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.     

A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

In this paper we study the fully nonlinear free boundary problem $$\left\{\begin{array}{ll}F(D^{2}u) = 1 & {\rm almost \, everywhere \, in}\, B_{1} \cap \Omega\\ |D^{2} u| \leqq K & {\rm almost \, everywhere \, in} \, B_{1} \setminus \Omega,\end{array}\right.$$ where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W ^2,n solutions are locally C ^1,1 inside B ₁. Under the extra condition that ${\Omega \supset \{D{u} \neq 0 \}}$ and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary ${\partial \Omega \cap B_1}$ .     

Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E ₀ > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L ²-critical nonlinear PDEs with nonlocal dispersion.     

Dynamics as a mechanism preventing the formation of finer and finer microstructure

We study the dynamics of pattern formation in the one-dimensional partial differential equation $$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$ proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u _x )² ?1)². What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: $$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$ Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]:

?While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (u _nv_n) of E[u,v] in the Sobolev space W ^{1 p}(0, 1)×L²(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W ₁ ^p(0,1)×L²(0,1) of all solutions (u(t),u_t(t)) with low initial energy as time t → ∞).

Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data — those where the strain field u _x has a transition layer structure; our analysis includes proofs that

?at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

?transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time t → ∞

?the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.

     

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

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.     

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].     

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.     

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 $$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$      

Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D ² u) =  f(X), based on the weakest and borderline integrability properties of the source function f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L ⁿ norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C ^1,α regularity estimates are also delivered when ${f\in L^{n+\varepsilon}}$ . The limiting upper borderline case, ${f \in L^\infty}$ , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that ${u \in C^{1,{\rm Log-Lip}}}$ , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the ${C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}$ norm of u based on the L ^n-ε norm of f, where ? is the Escauriaza universal constant. The exponent ${\frac{n-2\varepsilon}{n-\varepsilon}}$ is optimal. When the source function f lies in L ^q , n > q > n?ε, we also obtain the exact, improved sharp Hölder exponent of continuity.     

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.     

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 → + ∞.     

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.     

Regularity of Non-Newtonian Fluids

In this paper, we consider a non-Newtonian fluids with shear dependent viscosity in a bounded domain ${\Omega \subset \mathbb{R}^n, n = 2, 3}$ . For the power-law model with the viscosity as in (1.4), we show the global in time existence of a weak solution for ${q \geq \frac{11}{5}}$ when n = 3 (see Theorem 1.1), and the local in time existence of a weak solution for ${2 > q > \frac{3n}{n+2}}$ , when n = 2,3 (see Theorem 1.2).     

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 ${{K=K(\Omega, q)>0}}$ .     

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).     

Wavefronts for a cooperative tridiagonal system of differential equations

Consider the infinite system of nonlinear differential equations \(\dot u\) _n =f(_n?1, u_n, u_n+1),nε?, wherefεC ¹,D ₁ f > 0,D ₃ f>0, andf(0, 0, 0) = 0 =f(1, 1, 1). Existence of wavefronts—i.e., solutions of the formu _n (t) = U(n + ct), wherecε?,U(? ∞) = 0,U(+∞) = 1, andU is strictly increasing—is shown for functionsf which satisfy the condition: there existsa, 0<a<1, such thatf(x, x,x)<0 for 0<x<a andf(x, x, x) > 0 fora < x < 1.     

Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

The aim of this paper is to provide sufficient conditions for the existence of periodic solutions emerging from an upright position of small oscillations of a sleeping symmetrical gyrostat with equations of motion being α and β parameters satisfying Δ=α ²?4β>0 and $\beta-\frac{\alpha^{2}}{2}\pm \frac{\alpha \sqrt{\varDelta }}{2}<0$ , ε a small parameter and, F ₁ and F ₂ smooth periodic maps in the variable t in resonance p:q with some of the periodic solutions of the system for ε=0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.     

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.     

Well-Posedness of Nematic Liquid Crystal Flow in {L^{3}_{\rm uloc}(\mathbb{R}^3)}

In this paper, we establish the local well-posedness for the Cauchy problem of a simplified version of hydrodynamic flow of nematic liquid crystals in ${\mathbb{R}^3}$ for any initial data (u ₀, d ₀) having small ${L^{3}_{\rm uloc}}$ -norm of ${(u_{0}, \nabla d_{0})}$ . Here ${L^{3}_{\rm uloc}(\mathbb{R}^3)}$ is the space of uniformly locally L ³-integrable functions. For any initial data (u ₀, d ₀) with small ${\|(u_0, \nabla d_0)\|_{L^{3}(\mathbb{R}^3)}}$ , we show that there exists a unique, global solution to the problem under consideration which is smooth for t > 0 and has monotone deceasing L ³-energy for ${t \geqq 0}$ .