Global regularity for the 2D Oldroyd‐B model in the corotational case

This paper is dedicated to the Oldroyd‐B model with fractional dissipation (?Δ)^ατ for any α > 0. We establish the global smooth solutions to the Oldroyd‐B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. Moreover, in the Appendix, we provide some a priori estimates to the Oldroyd‐B model in the critical case, which may be useful and of interest for future improvement. Therefore, our result is closer to the resolution of the well‐known global regularity issue on the critical 2D Oldroyd‐B model. Copyright © 2016 John Wiley & Sons, Ltd.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

An Osgood type regularity criterion for the 2D Oldroyd‐B model

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, where denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2^j}.     

Global well‐posedness for free boundary problem of the Oldroyd‐B model fluid flow

In this paper, we prove the global well‐posedness of non‐Newtonian viscous fluid flow of the Oldroyd‐B model with free surface in a bounded domain of N‐dimensional Euclidean space . The assumption of the problem is that the initial data are small enough and orthogonal to rigid motions. Copyright © 2015 John Wiley & Sons, Ltd.     

Strong solutions of 3D compressible Oldroyd‐B fluids

This paper is concerned with a compressible viscoelastic fluids of Oldroyd‐B type. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. Moreover, we establish a blow‐up criterion for the strong solution in terms of the norm of the density tensor ρ and the norm of the symmetric tensor of constraints τ. All the results hold for the initial density vanishing from below. Copyright © 2012 John Wiley & Sons, Ltd.     

The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow

In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (?Δ)^αu in the space and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in [0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided . For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if , which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.     

Hypoellipticity and Local Solvability in Gevrey Classes

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s. We prove first that if P is s‐hypoelliptic then its transposed operator ^tP is s‐locally solvable, thus extending to the Gevrey classes the well‐known analogous result in the C^∞class. We prove also that if P is s‐hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s‐hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.     

Incompressible non‐Newtonian fluids: time asymptotic behaviour of weak solutions

We study the asymptotic behaviour in time of incompressible non‐Newtonian fluids in the whole space assuming that initial data also belong to L¹. Firstly, we consider the weak solution to the power‐law model with non‐zero external forces and we find the asymptotic behaviour in time of this solution in the same class of existence and uniqueness with p?. Secondly, we are interested in the asymptotic behaviour of weak solutions to the second grade model, and finally, we deal with the asymptotic behaviour in time of weak solutions to a simplified model of viscoelastic fluids of the Oldroyd type. Copyright © 2006 John Wiley & Sons, Ltd.     

A note on parameter free Π1‐induction and restricted exponentiation

We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})$ (= Boolean combinations of Σ₁) theorems of IΠ^?₁ in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ^?_{n + 1} is conservative over IΣ^?_n with respect to $\mathcal {B}(\Sigma _{n+1})$ sentences cannot be extended to Π_{n + 2} sentences. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Anomalous scaling for three‐dimensional Cahn‐Hilliard fronts

We prove the stability of the one‐dimensional kink solution of the Cahn‐Hilliard equation under d‐dimensional perturbations for d ≥ 3. We also establish a novel scaling behavior of the large‐time asymptotics of the solution. The leading asymptotics of the solution is characterized by a length scale proportional to t^1/3 instead of the usual t^1/2 scaling typical to parabolic problems. © 2004 Wiley Periodicals, Inc.     

Norm estimates for a Kakeya‐type maximal operator

We prove L^p → L^q estimates for the 2‐dimensional analog of the Kakeya maximal function. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L^∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.     

Global Solutions of Two‐Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem, and it is studied in this paper. We show global existence if the initial deformation gradient is close to the identity matrix in L² ∩ L^∞ and the initial velocity is small in L² and bounded in L^p for some p > 2. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd‐B model, the additional assumption on the initial velocity being bounded in L^p for some p > 2 may due to techniques we employed. The smallness assumption on the L² norm of the initial velocity is, however, natural for global well‐posedness. One of the key observations in the paper is that the velocity and the “ effective viscous flux” are sufficiently regular for positive time. The regularity of leads to a new approach for the pointwise estimate for the deformation gradient without using L^∞ bounds on the velocity gradients in spatial variables. © 2015 Wiley Periodicals, Inc.     

