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-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.     

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 ${\mu > \mu^*}$ .     

Elongational rheology of NIPAM-based hydrogels

Hydrogels of different composition based on the copolymerization of N-isopropyl acrylamide and surfmers of different chemical structure were tested in elongation using Hencky/real definitions for stress, strain, and strain rate, offering a more scientific insight into the effect of deformation on the properties. In a range between $\dot {\varepsilon }=10$ and 0.01 s $^{-1}$ , the material properties are independent of strain rate and show a very clear strain hardening with a “brittle” sudden fracture. The addition of surfmer increases the strain at break $\varepsilon _{\mathrm {H}}^{\max }$ and at the same time leads to a failure of hyperelastic models. The samples can be stretched up to Hencky strains $\varepsilon _{\mathrm {H}}^{\max }$ between 0.6 and 2.5, depending on the molecular structure, yielding linear Young’s moduli E $_{0}$ between 2,700 and 39,000 Pa. The strain-rate independence indicates an ideal rubberlike behavior and fracture in a brittle-like fashion. The resulting stress at break $\sigma _{\textrm max}$ can be correlated with $\varepsilon _{\mathrm {H}}^{\max } $ and $E_{0}$ as well as with the solid molar mass between the cross-linking points $M_{\mathrm {c}}^{\textrm {solids}} $ , derived from $E_{0}$ .     

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 ${\epsilon \to 0}$ .     

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

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

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 ${t \geqq 0}$ .     

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

On the Elastic Energy Density of Constrained Q-Tensor Models for Biaxial Nematics

Within the Landau–de Gennes theory, the order parameter describing a biaxial nematic liquid crystal assigns a symmetric traceless 3 × 3 matrix Q with three distinct eigenvalues to every point of the region Ω occupied by the system. In the constrained case of matrices Q with constant eigenvalues, the order parameter space is diffeomorphic to the eightfold quotient ${\mathbb{S}^3/\mathcal{H}}$ of the 3-sphere ${\mathbb{S}^3}$ , where ${\mathcal{H}}$ is the quaternion group, and a configuration of a biaxial nematic liquid crystal is described by a map from Ω to ${\mathbb{S}^3/\mathcal{H}}$ . We express the (simplest form of the) Landau–de Gennes elastic free-energy density as a density defined on maps ${q: \Omega \to \mathbb{S}^3}$ , whose functional dependence is restricted by the requirements that (1) it is well defined on the class of configuration maps from Ω to ${\mathbb{S}^3/\mathcal{H}}$ (residual symmetry) and (2) it is independent of arbitrary superposed rigid rotations (frame indifference). As an application of this representation, we then discuss some properties of the corresponding energy functional, including coercivity, lower semicontinuity and strong density of smooth maps. Other invariance properties are also considered. In the discussion, we take advantage of the identification of ${\mathbb{S}^3}$ with the Lie group of unit quaternions ${Sp(1) \cong SU(2)}$ and of the relations between quaternions and rotations in ${\mathbb{R}^3}$ and ${\mathbb{R}^4}$ .     

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.     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

Generalized Resolvent Estimates of the Stokes Equations with First Order Boundary Condition in a General Domain

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain ${\Omega}$ of the N-dimensional Eulidean space ${\mathbb{R}^N, N \geq 2}$ . This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter ${\lambda}$ varying in a sector ${\Sigma_{\sigma, \lambda_0} = \{\lambda \in \mathbb{C} \mid |\arg \lambda| < \pi-\sigma, \enskip |\lambda| \geq \lambda_0\}}$ , where ${0 < \sigma < \pi/2}$ and ${\lambda_0 \geq 1}$ . The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution ${p \in \hat{W}^1_{q, \Gamma}(\Omega)}$ to the variational problem: ${(\nabla p, \nabla \varphi) = (f, \nabla \varphi)}$ for any ${\varphi \in \hat W^1_{q', \Gamma}(\Omega)}$ . Here, ${1 < q < \infty, q' = q/(q-1), \hat W^1_{q, \Gamma}(\Omega)}$ is the closure of ${W^1_{q, \Gamma}(\Omega) = \{ p \in W^1_q(\Omega) \mid p|_\Gamma = 0\}}$ by the semi-norm ${\|\nabla \cdot \|_{L_q(\Omega)}}$ , and ${\Gamma}$ is the boundary of ${\Omega}$ . In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in ${(\lambda_0, \infty)}$ . Our assumption is satisfied for any ${q \in (1, \infty)}$ by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q =  2.     

Blast waves from cylindrical charges

Comparisons of explosives are often carried out using TNT equivalency which is based on data for spherical charges, despite the fact that many explosive charges are not spherical in shape, but cylindrical. Previous work has shown that it is possible to predict the over pressure and impulse from the curved surface of cylindrical charges using simple empirical formulae for the case when the length-to-diameter (L/D) ratio is greater or equal to 2/1. In this paper, by examining data for all length-to-diameter ratios, it is shown that it is possible to predict the peak over pressure, P, for any length-to-diameter ratio from the curved side of a bare cylindrical charge of explosive using the equation $P=K_PM(L/D)^{1/3}/R^3$ , where M is the mass of explosive, R the distance from the charge and $ K_P$ is an explosive-dependent constant. Further out where the cylindrical blast wave ‘heals’ into a spherical one, the more complex equation $P=C_1(Z^{\prime \prime })^{-3}+C_2(Z^{\prime \prime })^{-2}+C_3(Z^{\prime \prime })^{-1}$ gives a better fit to experimental data, where $ Z^{\prime \prime } = M^{1/3}(L/D)^{1/9}/D$ and $C_1,\, C_2 $ and $ C_3$ are explosive-dependent constants. The impulse is found to be independent of the L/D ratio.     

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.     

Global Regularity for the Two-Dimensional Anisotropic Boussinesq Equations with Vertical Dissipation

This paper establishes the global in time existence of classical solutions to the two-dimensional anisotropic Boussinesq equations with vertical dissipation. When only vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the ${L^\infty}$ -norm of the vertical velocity v and prove that ${\|v\|_{L^{r}}}$ with ${2\leqq r < \infty}$ does not grow faster than ${\sqrt{r \log r}}$ at any time as r increases. A delicate interpolation inequality connecting ${\|v\|_{L^\infty}}$ and ${\|v\|_{L^r}}$ then yields the desired global regularity.     

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

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

On the Stokes and Oseen Problems with Singular Data

We consider the steady Stokes and Oseen problems in bounded and exterior domains of ${\mathbb{R}^n}$ of class C ^k-1,1 (n = 2, 3; k ≥ 2). We prove existence and uniqueness of a very weak solution for boundary data a in ${W^{2-k-1/q,q} (\partial\Omega)}$ . If ${\Omega}$ is of class ${C^\infty}$ , we can assume a to be a distribution on ${\partial\Omega}$ .     

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 ${H^{\frac{N}{2}-1}}$ .