A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

We give simple proofs that a weak solution u of the Navier–Stokes equations with H ¹ initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L ^s (0, T;L ^r,∞(Ω)) or if its L ^s,∞(0, T;L ^r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.     

Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

We consider degenerate reaction diffusion equations of the form u _t ?=???u ^m ?+?f(x, u), where f(x, u) ~ a(x)u ^p with 1??? p m. We assume that a(x)?>?0 at least in some part of the spatial domain, so that ${u \equiv 0}$ is an unstable stationary solution. We prove that the unstable manifold of the solution ${u \equiv 0 }$ has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as ${t\to -\infty}$ while its support shrinks to an arbitrarily chosen point x* in the region where a(x)?>?0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.     

Decay estimates for some semilinear damped hyperbolic problems

Let Ω be a bounded open domain in R ⁿ , g ∶ R → R a non-decreasing continuous function such that g(0)=0 and h ε L _loc ¹ (R⁺; L ²(Ω)). Under suitable assumptions on g and h, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u ^′′ + Lu + g(u ^′) = h(t,x) as t → + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation $$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$ in R ⁺×Ω, u=0 on R ⁺×?Ω. For instance if \(g(s) = c\left| s \right|^{p - 1} s + d\left| s \right|^{q - 1} s\) with c, d>0 and 1 < p≦q, (n?2)q≦n+2, then if \(h \in L^\infty (R + ;L^2 (\Omega ))\) , all solutions are bounded in the energy space for t≧0 and if u, v are two such solutions, the energy norm of u(t) ? v(t) decays like t ^?1/p?1 as t → + ∞.     

Steady Flow of a Navier-Stokes Fluid Around a Rotating Obstacle

Let ? be a body immersed in a Navier-Stokes liquid ? that fills the whole space. Assume that ? rotates with prescribed constant angular velocity ω. We show that if the magnitude of ω is not “too large”, there exists one and only one corresponding steady motion of ? such that the velocity field v(x) and its gradient grad?v(x) decay like |x|^?1 and |x|^?2, respectively. Moreover, the pressure field p(x) and its gradient grad?p(x) decay like |x|^?2 and |x|^?3, respectively. These solutions are “physically reasonable” in the sense of Finn. In particular, they are unique and satisfy the energy equation. This result is relevant to several applications, including sedimentation of heavy particles in a viscous liquid.     

Regularity for Energy-Minimizing Area-Preserving Deformations

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem $$\inf_{\mathcal{K}}\int_{\varOmega}|\nabla \textbf {v}|^2, \quad\varOmega\subset \mathbb {R}^2 $$ as the Lagrange multiplier corresponding to the incompressibility constraint det?v=1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ^?=u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L ² norm of the pressure. As a by-product we conclude that $\textbf {u}\in C^{\frac{1}{2}}_{\textrm {loc}}(\varOmega)$ if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative.     

Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier–Stokes equations with a Navier slip boundary condition on general compact and smooth domains in R ³. We first obtain the higher order regularity estimates for the solutions to Prandtl’s equation boundary layers. Furthermore, we prove that the strong solution to Navier–Stokes equations converges to the Eulerian one in C([0, T]; H ¹(Ω)) and ${L^\infty((0,T) \times \Omega)}$ , where T is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.     

Periodic solutions to some problems of n-body type

We prove the existence of at least one T-periodic solution to a dynamical system of the type $$ - m_i \ddot u_i = \sum\limits_{j = 1,j \ne i}^n {\triangledown V_{ij} (u_i - u_j ,{\text{ }}t)}$$ (1) where the potentials V _ij are T-periodic in t and singular at the origin, u _i ε R ^k i=1, ..., n, and k≧3. We also provide estimates on the H ¹ norm of this solution. The proofs are based on a variant of the Ljusternik-Schnirelman method. The results here generalize to the n-body problem some results obtained by Bahri & Rabinowitz on the 3-body problem in [6].     

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F ^T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.     

A symmetry analysis of some classes of evolutionary nonlinear (2+1)-diffusion equations with variable diffusivity

In this paper the (2+1)-nonlinear diffusion equation u _t ?div(f(u)grad u)=0 with variable diffusivity is considered. Using the Lie method, a complete symmetry classification of the equation is presented. Reductions, via two-dimensional Lie subalgebras of the equation, to first- or second-order ordinary differential equations are given. In a few interesting cases exact solutions are presented.     

