Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Potentials for tangent tensor fields on spheroids

In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let \(\hat n\) be the outward unit normal to S, ▽ _S the surface gradient on S, I _S the metric tensor on S, g_ij the four covariant components of I _S (i,j = 1, 2), h _ij the four covariant components of -\(\hat n\)xI _S , and D _i covariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ? and ψ on S, unique to within additive constants, such that \(u = \nabla _s \varphi - \hat n \times \nabla _s \psi \); the covariant components of u are \(u_i = D_i \varphi + h_i^j D_j \psi \). This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are

$$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$     

Bistable Boundary Reactions in Two Dimensions

In a bounded domain \({\Omega \subset \mathbb R^2}\) with smooth boundary we consider the problem

$\Delta u = 0 \quad {\rm{in }}\, \Omega, \qquad \frac{\partial u}{\partial \nu} = \frac1\varepsilon f(u) \quad {\rm{on }}\,\partial\Omega,$

where ν is the unit normal exterior vector, ε > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(π u) or f(u) = (1 ? u ²)u. We construct solutions that develop multiple transitions from ?1 to 1 and vice-versa along a connected component of the boundary ?Ω. We also construct an explicit solution when Ω is a disk and f(u) = sin(π u).

     

Effects of Fuel Lewis Number on Localised Forced Ignition of Globally Stoichiometric Stratified Mixtures: a Numerical Investigation

The influences of fuel Lewis number Le_F on localised forced ignition of globally stoichiometric stratified mixtures have been analysed using three-dimensional compressible Direct Numerical Simulations (DNS) for cases with Le_F ranging from 0.8 to 1.2. The globally stoichiometric stratified mixtures with different values of root-mean-square (rms) equivalence ratio fluctuation (i.e. ?^′= 0.2, 0.4 and 0.6) and the Taylor micro-scale l_? of equivalence ratio ? variation (i.e. l_?/l_f= 2.1, 5.5 and 8.3 with l_f being the Zel’dovich flame thickness of the stoichiometric laminar premixed flame) have been considered for different initial rms values of turbulent velocity u^′. A pseudo-spectral method is used to initialise the equivalence ratio variation following a presumed bi-modal distribution for prescribed values of ?^′ and l_?/l_f for global mean equivalence ratio 〈?〉=1.0. The localised ignition is accounted for by a source term in the energy transport equation that deposits energy for a stipulated time interval. It has been observed that the maximum values of temperature and the fuel reaction rate magnitude increase with decreasing Le_F during the period of external energy deposition. The initial values of Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f have been found to have significant effects on the extent of burning of the stratified mixtures following localised ignition. For a given value of u^′/S_b(?=1), the extent of burning decreases with increasing Le_F. An increase in u^′ leads to a monotonic reduction in the burned gas mass for all values of Le_F in all stratified mixture cases but an opposite trend is observed for the Le_F=0.8 homogeneous mixture. It has been found that an increase in ?^′ has adverse effects on the burned gas mass, whereas the effects of l_?/l_f on the extent of burning are non-monotonic and dependent on ?^′ and Le_F. Detailed physical explanations have been provided for the observed Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f dependences.     

The stress state of a plate subjected to a piecewise homogeneous load

We consider the stress-strain state of a plate having a doubly connected domain S bounded from the outside by a circle of radius R and from the inside by an ellipse with two rectilinear cuts. The cuts lie symmetrically on the x-axis. The plate is subjected to various forces: the hole contour (the ellipse) is under the action of uniformly distributed forces of intensity q, and the cut shores are free of loads; at the points ±ib of the imaginary axis, the plate is under the action of a lumped force P.The solution of the problem is reduced to determining two analytic functions φ(z) and ψ(z) satisfying certain boundary conditions (depending on the type of the acting loads).We use the Kolosov-Muskhelishvili method to reduce the problem to a system of linear algebraic equations for the coefficients in the expansions of the functions φ(z) and ψ(z). The solution thus obtained is illustrated by numerical examples.     

Entire Solutions to Equivariant Elliptic Systems with Variational Structure

We consider the system Δu ? W _u (u) = 0, where \({u : \mathbb{R}^n \to \mathbb{R}^n}\) , for a class of potentials \({W : \mathbb{R}^n \to \mathbb{R}}\) that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial G-equivariant entire solutions connecting the global minima of W along certain directions at infinity.     