DAG‐Width and Circumference of Digraphs

We prove that every digraph of circumference l has DAG‐width at most l. This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014).¹ As a consequence of this result we deduce that the k‐linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in J. Bang‐Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak k‐linkage problem (where we ask for arc‐disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP‐hard on digraphs of DAG‐width at most 5.     

Abstract wave equations with acoustic boundary conditions

We define an abstract setting to treat wave equations equipped with time‐dependent acoustic boundary conditions on bounded domains of R ⁿ . We prove a well‐posedness result and develop a spectral theory which also allows to prove a conjecture proposed in [13]. Concrete problems are also discussed. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic expansions in n−1 for percolation critical values on the n‐Cube and ℤn

We use the lace expansion to prove that the critical values for nearest‐neighbor bond percolation on the n‐cube {0, 1}ⁿ and on the integer lattice ?ⁿ have asymptotic expansions, with rational coefficients, to all orders in powers of n^?1. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Reduction of the adjoint representation of sl(2,1): generators of primitive ideals

We present a reduction of the adjoint representation of the Lie superalge-bra,sl(2,1) and a study of the quotient algebra B(^c,k)= u/u(C?c)+u(D?kc), where c,k are two complex numbers. Under some additional conditions, we prove that every irreducible infinite dimensional representation of B(^c,k) is faithful, and that B(^C,K) is a primitive algebra. We give explicitly a set of generators of primitive degenerate ideal of infinite codimension. Essentially we prove that any minimal primitive ideal of u(sl(2,1)) is generated, as a 2-sided ideal, by its intersection with the algebra of gg-iuvariants.     

A lower‐epiperimetric inequality for area‐minimizing surfaces

The epiperimetric inequality introduced by E. R. Reifenberg in [3] gives a rate of decay at a point for the decreasing k‐density of area of an area‐minimizing integral k‐cycle. While dilating the cycle at that point, this rate of decay holds once the configuration is close to a tangent cone configuration and above the limiting density corresponding to that configuration. This is why we propose to call the Reifenberg epiperimetric inequality an upper‐epiperimetric inequality. A direct consequence of this upper‐epiperimetric inequality is the statement that any point possesses a unique tangent cone. The upper‐epiperimetric inequality was proven by B. White in [5] for area‐minimizing 2‐cycles in ?ⁿ. In the present paper we introduce the notion of a lower‐epiperimetric inequality. This inequality gives this time a rate of decay for the decreasing k‐density of area of an area‐minimizing integral k‐cycle, while dilating the cycle at a point once the configuration is close to a tangent cone configuration and below the limiting density corresponding to that configuration. Our main result in the present paper is to prove the lower‐epiperimetric inequality for area‐minimizing 2‐cycles in ?ⁿ. As a consequence of this inequality we prove the “splitting before tilting” phenomenon for calibrated 2‐rectifiable cycles, which plays a crucial role in the proof of the regularity of 1‐1 integral currents in [4]. © 2004 Wiley Periodicals, Inc.     

Crack Singularities for General Elliptic Systems

We consider general homogeneous Agmon‐Douglis‐Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two‐dimensional domain. We prove that the singular functions expressed in polar coordinates (r, θ) near the crack tip all have the form r^{k + 1/2}φ(θ) with k ≥ 0 integer, with the possible exception of a finite number of singularities of the form r^k log r φ(θ). We also prove results about singularities in the case when the boundary conditions on the two sides of the crack are not the same, and in particular in mixed Dirichlet‐Neumann boundary value problems for strongly coercive systems: in the latter case, we prove that the exponents of singularity have the form with real η and integer k. This is valid for general anisotropic elasticity too.