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The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher--Kolmogorov--Petrovsky--Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event. [ABSTRACT FROM AUTHOR]

We study the steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions, we establish the existence of a boundary layer solution for both linear and nonlinear Boltzmann equations in half-space with diffusive reflection boundary condition in L∞x,v when the far-field Mach number of the Maxwellian is zero. The continuity and the special decay of the solution are obtained. The uniqueness is established under some constraint conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain (X, Y) ∈ [0, L] x ℝ+) with a moving flat boundary {Y=0}. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the same order in term of the reciprocal of a small parameter ε; the tangential component of the magnetic field does not degenerate near the boundary; and the ratio of the strength of the tangential component of the magnetic field and the tangential component of velocity is suitably small. And the error terms are estimated in an L∞ sense. [ABSTRACT FROM AUTHOR]

We homogenize the Boltzmann-Poisson system where the background medium is given by a periodic permittivity and a periodic charge concentration. The domain is the half-space with a periodic charge concentration on the boundary. Hence the domain consists of cells in R³ that are periodically repeated in two dimensions and unbounded in the third dimension. We obtain formal results for this homogenization problem. We prove the existence and uniqueness of the solution of the Laplace and Poisson problems in the fast variables with periodic and surface charge boundary conditions generating an electric field at infinity, obtaining formal solutions for the potential in terms of Magnus expansions for the case where the diagonal permittivity matrix depends on the vertical fast variable. Further on, splitting the potential into a stationary part and a self-consistent part, performing the two-scale homogenization expansions for the Poisson and the Boltzmann equations, and applying a solvability condition, we arrive at the drift-diffusion equations for the boundary-layer problem. [ABSTRACT FROM AUTHOR]

Boundary-value problems involving the linear differential equation εy" - xy' + y = 0 have surprising properties as ε → 0. We examine this equation from eight points of view, showing how it sheds light on aspects of numerical analysis (backward error analysis and illconditioning), asymptotics (boundary layer analysis), dynamical systems (slow manifolds), ODE theory (Sturm--Liouville operators), spectral theory (eigenvalues and pseudospectra), sensitivity analysis (adjoints and SVD), physics (ghost solutions), and PDE theory (Lewy nonexistence). [ABSTRACT FROM AUTHOR]

In this paper, we prove the global existence of solutions with analytic regularity to the 2D magnetohydrodynamic (MHD) boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multiscale expansion in [D. Gérard-Varet and M. Prestipino, Z. Angew. Math. Phys., 68 (2017), 76]. The analysis shows that the combined effect of the magnetic diffusivity and transverse magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global-in-time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile. [ABSTRACT FROM AUTHOR]

While sliding mode observers (SMOs) using discontinuous relays are widely analyzed, most SMOs are implemented computationally using a continuous approximation of the discontinuous relays. This results in the formation of a boundary layer in a neighborhood of the sliding manifold in the observer error space. In this paper, a convex programming method for constructing boundary-layer SMOs (BL-SMOs) is presented for a wide class of nonlinear systems characterized by incremental quadratic constraints. Specifically, the observer gains are computed by solving a set of linear matrix inequalities and ultimate bounds on the state of a synthetic descriptor system. A novel application of filtering using smooth window functions is leveraged to reconstruct the unknown exogenous inputs to prescribed accuracy. Two numerical examples are presented to illustrate the effectiveness of our proposed approach. [ABSTRACT FROM AUTHOR]

For compact domains with smooth boundaries, we present a surface spline approximation scheme that delivers rates in Lp that are optimal for linear approximation in this setting. This scheme can overcome the boundary effects, observed by Johnson [Constr. Approx., 14 (1998), pp. 429--438], by placing centers with greater density near the boundary. It owes its success to an integral identity employing a minimal number of boundary layer potentials, which, in turn, is derived from the boundary layer potential solution to the Dirichlet problem for the m-fold Laplacian. Furthermore, this integral identity is shown to be the "native space extension" of the target function. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of boundary layer solutions for a viscous hyperbolic system transformed via a Cole-Hopf transformation from a singular chemotactic system modeling the initiation of tumor angiogenesis proposed in [H. A. Levine, B. Sleeman, and M. Nilsen-Hamilton, Math. Biosci., 168 (2000), pp. 71-115]. It was previously shown in [Q. Hou, Z. Wang, and K. Zhao, J. Differential Equations, 261 (2016), pp. 5035-5070] that when prescribed with Dirichlet boundary conditions, the system possesses boundary layers at the boundaries in an bounded interval (0, 1) as the chemical diffusion rate (denoted by ε > 0) is small. This paper proceeds to prove the stability of boundary layer solutions and identify the precise structure of boundary layer solutions. Roughly speaking, we justify that the solution with ε > 0 converges to the solution with ε = 0 (outer layer solution) plus the inner layer solution with the optimal rate at order of O(ε½) as ε → 0, where the outer and inner layer solutions are well determined and the relation between outer and inner layer solutions can be explicitly identified. Finally we transfer the results to the original pretransformed chemotaxis system and discuss the implications of our results. [ABSTRACT FROM AUTHOR]