Regularity and Singularities of Optimal Convex Shapes in the Plane

We focus here on the analysis of the regularity or singularity of solutions Ω ₀ to shape optimization problems among convex planar sets, namely:

$J(\Omega_{0})={\rm min} \{J(\Omega), \Omega \quad {\rm convex},\Omega \in \mathcal{S}_{\rm ad}\},$

where \({\mathcal{S}_{\rm ad}}\) is a set of 2-dimensional admissible shapes and \({J:\mathcal{S}_{\rm ad}\rightarrow\mathbb{R}}\) is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results:

1. i)
   under a suitable convexity property of the functional J, we prove that Ω ₀ is a W ^2,p -set, \({p\in[1, \infty]}\). This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P(Ω), where R(Ω) = F(|Ω|, E _f (Ω), λ₁(Ω)) involves the area |Ω|, the Dirichlet energy E _f (Ω) or the first eigenvalue of the Laplace–Dirichlet operator λ₁(Ω), and P(Ω) is the perimeter of Ω;
    

1. ii)
   under a suitable concavity assumption on the functional J, we prove that Ω ₀ is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω) ? P(Ω), with the same notations as above.
    

     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Power-law creep and residual stresses in a carbopol gel

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ) ^α , with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ? 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.     

On the orders of the best approximations of integrals of functions by integrals of rank <Emphasis Type="Italic">σ</Emphasis>

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ? is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^? .     

Inviscid Models Generalizing the Two-dimensional Euler and the Surface Quasi-geostrophic Equations

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u _j of the velocity field u is determined by the scalar θ through \({u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}\) , where \({\mathcal{R}}\) is a Riesz transform and Λ = (?Δ)^1/2. The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound \({\|\nabla u||_{L^\infty}}\) for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log(I + log(I ? Δ))) ^γ with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ ^β with 0 ≦ β ≦ 1 is also obtained.     

Does Density Ratio Significantly Affect Turbulent Flame Speed?

In order to experimentally study whether or not the density ratio σ substantially affects flame displacement speed at low and moderate turbulent intensities, two stoichiometric methane/oxygen/nitrogen mixtures characterized by the same laminar flame speed S _L = 0.36 m/s, but substantially different σ were designed using (i) preheating from T _u = 298 to 423 K in order to increase S _L , but to decrease σ, and (ii) dilution with nitrogen in order to further decrease σ and to reduce S _L back to the initial value. As a result, the density ratio was reduced from 7.52 to 4.95. In both reference and preheated/diluted cases, direct images of statistically spherical laminar and turbulent flames that expanded after spark ignition in the center of a large 3D cruciform burner were recorded and processed in order to evaluate the mean flame radius \(\bar {R}_{f}\left (t \right )\) and flame displacement speed \(S_{t}=\sigma ^{-1}{d\bar {R}_{f}} \left / \right . {dt}\) with respect to unburned gas. The use of two counter-rotating fans and perforated plates for near-isotropic turbulence generation allowed us to vary the rms turbulent velocity \(u^{\prime }\) by changing the fan frequency. In this study, \(u^{\prime }\) was varied from 0.14 to 1.39 m/s. For each set of initial conditions (two different mixture compositions, two different temperatures T _u , and six different \(u^{\prime })\), five (respectively, three) statistically equivalent runs were performed in turbulent (respectively, laminar) environment. The obtained experimental data do not show any significant effect of the density ratio on S _t . Moreover, the flame displacement speeds measured at u′/S _L = 0.4 are close to the laminar flame speeds in all investigated cases. These results imply, in particular, a minor effect of the density ratio on flame displacement speed in spark ignition engines and support simulations of the engine combustion using models that (i) do not allow for effects of the density ratio on S _t and (ii) have been validated against experimental data obtained under the room conditions, i.e. at higher σ.     

Planar Vector Fields with a Given Set of Orbits

We determine all the \({\mathcal{C}^1}\) planar vector fields with a given set of orbits of the form y ? y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve \({g(x, y) = \sum_{j=0}^S a_j(x) y^{S-j}= 0}\) with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S.     

Regularity of Solutions to the Navier-Stokes Equations for Compressible Barotropic Flows on a Polygon

The Navier-Stokes system for a steady-state barotropic nonlinear compressible viscous flow, with an inflow boundary condition, is studied on a polygon D. A unique existence for the solution of the system is established. It is shown that the lowest order corner singularity of the nonlinear system is the same as that of the Laplacian in suitable L ^q spaces. Let ω be the interior angle of a vertex P of D. If \(\) and \(\), then the velocity u is split into singular and regular parts near the vertex P. If α < 2 and \(\) or if α > 2 and 2 < q < ∞&;, it is shown that u∈ (H ^{2, q} (D))².     

Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l ³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is any portion of?ω withlength γ ₀>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ ₀, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ ₀) is contained in a generatrix ofS. We show that, if the applied body force density isO(? ²) with respect to?, the fieldu(?)=(u _i (?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH ¹(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts _?1 ¹ u dx ₃, which belongs to the spaceV _F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η _i ) ∈V _F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified.     

Eventual C∞-regularity and concavity for flows in one-dimensional porous media

We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation $$\begin{gathered} {\text{ }}u_t = \left( {u^m } \right)_{xx} {\text{ in }}Q = \mathbb{R} \times \left( {{\text{0,}}\infty } \right){\text{,}} \hfill \\ u\left( {x{\text{,0}}} \right) = u_{\text{0}} \left( x \right){\text{ for }}x \in \mathbb{R}{\text{,}} \hfill \\ \end{gathered}$$ with m > 1 and, u ₀a continuous, nonnegative function. It is well known that, across a moving interface x=ζ(t) of the solution u(x, t), the derivatives v tand v _x of the pressure v = (m/(m?1)) u ^m?1 have jump discontinuities. We prove that each moving part of the interface is a C ^∞curve and that v is C ^∞on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t→∞.     

The Viscosity Method for the Homogenization of Soft Inclusions

In this paper, we consider periodic soft inclusions T _ε with periodicity ε, where the solution, u _ε , satisfies semi-linear elliptic equations of non-divergence in \({\Omega_{\epsilon}=\Omega\setminus \overline{T}_\epsilon}\) with Neumann data on \({\partial T^{\mathfrak a}}\). The difficulty lies in the non-divergence structure of the operator where the standard energy method, which is based on the divergence theorem, cannot be applied. The main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of u _ε , referred to as u, as ε approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of a second corrector is then developed to show that the limit, u, is the viscosity solution of a homogenized equation.     

Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2?, clamped along their entire lateral face, all having the same middle surfaceS=φ(?ω) ?R ³, whereω ?R ² is a bounded and connected open set with a Lipschitz-continuous boundaryγ, andφ ∈l ³ ( $\overline \omega$ ;R ³). We make an essential geometrical assumption on the middle surfaceS, which is satisfied ifγ andφ are smooth enough andS is “uniformly elliptic”, in the sense that the two principal radii of curvature are either both>0 at all points ofS, or both<0 at all points ofS. We show that, if the applied body force density isO(1) with respect to?, the fieldtu(?)=(u _i(?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, one “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges inH ¹(Ω)×H ¹(Ω)×L ²(Ω) as?→0 to a limitu, which is independent of the transverse variable. Furthermore, the averageξ=1/2ε _?1 ¹ u dx ₃, which belongs to the space $$V_M (\omega ) = H_0^1 (\omega ) \times H_0^1 (\omega ) \times L^2 (\omega ),$$ satisfies the (scaled) two-dimensional equations of a “membrane shell” viz., $$\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta )\gamma _{\alpha \beta } (\eta ) \sqrt \alpha dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\}\eta _i \sqrt a dy$$ for allη=(η _i) εV _M(ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\gamma _{\alpha \beta } (\eta ) = \frac{1}{2}\left( {\partial _{\alpha \eta \beta } + \partial _{\beta \eta \alpha } } \right) - \Gamma _{\alpha \beta }^\sigma \eta _\sigma - b_{\alpha \beta \eta 3} $$ are the components of the linearized change of metric tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_{\alpha \beta }$ are the components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “membrane shell” are therefore justified.     

Perspectives on the Phenomenology and Modeling of Boundary Layer Transition

The characteristics of the coherent structures in a strongly decelerated large-velocity-defect boundary layer are analysed by direct numerical simulation. The simulated boundary layer starts as a zero-pressure-gradient boundary layer, decelerates under a strong adverse pressure gradient, and separates near the end of the domain, in the form of a very thin separation bubble. The Reynolds number at separation is R e _?? =3912 and the shape factor H=3.43. The three-dimensional spatial correlations of (u, u) and (u, v) are investigated and compared to those of a zero-pressure-gradient boundary layer and another strongly decelerated boundary layer. These velocity pairs lose coherence in the streamwise and spanwise directions as the velocity defect increases. In the outer region, the shape of the correlations suggest that large-scale u structures are less streamwise elongated and more inclined with respect to the wall in large-defect boundary layers. The three-dimensional properties of sweeps and ejections are characterized for the first time in both the zero-pressure-gradient and adverse-pressure-gradient boundary layers, following the method of Lozano-Durán et al. (J. Fluid Mech. 694, 100–130, [2012]). Although longer sweeps and ejections are found in the zero-pressure-gradient boundary layer, with ejections reaching streamwise lengths of 5 boundary layer thicknesses, the sweeps and ejections tend to be bigger in the adverse-pressure-gradient boundary layer. Moreover, small near-wall sweeps and ejections are much less numerous in the large-defect boundary layer. Large sweeps and ejections that reach the wall region (wall-attached) are also less numerous, less streamwise elongated and they occupy less space than in the zero-pressure-gradient boundary layer.