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.     

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.     

Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights

Let A be a positive self-adjoint elliptic operator of order 2m on a bounded open set Ω ?? ^k . We consider the variational eigenvalue problem (P) $$\mathcal{A}u = \lambda r{\text{(}}x{\text{)}}u,{\text{ }}x \in \Omega ,$$ , with Dirichlet or Neumann boundary conditions; here the “weight” r is a real-valued function on Ω which is allowed to change sign in Ω or to be discontinuous. Such problems occur naturally in the study of many nonlinear elliptic equations. In an earlier work [Trans. Amer. Math. Soc. 295 (1986), pp. 305–324], we have determined the leading term for the asymptotics of the eigenvalues λ of (P). In the present paper, we obtain, under more stringent assumptions, the corresponding remainder estimates. More precisely, let N ^±(λ) be the number of positive (respectively, negative) eigenvalues of (P) less than λ>0 (respectively, greater than λ<0); set r _± = max (±r, 0) and \(\Omega _ \pm = {\text{\{ }}x \in \Omega :r{\text{(}}x{\text{)}} \gtrless {\text{0\} }}\) . We show that $$N^ \pm {\text{(}}\lambda {\text{) = }}\mathop \smallint \limits_{\Omega _ \pm } {\text{(}}\lambda r{\text{(}}x{\text{))}}^{\frac{k}{{{\text{2}}m}}} {\text{ }}\mu \prime _\mathcal{A} {\text{(}}x{\text{) }}dx + 0{\text{(}}\left| \lambda \right|^{\frac{{k - 1}}{{{\text{2}}m}} + \delta } {\text{) as }}\lambda \to \pm \infty {\text{,}}$$ , where δ>0 and μ^′ _A (x) is the Browder-Gårding density associated with the principal part of A. How small δ can be chosen depends on the “regularity” of the leading coefficients of A, r _±, and of the boundary of Ω _±. These results seem to be new even for positive weights.     

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).     

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L ^p -condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables ${u \in L^p}$ and ${u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}$ , then the blow-up does not occur, provided ${\alpha > N/2}$ or ${-1 < \alpha \leq N\,/p}$ . This includes the L ³ case natural for the Navier–Stokes equations. For ${\alpha = N\,/2}$ we exclude profiles with asymptotic power bounds of the form ${ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}$ . Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant.     

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.     

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.     

Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.     

Pressure drop in alternating curved tubes

Pressure drop measurements in the laminar and turbulent regions for water flowing through an alternating curved circular tube (x=h sin 2πz/λ) are presented. Using the minimum radius of curvature of this curved tube in place of that of the toroidally curved one in calculating the Dean number (ND=Re(D/2R _c )², it is found that the resulting Dean number can help in characterizing this flow. Also, the ratio between the height and length of the tube waves which represents the degree of waveness affects significantly the pressure drop and the transition Dean number. The following correlations have been found:

1. For laminar flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.03,\operatorname{Re}< 2000.$$
2. For turbulent flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.005,2000< \operatorname{Re}< 15000.$$
3. The transition Dean number: $$ND_{crit} = 5.012 \times 10^3 \left( {\frac{D}{{2R}}} \right)^{2.1} ,0.0111< {D \mathord{\left/ {\vphantom {D {2R_c }}} \right. \kern-\nulldelimiterspace} {2R_c }}< 0.71.$$

     

Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

In this paper, we consider the generalized Navier?CStokes equations where the space domain is ${\mathbb{T}^N}$ or ${\mathbb{R}^N, N\geq3}$ . The generalized Navier?CStokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier?CStokes equations by the more general operator (???) ^?? with ${\alpha\in (\frac{1}{2},\frac{N+2}{4})}$ . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in ${H^s, s\in[-\alpha,0]}$ , if ${1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}$ , if ${\frac{1}{2} < \alpha\leq 1}$ . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier?CStokes equation is local well-posed for a large set of the initial data in H ^?1+, exhibiting a gain of ${\frac{N}{2}-}$ derivatives with respect to the critical Hilbert space ${H^{\frac{N}{2}-1}}$ .     

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.     

Hölder Continuity and Injectivity of Optimal Maps

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f ⁺(x) is bounded away from zero and infinity in an open region ${U' \subset \mathbf{R}^n}$ , and the target density f ^?(y) is bounded away from zero and infinity on its support ${\overline{V} \subset \mathbf{R}^n}$ , which is strongly c-convex with respect to U′, and the transportation cost c satisfies the ${\mathbf{A3}_{\rm w}}$ condition of Trudinger and Wang (Ann Sc Norm Super Pisa Cl Sci 5, 8(1):143–174, 2009), we deduce the local Hölder continuity and injectivity of the optimal map inside U′ (so that the associated potential u belongs to ${C^{1,\alpha}_{loc}(U')}$ ). Here the exponent α > 0 depends only on the dimension and the bounds on the densities, but not on c. Our result provides a crucial step in the low/interior regularity setting: in a sequel (Figalli et al., J Eur Math Soc (JEMS), 1131–1166, 2013), we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. Three key tools are introduced in the present paper. Namely, we first find a transformation that under ${\mathbf{A3}_{\rm w}}$ makes c-convex functions level-set convex (as was also obtained independently from us by Liu (Calc Var Partial Diff Eq 34:435–451, 2009)). We then derive new Alexandrov type estimates for the level-set convex c-convex functions, and a topological lemma showing that optimal maps do not mix the interior with the boundary. This topological lemma, which does not require ${\mathbf{A3}_{\rm w}}$ , is needed by Figalli and Loeper (Calc Var Partial Diff Eq 35:537–550, 2009) to conclude the continuity of optimal maps in two dimensions. In higher dimensions, if the densities f ^± are Hölder continuous, our result permits continuous differentiability of the map inside U′ (in fact, ${C^{2,\alpha}_{loc}}$ regularity of the associated potential) to be deduced from the work of Liu et al. (Comm Partial Diff Eq 35(1):165–184, 2010).     

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.     

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.     

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.     

Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) _λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

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 ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .