain regularizing effects of the semigroup are established. In the case that L is in divergence form it is shown that under generic conditions orbits tend, as t → + ∞, to some equilibrium and that the stable equilibria attract essentially (Baire category) the whole space L²(B).     

LARGE TIME BEHAVIOR OF SOLUTIONS TO SOME DEGENERATE PARABOLIC EQUATIONS

The purpose of this paper is to study the limit in L ¹(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form u_t ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Two-sided estimates of a solution of the Neumann problem as t ← ∠ for a second-order Quasilinear Parabolic Equation

We establish exact upper and lower bounds as t ← ∠ for the norm ‖u(·, t)‖ _{L ∞(Ω)} of a solution of the Neumann problem for a second-order quasilinear parabolic equation in the region D=Ω×{>0}, where Ω is a region with noncompact boundary.     

The Cauchy problem for quasilinear SG‐hyperbolic systems

We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × ?ⁿ and presenting a linear growth for |x | → ∞. We prove well‐posedness in the Schwartz space ?? (?ⁿ ). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of semi-linear parabolic systems in diffusion–reaction problems

In this paper, we investigate problems of approximation for the solution of a system of coupled semi-linear parabolic partial differential equations that model diffusion-reaction problems in chemical engineering. Given that the solutions belong to H^s (0, ∞), we consider finite-element approximations on bounded domains (0, R(h)) such that lim_h→0[R(h)] = ∞. Optimal convergence estimates are found to depend on the asymptotic behaviour of the solution and its regularity near t = 0. In the L²-norm, they cannot exceed an order of O((;h²/t^3/4) + h²[In h]²). For that purpose, a Wheeler-type argument is also generalized to non-coercive elliptic forms. Fully discrete schemes that preserve the positivity of the solutions are also considered. Due to the singularity at t = 0, they lead to estimates of the order O(τ^1/4 + h²/t^3/4).     

Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems

We consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions in a scattering regime. We assume the matrix‐valued coefficients are analytic in the space variable, and we further suppose that the corresponding dispersion relation admits real‐valued modes only with one‐dimensional polarization subspaces. Hence a BKW‐type analysis of the solutions is possible. We typically consider time‐dependent solutions to the PDE that are carried asymptotically in the past and as x → ?∞ along one mode only and determine the piece of the solution that is carried for x → +∞ along some other mode in the future. Because of the assumed nondegeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the Landau‐Zener mechanism. We completely elucidate the spacetime properties of the leading term of this exponentially small wave, when the semiclassical parameter is small, for large values of x and t, when some avoided crossing of finite width takes place between the involved modes. © 2006 Wiley Periodicals, Inc.     

The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients

We consider the Dirichlet problem for non‐divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi‐linear parabolic equations in a bounded domain. In particular, if the boundary is of class C^1,δ , δ ∈ [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

Reconstruction of kernel depending also on space variable

The paper deals with the reconstruction of the convolution kernel, together with the solution, in a mixed linear evolution system of hyperbolic type. This problem describes uniaxial deformations u of a cylindrical domain (0,π) × Ω, which is filled with a linear viscoelastic solid whose material properties are supposed to be uniform on Ω‐sections perpendicular to the x axis. Various types of boundary conditions in [0,T] × {0,π} × Ω are prescribed, whereas Dirichlet conditions are assumed in [0,T] × (0,π) × ∂Ω. To reconstruct both u and k, we suppose of knowing for any time tand any x ∈ (0,π) the flux of the viscoelastic stress vector through the boundary of the Ω‐section. The main novelty is that the unknown kernel k is allowed to depend, not only on the time variable t but also on the space variable x.     

Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta‐Kawashima condition. We show that these solutions approach a constant equilibrium state in the L^p‐norm at a rate O(t^{? (m/2)(1 ? 1/p)}) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman‐Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem. © 2007 Wiley Periodicals, Inc.     

Some existence and nonexistence theorems for solutions of degenerate parabolic equations

The initial and boundary value problem for the degenerate parabolic equation v_t = Δ(?(v)) + F(v) in the cylinder $\text{Ω × ¦0, ∞), Ω ? R}{}^{\mathrm{n}}$ bounded, for a certain class of point functions ? satisfying ?′(v) ? 0 (e.g., $\text{?(v) = ¦v¦}{}^{\mathrm{m}}\text{sign}\text{v}$) is considered. In the case that F(v) sign $\text{v ? C(1 + ¦?(v)¦}{}^{\mathrm{\alpha }}\text{), α < 1}$, the equation has a global time solution. The same is true for α = 1 provided the measure of Ω is sufficiently small. In the case that $\text{F(v)}\text{?(v)}$ is nondecreasing a condition is given on the initial state v(x, 0) which implies that the solution must blow up in finite time. The existence of such initial states is discussed.     

On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

Critical behavior of nonintersecting Brownian motions at a tacnode

We study a model of n one‐dimensional, nonintersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time‐space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new family of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new Riemann‐Hilbert problem of size 4 × 4. We prove solvability of the Riemann‐Hilbert problem and establish a remarkable connection with the Hastings‐McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the nonintersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class. © 2011 Wiley Periodicals, Inc     

Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ ([0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. Under appropriate conditions upon the given functions F and φ, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ? × X × ?^k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solutions of the Laplace equation with a nonlinear dynamical boundary condition

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝ^N, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.