Parabolic Systems with p, q-Growth: A Variational Approach

We consider the evolution problem associated with a convex integrand ${f : \mathbb{R}^{Nn}\to [0,\infty)}$ satisfying a non-standard p, q-growth assumption. To establish the existence of solutions we introduce the concept of variational solutions. In contrast to weak solutions, that is, mappings ${u\colon \Omega_T \to \mathbb{R}^n}$ which solve $$ \partial_tu-{\rm div} Df(Du)=0 $$ weakly in ${\Omega_T}$ , variational solutions exist under a much weaker assumption on the gap q ? p. Here, we prove the existence of variational solutions provided the integrand f is strictly convex and $$\frac{2n}{n+2} < p \le q < p+1.$$ These variational solutions turn out to be unique under certain mild additional assumptions on the data. Moreover, if the gap satisfies the natural stronger assumption $$ 2\le p \le q < p+ {\rm min}\big \{1,\frac{4}{n} \big \},$$ we show that variational solutions are actually weak solutions. This means that solutions u admit the necessary higher integrability of the spatial derivative Du to satisfy the parabolic system in the weak sense, that is, we prove that $$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$      

A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.     

Generalized Flows Satisfying Spatial Boundary Conditions

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

Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A ₁(x, D) and A ₂(x, D) be differential operators of the first order acting on l-vector functions ${u= (u_1, \ldots, u_l)}$ in a bounded domain ${\Omega \subset \mathbb{R}^{n}}$ with the smooth boundary ${\partial\Omega}$ . We assume that the H ¹-norm ${\|u\|_{H^{1}(\Omega)}}$ is equivalent to ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ and ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ , where B _i  = B _i (x, ν) is the trace operator onto ${\partial\Omega}$ associated with A _i (x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ${\partial\Omega}$ ). Furthermore, we impose on A ₁ and A ₂ a cancellation property such as ${A_1A_2^{\prime}=0}$ and ${A_2A_1^{\prime}=0}$ , where ${A^{\prime}_i}$ is the formal adjoint differential operator of A _i (i = 1, 2). Suppose that ${\{u_m\}_{m=1}^{\infty}}$ and ${\{v_m\}_{m=1}^{\infty}}$ converge to u and v weakly in ${L^2(\Omega)}$ , respectively. Assume also that ${\{A_{1}u_m\}_{m=1}^{\infty}}$ and ${\{A_{2}v_{m}\}_{m=1}^{\infty}}$ are bounded in ${L^{2}(\Omega)}$ . If either ${\{B_{1}u_m\}_{m=1}^{\infty}}$ or ${\{B_{2}v_m\}_{m=1}^{\infty}}$ is bounded in ${H^{\frac{1}{2}}(\partial\Omega)}$ , then it holds that ${\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}$ . We also discuss a corresponding result on compact Riemannian manifolds with boundary.     

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

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

A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios <???>?=?0.7 and <???>?=?1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y _F and mixture fraction ξ (i.e. $\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })$ for Reynolds Averaged Navier Stokes simulations where $\overline q $ , $\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }$ and $q^{\prime\prime}= q-\tilde{q}$ are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ may need to be solved. The statistical behaviours of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T ₁ , reaction rate T ₄ and molecular dissipation $\left( {-D_2 } \right))$ have been analysed in detail. The contribution of T ₁ remains important for all cases considered here. The term T ₄ acts as a major contributor in <???>?=?1.0 cases, but plays a relatively less important role in <???>?=?0.7 cases, whereas the term $\left( {-D_2 } \right)$ acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T ₁ , T ₄ and $\left( {-D_2 } \right)$ have been addressed in detail. A model has been identified for the turbulent transport term T ₁ which satisfactorily predicts the corresponding term obtained from DNS data. The models for T ₄ , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T ₄ obtained from DNS data. It has been found that existing algebraic models for $D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }$ for the closure of the $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ transport equation.     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

On the effect of non-smooth Coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-DoF-system

This article deals with self-excited vibrations, attractivity of stationary solutions, and the corresponding bifurcation behavior of two-dimensional differential inclusions of the type $\mathbf{M}\mathbf{q}'' + \mathbf{D}\mathbf{q}' + (\mathbf{K} + \bar{\mu}\mathbf{N})\mathbf{q} \in-\mathbf{R}\operatorname{Sign}(\mathbf{q}')$ . For the smooth case R=0, the equilibrium may become unstable due to non-conservative positional forces stemming from the circulatory matrix N. This type of instability is usually referred to as flutter instability and the loss of stability is related to a Hopf bifurcation of the steady state, which occurs for a critical parameter $\bar{\mu}= \bar{\mu}_{\mathrm{crit}}$ . For R≠0, the steady state is a set of equilibria, which turns out to be attractive for all values of the bifurcation parameter $\bar{\mu}$ . Depending on $\bar{\mu}$ , the basin of attraction of the equilibrium set can be infinite or finite. The transition from an infinite to a finite basin of attraction occurs at the stability threshold $\bar{\mu}_{\mathrm{crit}}$ of the underlying smooth problem. For the finite basin of attraction, its size is proportional to the Coulomb friction and inverse-proportional to $(\bar{\mu}- \bar{\mu}_{\mathrm{crit}})$ . By adding Coulomb damping the notion of steady state stability for the smooth problem is replaced by the question whether the basin of attraction of the steady state is infinite or finite. Simultaneously, the local Hopf-bifurcation is replaced by a global bifurcation. This implies that in the presence of Coulomb damping the occurrence of self-excited vibrations can only be investigated with regard to the perturbation level.     

Phragmén-Lindelöf theorems for nonlinear elliptic equations

We obtain theorems of Phragmén-Lindelöf type for the following classes of elliptic partial differential inequalities in an arbitrary unbounded domain \(\Omega \subseteq \mathbb{R}^n ,{\text{ }}n \geqq 2\) (A.1) $$\sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} 9(x)\frac{{\partial u}}{{\partial xj}}} \right)} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a _ij are elliptic in Ω and b _i ε L^∞(Ω) and where also a _ij are uniformly elliptic and Holder continuous at infinity and b _i = O(|x|⁺¹) as x → ∞; (A.2) $${\text{(A}}{\text{.2) }}\sum\limits_{i,j = 1}^n {a_{ij} (x,{\text{ }}u,{\text{ }}\nabla u)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a_ijare uniformly elliptic in Ω and b _iε L^∞(Ω); and finally (A.3) $${\text{div(}}\nabla u^p \nabla u {\text{)}} \geqq f{\text{(}}u{\text{), }}p > - 1,$$ where the operator on the left is the so-called P-Laplacian. The function f is always supposed positive and continuous. Moreover u is assumed throughout to be in the natural weak Sobolev space corresponding to the particular inequality under consideration, namely u ε. W _loc ^1,2 (Ω) ∩L _loc ^t8 (Ω) for (A.I), W _loc ^2,n(Ω) for (A.2), and W _loc ^1,p+2 (Ω) ∩ L _loc ^t8 (Ω) for (A.3). As a consequence of our results we obtain both non-existence and Liouville theorems, as well as existence theorems for (A.1).     

Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, ${\Omega \subset \mathbb{R}^2}$ is a bounded domain, ${\Omega_i^{+}}$ and ${\Omega_j^{-}}$ are mutually disjoint subdomains of Ω and ${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$ are characteristic functions of ${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C ¹-stable critical point of Kirchhoff–Routh function ${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$ with ${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$ and ${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity ${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$ . The case that n = 0 can be dealt with in the same way as well by taking each ${\Omega_j^{-}}$ as an empty set and set ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .     

First-principles study of full a-dislocations in pure magnesium

Full a-dislocations on the (0001) basal plane, \((10\bar 10)\) prismatic plane, and \((10\bar 11)\) and \((10\bar 12)\) pyramidal planes in pure magnesium are investigated by using the Peierls-Nabarro model combined with generalized stacking fault (GSF) energies from first-principles calculations. The results show that the \(\left( {10\bar 11} \right)\left\langle {11\bar 20} \right\rangle\) and \(\left( {10\bar 12} \right)\left\langle {11\bar 20} \right\rangle\) slip modes have nearly the same GSF energy barriers, which are obviously larger than the GSF energy barriers of the \(\left( {0001} \right)\left\langle {11\bar 20} \right\rangle\) and \(\left( {10\bar 10} \right)\left\langle {11\bar 20} \right\rangle\) slip modes. For both edge and screw full dislocations, the maximum dislocation densities, Peierls energies, and stresses of dislocations on the \((10\bar 10)\) , (0001), \((10\bar 11)\) , and \((10\bar 12)\) planes eventually increase. Moreover, the Peierls energies and the stresses of screw full dislocations are always lower than those of edge full dislocations for all slip systems. Dislocations on the \((10\bar 11)\) and \((10\bar 12)\) pyramidal planes possess smaller core energies, while the \((10\bar 10)\) prismatic plane has the largest ones, implying that the formation of full dislocations on the \((10\bar 10)\) plane is more difficult.     

