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 ${L^\infty(\Omega)}$ .     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

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.     

A well posed problem in singular Fickian diffusion

We prove that the problem of solving $$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$ with initial conditionu(x, 0)=φ(x) and flux conditions at infinity \(\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)\) , admits a unique solution \(u \in C^\infty \{ - \infty< x< \infty ,0< t< T\} \) for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L _loc ^∞ [0, ∞) The maximal existence time is given by $$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$ This mixed problem is ill posed for anym outside the above specified range.     

Temperature solutions due to time-dependent moving-line-heat sources

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-line-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) $\dot Q_1 (t) = \dot Q_0 \exp ( - \lambda t)$ , (ii) $\dot Q_2 (t) = \dot Q_0 (t/t^ \star )\exp ( - \lambda t)$ , and $\dot Q_3 (t) = \dot Q_0 [1 + a\cos (\omega t)]$ , whereλ andω are real parameters andt? characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ (α,x;b) and its decompositionsC _Γ andS _Γ. It is also demonstrated that the present analysis covers the classical temperature solution of a constant strength source under quasi-steady-state situations.     

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.     

Compressible Navier–Stokes Equations on Thin Domains

We consider the barotropic Navier–Stokes system describing the motion of a compressible viscous fluid confined to a straight layer \({\Omega_\varepsilon = \omega\times (0, \varepsilon)}\) , where ω is a particular 2-D domain (a periodic cell, bounded domain or the whole 2-D space). We show that the weak solutions in the 3D domain converge to a (strong) solutions of the 2-D Navier–Stokes system on ω as \({\varepsilon \to 0}\) on the maximal life time of the strong solution.     

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.     

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.     

A minimization problem related to the estimation of a lower bound for the temperature in a free boundary value problem related to the combustion of a solid

This paper investigates the least time τ_* of the first zero of the bounded solution to an initial boundary value problem for the heat equation. The heat equation is considered in the domain $$\left\{ {(x,t)| - \infty< x< s(t),0< t \leqslant T} \right\}$$ . The initial conditionu(x, 0)=φ(x) and the boundary conditionu _x (s(t),t)=?R are specified. Let τ=τ(φ,R, s) denote the first zero ofu onx=s(t), that is,u(s(τ), τ)=0. Let τ_*=min τ, where the minimum is taken over a class of functionss=s(t). The existence of τ_* is demonstrated, and a generalization of the problem is discussed.     

Viscous ring modes in vortices with axial jet

In this paper, we show the existence of new families of linear eigenmodes in vortices with axial jet. These modes are viscous in nature and concentrated in a ring around the vortex at the critical radial location r _c  > 0 where ${m\Omega '_c + kW'_c=0}$ where ${\Omega_c'}$ and ${W_c'}$ are the radial derivative at r _c of the angular and axial velocity of the vortex. Using a large Reynolds-number asymptotic approach for an arbitrary axisymmetrical vortex with axial flow, both the complex frequency and the spatial structure of the eigenmodes are obtained for any azimuthal and axial wave number. The asymptotic predictions are compared to numerical results for the q-vortex and a good agreement is demonstrated. We show that for sufficiently large Reynolds numbers, a necessary and sufficient condition of instability of viscous ring modes is that there exists a location r _c where ${\Omega_c\Omega_c'[r_c\Omega_c'(2\Omega_c+r_c\Omega'_c)+(W_c')^2]<0}$ and ${W_c'\neq0}$ , which also corresponds to the condition of inviscid instability obtained by Leibovich and Stewartson (J Fluid Mech 126:335–356, 1983).     

On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ ₀=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .     

A hereditary partial differential equation with applications in the theory of simple fluids

The linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation $$S + p\delta = 2\int\limits_0^\infty {m(s)(E(t - s} ) - E(t))ds$$ is considered. Here S is the stress tensor, δ the Kronecker delta, p the constitutively indeterminate mean normal stress, E the infinitesimal strain tensor, and m(s) a material function. The shear relaxation modulus G is defined as (i) $$G(s) = \int\limits_\infty ^s {m(\xi )d\xi .}$$ In this paper it is shown that if G satisfies the assumptions (i) $$G \in C^2 [0,\infty ),{\text{ }}G(s) \to 0{\text{ as }}s \to \infty,$$ (ii) $$( - 1)^k \frac{{d^k G(s)}}{{ds^k }} > 0,{\text{ }}k = 0,1,$$ (iii) $$G''(s) \geqq 0,$$ then the rest state of the fluid is stable in an appropriate “fading memory” norm. The additional assumption (iv) $$ - \int\limits_0^\infty {G'} (s)s^2 ds < \infty$$ yields asymptotic stability.     

Existence of Bubbling Solutions for Chern–Simons Model on a Torus

We study the existence of bubbling solutions for the the following Chern–Simons–Higgs equation: $$\Delta u +\frac1{\varepsilon^2} {\rm e}^u(1-{\rm e}^u) = 4\pi \sum_{i=1}^{2k}\delta_{p_i},\quad \text{in}\,\Omega,$$ where Ω is a torus. If k = 1, for any critical point q of the associated sum of the Green functions, we introduce a quantity D(q) (see (1.11) below). We show that for any non-degenerate critical point q with D(q) < 0, the above problem has a solution u _ε satisfying that ε → 0, u _ε blows up at q. The calculations in this paper also show that, if a sequence of solutions u _ε blows up at q as ε → 0, then q must be a critical point of the associated sum of the Green functions, and ${D(q) \leqq 0}$ . So, the condition D(q) < 0 is almost necessary to obtain our result. We also construct solutions with k bubbles for ${k \geqq 2}$ .     

On Saint-Venant’s Problem with Concentrated Loads

The classical Saint-Venant problem is to find a solution of the traction problem of elastostatics in a finite cylinder ?? loaded over its bases. We prove that the problem has a unique solution for equilibrated surface forces $\hat{ \boldsymbol { s}}\in W^{-1,q}(\partial\Omega)$ , with q??(2?? ₀,+??) for some positive ? ₀ depending on ??. Hence $\hat{ \boldsymbol { s}}$ can model force acting on ???, concentrated on sets of zero Lebesgue surface measure of ???. Moreover, if $\hat{ \boldsymbol { s}}$ is equilibrated on each basis, we give a simple proof of the Toupin estimate expressing Saint-Venant??s principle.     

A Concentration Phenomenon for Semilinear Elliptic Equations

For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q _n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q _n  > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u _n is a nontrivial solution corresponding to Q _n , then the sequence (u _n ) concentrates at x ₀ with respect to the H ¹ and certain L ^q -norms. We also show that if the sets {Q _n  > 0} shrink to two points and u _n are ground state solutions, then they concentrate at one of these points.     

Radiale Stoffdispersionskoeffizienten in durchströmten keramischen Schäumen

The radial mass dispersion coefficient of ceramic foam structures with pore finesses of 10, 20, 30 and 50 PPI (pores per inch) in convective flow is determined by means of the injection method (methane as tracer gas). For empty tube velocities of ca. 0.6–6 m/s we obtain $$Pe_p = u_0 \cdot p/D_r = 8 ( \pm 30\% )$$ wherep: average pore diameter.     

Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits

We consider the singularly perturbed system $\dot x$ =εf(x,y,ε,λ), $\dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold y=0 and that y ₀(t) is a homoclinic solution of $\dot y$ =g(0,y,0,0). Under an additional condition, we show that there is a curve in the (ε,λ) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil'nikov saddle-focus homoclinic orbits.