Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation

A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|_∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.     

Homeomorphisms of Bounded Variation

We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher dimensions we show that f ⁻¹ is of bounded variation provided that f ϵ W ^1,1(Ω; R ⁿ ) is a homeomorphism with |Df| in the Lorentz space L ^n-1,1(Ω). Dedicated to Tadeusz Iwaniec on his 60th birthday     

A Remark on the Existence of 2-D Steady Navier–Stokes Flow in Bounded Symmetric Domain Under General Outflow Condition

Let Ω be a 2-dimensional bounded domain, symmetric with respect to the x₂-axis. The boundary has several connected components, intersecting the x₂-axis. The boundary value is symmetric with respect to the x₂-axis satisfying the general outflow condition. The existence of the symmetric solution to the steady Navier–Stokes equations was established by Amick [2] and Fujita [4]. Fujita [4] proved a key lemma concerning the solenoidal extension of the boundary value by virtual drain method. In this note, we give a different proof via elementary approach by means of the stream function.     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

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

A Density Result in Two-Dimensional Linearized Elasticity,and Applications

We show that in a two-dimensional bounded open set whose complement has a finite number of connected components, the vector fields uH ¹ (Ωℝ²) are dense in the space of fields whose symmetrized gradient e(u) is in L ² (Ωℝ⁴). This allows us to show the continuity of some linearized elasticity problems with respect to variations of the set, with applications to shape optimization or the study of crack evolution. (Accepted September 18, 2002) Published online February 4, 2003 Commmunicated by V. Šverák     

Singular Limits of a Two-Dimensional Boundary Value Problem Arising in Corrosion Modelling

We consider the boundary value problem where Ω is a smooth and bounded domain in ℝ² and λ > 0. We prove that for any integer k ≧ 1 there exist at least two solutions u _λ with the property that the boundary flux satisfies up to subsequences λ → 0, where the ξ _j are points of ∂Ω ordered clockwise in j.     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

On the Time Decay of the Solutions of the Navier–Stokes System

We investigate the relationship between the time decay of the solutions u of the Navier–Stokes system on a bounded open subset of and the time decay of the right-hand sides f. In suitable function spaces, we prove that u always inherits at least part of the decay of f, up to exponential, and that the decay properties of u depend only upon the amount and type (e.g., exponential, or power-like) of decay of f. This is done by first making clear what is meant by “type” and “amount” of decay and by next elaborating upon recent abstract results pointing to the fact that, in linear and nonlinear PDEs, the decay of the solutions is often intimately related to the Fredholmness of the differential operator. This work was done while the second author was visiting the Bernoulli Center, EPFL, Switzerland, whose support is gratefully acknowledged.     

Existence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions

c ). To begin with, we assume that the flux-function f(u) is piecewise genuinely nonlinear, in the sense that it exhibits finitely many (at most p, say) points of lack of genuine nonlinearity along each wave curve. Importantly, our analysis applies to arbitrary large p, in the sense that the constant c restricting the total variation is independent of p. Second, by an approximation argument, we prove that the existence theory above extends to general flux-functions f(u) that can be approached by a sequence of piecewise genuinely nonlinear flux-functions f ^ε(u). The main contribution in this paper is the derivation of uniform estimates for the wave curves and wave interactions (which are entirely independent of the properties of the flux-function) together with a new wave interaction potential which is decreasing in time and is a fully local functional depending upon the angle made by any two propagating discontinuities. Our existence theory applies, for instance, to the p-system of gas dynamics for general pressure-laws p=p(v) satisfying solely the hyperbolicity condition p′(v)<0 but no convexity assumption. (Accepted December 30, 2002) Published online April 23, 2003 Communicated by C. M. Dafermos     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.     

A New Approach to the Existence of Weak Solutions of the Steady Navier–Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries

We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ¹, . . . , Γ ^M and N − M bounded components Γ ^M+1, . . . , Γ ^N . We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L ³-extension to Ω with the gradient in L ²(Ω)^3×3. We assume that the fluxes of α through the bounded components Γ ^M+1, . . . , Γ ^N of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ¹, . . . , Γ ^M .     

Global Existence for Reaction‐Diffusion Systems Modelling Ignition

The pair of parabolic equations , , with a>0 and b>0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. Of particular interest is the case where , which appears in the Frank‐Kamenetskii approximation to Arrhenius‐type reactions. We show here that for a large choice of the nonlinearity f(u,v) in (1), (2)(including the model case (3)), the corresponding initial‐value problem for (1), (2) in the whole space with bounded initial data has a solution which exists for all times. Finite‐time blow‐up might occur, though, for other choices of function f(u,v), and we discuss here a linear example which strongly hints at such behaviour. (Accepted September 16, 1996)     

A Remark on the Invariance of Solenoidal Vectors in Elastostatics

The linear elastostatic displacement boundary-value problem is considered for a bounded simply connected region Ω in R ⁿ in the case of a homogeneous isotropic medium. For n = 2 it is shown that if all solenoidal forcing terms result in solenoidal displacements, then Ω is a disk. It is likely that the result is true for n ≥ 3 but that problem is not resolved. This revised version was published online in August 2006 with corrections to the Cover Date.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

Weak Continuity and Lower Semicontinuity Results for Determinants

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

Removable Singularities for Fully Nonlinear Elliptic Equations

We obtain a removability result for the fully nonlinear uniformly elliptic equations F(D ² u)+f(u)=0. The main theorem states that every solution to the equation in a punctured ball (without any restrictions on the behaviour near the centre of the ball) is extendable to the solution in the entire ball provided the function f satisfies certain sharp conditions depending on F. Previously such results were known for linear and quasilinear operators F. In comparison with the semi- or quasilinear theory the techniques for the fully nonlinear equations are new and based on the use of the viscosity notion of generalised solution rather than the distributional or the weak solutions. Accepted May 3, 2000?Published online November 16, 2000     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

A class of quasilinear differential inequalities whose solutions are ultimately constant

Let be a domain in R ⁿ with compact complement and let T be a quasilinear elliptic or degenerate elliptic operator associated with functions u C ²(). This paper is a study of solutions of (sgn u) Tuf(¦u¦, ¦ grad u¦) where f belongs to a class of functions here termed bifurcation functions. The main condition on f is that uniqueness fails for the ordinary differential equation y'=f(y, y) with the initial condition y(0)=y(0)=0. The conclusion is that u is constant for large ¦x¦ and hence, under mild supplementary hypotheses, u has compact support. Examples show that the results fail if the assumptions on f are only slightly weakened, so that the class of f is, essentially, the largest class for which the results can be stated truly.To James Serrin, in celebration of his 60^th birthday