An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study)

Transition to turbulence in axially symmetrical laminar pipe flows with periodic time dependence classified as pure oscillating and pulsatile (pulsating) ones is the concern of the paper. The current state of art on the transitional characteristics of pulsatile and oscillating pipe flows is introduced with a particular attention to the utilized terminology and methodology. Transition from laminar to turbulent regime is usually described by the presence of the disturbed flow with small amplitude perturbations followed by the growth of turbulent bursts. The visual treatment of velocity waveforms is therefore a preferred inspection method. The observation of turbulent bursts first in the decelerating phase and covering the whole cycle of oscillation are used to define the critical states of the start and end of transition, respectively. A correlation study referring to the available experimental data of the literature particularly at the start of transition are presented in terms of the governing periodic flow parameters. In this respect critical oscillating and time averaged Reynolds numbers at the start of transition; Re _os,crit and Re _ta,crit are expressed as a major function of Womersley number, $\sqrt {\omega ^\prime } $ defined as dimensionless frequency of oscillation, f. The correlation study indicates that in oscillating flows, an increase in Re _os,crit with increasing magnitudes of $\sqrt {\omega ^\prime } $ is observed in the covered range of $1<\sqrt {\omega ^\prime } <72$ . The proposed equation (Eq. 7), ${\rm{Re}}_{os,crit} ={\rm{Re}}_{os,crit} \left( {\sqrt {\omega ^\prime } } \right)$ , can be utilized to estimate the critical magnitude of $\sqrt {\omega ^\prime }$ at the start of transition with an accuracy of ±12?% in the range of $\sqrt {\omega ^\prime } <41$ . However in pulsatile flows, the influence of $\sqrt {\omega ^\prime }$ on Re _ta,crit seems to be different in the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ . Furthermore there is rather insufficient experimental data in pulsatile flows considering interactive influences of $\sqrt {\omega ^\prime } $ and velocity amplitude ratio, A ₁. For the purpose, the measurements conducted at the start of transition of a laminar sinusoidal pulsatile pipe flow test case covering the range of 0.21<?A ₁?<0.95 with $\sqrt {\omega ^\prime } <8$ are evaluated. In conformity with the literature, the start of transition corresponds to the observation of first turbulent bursts in the decelerating phase of oscillation. The measured data indicate that increase in $\sqrt {\omega ^\prime } $ is associated with an increase in Re _ta,crit up to $\sqrt {\omega ^\prime } =3.85$ while a decrease in Re _ta,crit is observed with an increase in $\sqrt {\omega ^\prime } $ for $\sqrt {{\omega }'} >3.85$ . Eventually updated portrait is pointing out the need for further measurements on i) the end of transition both in oscillating and pulsatile flows with the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ , and ii) the interactive influences of $\sqrt {\omega ^\prime } $ and A ₁ on Re _ta,crit in pulsatile flows with the range of $\sqrt {\omega ^\prime } >8$ .     

Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems

We consider the following nonlinear Schrödinger system in ${\mathbb{R}^3}$ $$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$ where P(r) and Q(r) are positive radial potentials, ${\mu > 0, \nu > 0}$ and ${\beta \in \mathbb{R}}$ is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.     

Real Solutions to the Nonlinear Helmholtz Equation with Local Nonlinearity

In this paper, we study real solutions of the nonlinear Helmholtz equation $$- \Delta u - k^2 u = f(x,u),\quad x\in \mathbb{R}^N$$ satisfying the asymptotic conditions $$u(x)=O\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2u(x)=o\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm as}\, r=|x| \to\infty.$$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein–Gordon equations with arbitrarily large frequency.     

Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ)=U(γ ₁,γ ₂,γ ₃). This motion subject to the constraint 〈ν,ω〉=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω ₁,ω ₂,ω ₃) is the angular velocity and 〈?,?〉 is the inner product of $\mathbb{R}^{3}$ . We provide the following new integrable cases. (i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that $$U=-\frac{1}{2I_1I_2}\bigl(I_1\mu^2_1+I_2 \mu^2_2\bigr), $$ where I ₁,I ₂, and I ₃ are the principal moments of inertia of the body, μ ₁ and μ ₂ are solutions of the first-order partial differential equation $$\gamma_3 \biggl(\frac{\partial\mu_1}{\partial\gamma_2}- \frac{\partial\mu_2}{\partial \gamma_1} \biggr)- \gamma_2\frac{\partial \mu_1}{\partial\gamma_3}+\gamma_1\frac{\partial\mu_2}{\partial \gamma_3}=0. $$ (ii) The Veselova problem is integrable for the potential $$U=-\frac{\varPsi^2_1+\varPsi^2_2}{2(I_1\gamma^2_2+I_2\gamma^2_1)}, $$ where Ψ ₁ and Ψ ₂ are the solutions of the first-order partial differential equation where $p=\sqrt{I_{1}I_{2}I_{3} (\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}} )}$ . Also it is integrable when the potential U is a solution of the second-order partial differential equation where $\tau_{2}=I_{1}\gamma^{2}_{1}+I_{2}\gamma^{2}_{2}+I_{3}\gamma^{2}_{3}$ and $\tau_{3}=\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}}$ . Moreover, we show that these integrable cases contain as a particular case the previous known results.     

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

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

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II 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? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.