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C _b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L ³ spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L ³ spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.     

Solution of three-dimensional problems for thick elastic shells by the method of reference surfaces

A new method for solving the elasticity problem for thick and thin shells is proposed. The method is based on the concept of reference surfaces inside the shell. According to this method, N reference surfaces are introduced in the body of the shell so that they are parallel to the midsurface and located at the Chebyshev polynomial nodes, which permits taking the displacement vectors u ¹, u ², …, u ^N of these surfaces for the desired functions. This choice of the desired functions allows one to represent the resolving equations of the proposed theory of higher-order shells in a sufficiently concise form and obtain deformation relations which permit describing the shell displacements as motions of a rigid body.     

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.     

Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m ₀ of vanishing electric displacements, D(m ₀)=0, on the unit sphere of propagation directions m ²=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors D _α (m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m ₀ and can be described by the Poincaré indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.     

Classification of the structure of positive radial solutions to\Delta u + K(|x|)u^p = 0inR n

Positive radial solutions to the semilinear elliptic equation \(\Delta u + K(|x|)u^p = 0\) inR ⁿ are studied, wherep > 1,n > 2 andK ≧ 0. It is shown that, under a general condition onK(r) andp, the structure of positive radial solutions becomes one of three types. We give sharp criteria to classify the type of the structure, and apply the result to the conformal scalar curvature equation.     

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 gradient based iterative algorithm for solving structural dynamics model updating problems

The procedure of updating an existing but inaccurate model is an essential step toward establishing an effective model. Updating damping and stiffness matrices simultaneously with measured modal data can be mathematically formulated as following two problems. Problem 1: Let M _a ∈SR ^n×n be the analytical mass matrix, and Λ=diag{λ ₁,…,λ _p }∈C ^p×p , X=[x ₁,…,x _p ]∈C ^n×p be the measured eigenvalue and eigenvector matrices, where rank(X)=p, p<n and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in\nobreak{\mathbf{C}} $ , $x_{2j} = \bar{x}_{2j-1} \in{\mathbf{C}}^{n} $ for j=1,…,l, and λ _k ∈R, x _k ∈R ⁿ for k=2l+1,…,p. Find real-valued symmetric matrices D and K such that M _a XΛ ²+DXΛ+KX=0. Problem 2: Let D _a ,K _a ∈SR ^n×n be the analytical damping and stiffness matrices. Find $(\hat{D}, \hat{K}) \in\mathbf{S}_{\mathbf{E}}$ such that $\| \hat{D}-D_{a} \|^{2}+\| \hat{K}-K_{a} \|^{2}= \min_{(D,K) \in \mathbf{S}_{\mathbf{E}}}(\| D-D_{a} \|^{2} +\|K-K_{a} \|^{2})$ , where S _E is the solution set of Problem 1 and ∥?∥ is the Frobenius norm. In this paper, a gradient based iterative (GI) algorithm is constructed to solve Problems 1 and 2. A sufficient condition for the convergence of the iterative method is derived and the range of the convergence factor is given to guarantee that the iterative solutions consistently converge to the unique minimum Frobenius norm symmetric solution of Problem 2 when a suitable initial symmetric matrix pair is chosen. The algorithm proposed requires less storage capacity than the existing numerical ones and is numerically reliable as only matrix manipulation is required. Two numerical examples show that the introduced iterative algorithm is quite efficient.     

Quasilinear and Hessian Type Equations with Exponential Reaction and Measure Data

We prove existence results concerning equations of the type \({-\Delta_pu=P(u)+\mu}\) for p > 1 and F _k [?u] = P(u) + μ with \({1 \leqq k < \frac{N}{2}}\) in a bounded domain Ω or the whole \({\mathbb{R}^N}\) , where μ is a positive Radon measure and \({P(u)\sim e^{au^\beta}}\) with a > 0 and \({\beta \geqq 1}\) . Sufficient conditions for existence are expressed in terms of the fractional maximal potential of μ. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of μ. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form \({u={\bf W}_{\alpha, p}^R[P(u)]+f}\) in \({\mathbb{R}^N}\) , where \({0 < R \leqq \infty}\) and f is a positive integrable function.     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.