Classical inf-sup stable mixed finite elements for the incompressible (Navier-)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. However, a modification only in the right-hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor-Hood and mini elements, which have continuous discrete pressures. For the modification of the righthand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H(div)-conforming ux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a priori error estimates. Numerical examples for the incompressible Stokes and Navier- Stokes equations confirm that the new pressure-robust Taylor{Hood and mini elements converge with optimal order and outperform significantly the classical versions of those elements when the continuous pressure is comparably large. [ABSTRACT FROM AUTHOR]

A computational and asymptotic analysis of the solutions of Carrier's problem is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to construct asymptotic solutions to the problem. This asymptotic approach explains the bifurcation structure identified numerically, and its predictions of the bifurcation points are in excellent agreement with the numerical results. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender and Orszag [Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York, 1999] is incorrect. [ABSTRACT FROM AUTHOR]

The regular Cauchy--Born method is a useful and efficient tool for analyzing bulk properties of materials in the absence of defects. However, the method normally fails to capture surface effects, which are essential to determining material properties at small length scales. In this paper, we present a corrector method that improves upon the prediction for material behavior from the Cauchy--Born method over a small boundary layer at the surface of a one-dimensional (1D) material by capturing the missed surface effects. We justify the separation of the problem into a bulk response and a localized surface correction by establishing an error estimate, which vanishes in the long wavelength limit. [ABSTRACT FROM AUTHOR]

A method is presented to generate surface boundary layers. The novelty of the method consists in a uniform treatment of the concave and convex regions based on offsetting the initial front. In particular, concave situations are enforced to respect the entropy satisfying solution. Local solutions of the offset of edges in the planes and triangles in the three dimensional space are also given, showing the relationship between normal computation, distance extrusion, and the generalized Voronoi diagram. The boundary layer mesh generation is cast as a weak solution of the Eikonal equation computed locally, layer by layer, to mimic the hyperbolic character of the Eikonal equation, augmented by a possibly nonbijective mapping between layers. This provides a theoretical framework to rely on for practical answers such as configurations where the adjacent topologies to a boundary layer may be curved, or at least not aligned with the offset direction, or if the boundary layer mesh must be generated as a mapping between the first and last layer or as a composition of mappings for each layer. Numerous examples illustrate the potential of the proposed solution. [ABSTRACT FROM AUTHOR]

We investigate the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovskii-Piskunov-type models in the presence of a steady ce llular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large Peclet number, Pe ≫ 1) and arbitrary reaction rate (arbitrary Damko hler number Da). We identify three regimes corresponding to the distinguished limits Da = 0(Pe-1), Da = 0((logPe)-1), and Da = O(Pe) and, in each regime, obtain the front speed in terms of a different nontrivial function of the relevant combination of Pe and Da. Closed-form expressions for the speed, characterized by power- law and logarithmic dependences on Da and Pe and valid in intermediate regimes, are deduced as limiting cases. Taken together, our asymptotic results provide a complete description of the complex dependence of the front speed on Da for Pe ≫ 1. They are confirmed by numerical solutions of the eigenvalue problem determining the front speed, and illustrated by a number of numerical simulations of the advection-diffusion-reaction equation. [ABSTRACT FROM AUTHOR]

We present a new class of well-conditioned integral equations for the solution of two and three dimensional scattering problems by homogeneous penetrable scatterers. Our novel boundary integral equations result from using regularizing operators which are suitable approximations of the admittance operators that map the transmission boundary conditions to the exterior and, respectively, interior Cauchy data on the interface between the media. We refer to these regularized boundary integral equations as generalized combined source integral equations (GCSIE). The admittance operators can be expressed in terms of Dirichlet-to-Neumann operators and their inverses. We construct our regularizing operators in terms of simple boundary layer operators with complex wavenumbers. The choice of complex wavenumbers in the definition of the regularizing operators ensures the unique solvability of the GCSIE. The GCSIE are shown to be integral equations of the second kind for regular enough interfaces of material discontinuity. Notably, the novel GCSIE have better spectral properties than the classical combined field integral equations. [ABSTRACT FROM AUTHOR]

Anisotropically refined mixed finite elements are beneficial for the resolution of local features such as boundary layers. Unfortunately, the stability of the resulting scheme is highly sensitive to the aspect ratio of the elements. Previous analysis revealed that the degeneration arises from a relatively small number of spurious (piecewise constant) pressure modes. The present article is concerned with resolving the problem of how to suppress the spurious pressure modes in order to restore stability yet at the same time not incur any deterioration in the approximation properties of the reduced pressure space. Two results are presented. The first gives the minimal constraints on the pressure space needed to restore stability with respect to aspect ratio and shows that the approximation properties of the constrained pressure space and the unconstrained pressure space are essentially identical. Alternatively, one can impose the constraint weakly through the use of a stabilized finite element scheme. A second result shows that the stabilized finite element scheme is robust with respect to the aspect ratio of the elements and produces an approximation that satisfies an error bound of the same type to the mixed finite element scheme using the constrained space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of the boundary layer in the small viscosity limit of the compressible isentropic Navier--Stokes equations with nonslip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash-Moser-Hörmander iteration scheme and further develop the energy method introduced in [R. Alexander et al., J. Amer. Math. Soc., DOI:S0894-0347(2014)00813-4] to obtain the well-posedness of the equations locally in time. [ABSTRACT FROM AUTHOR]

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, if the initial datum u0 is monotone on a number of intervals (on some strictly increasing, on some strictly decreasing) and analytic on the complement of these intervals, we show that the local existence and uniqueness hold. The same result is true for the hydrostatic Euler equations if we assume these conditions for the initial vorticity ω0 = ∂y u0. [ABSTRACT FROM AUTHOR]

We use a recently developed method [O. Costin, M. Huang, and W. Schlag, Nonlin-earity, 25 (2012), pp. 125-164], [O. Costin, M. Huang, and S. Tanveer, Duke Math. J., 163 (2014), pp. 665-704] to find accurate analytic approximations with rigorous error bounds for the classic similarity solution of Blasius of the boundary layer equation in fluid mechanics, the two-point boundary value problem f'" + f f" = 0 with f(0) = f'(0) = 0 and limx→∞ f'(x) = 1. The approximation is given in terms of a polynomial in [0,5/2] and in terms of the error function in [5/2, ∞). The two representations for the solution in different domains match at x = 5/2 determining all free parameters in the problem, in particular f"(0) = 0.469600 ± 0.000022 at the wall. The method can in principle provide approximations to any desired accuracy for this or wide classes of linear or nonlinear differential equations with initial or boundary value conditions. The analysis relies on controlling the errors in the approximation through contraction mapping arguments, using energy bounds for the Green's function of the linearized problem. [ABSTRACT FROM AUTHOR]

We present an asymptotic study of the Dirichlet to Neumann map of high contrast composite media with perfectly conducting inclusions that are close to touching. The result is an explicit characterization of the map in the asymptotic limit of the distance between the particles tending to zero. Known homogenization results state that the effective conductivity of the high contrast composites is determined by resistor networks with topology defined by the geometric distribution of the inclusions. The result in this paper is stronger than homogenization because it applies to arbitrary boundary conditions. In particular, it accounts for the coupling of boundary current flow induced by oscillatory boundary potentials and the flow in the network. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of circularly symmetric solutions to the initial boundary value problem of the nonisentropic compressible Navier--Stokes equations in a two-dimensional exterior domain with impermeable boundary conditions when the viscosities and the heat conduction coefficient tend to zero at the same rate. By multiscale analysis, we obtain that away from the boundary the nonisentropic compressible viscous flow can be approximated by the corresponding inviscid flow, and near the boundary there are boundary layers for the angular velocity, density, and temperature in the leading order expansions of solutions, while the radial velocity and pressure do not have boundary layers in the leading order. The boundary layers of velocity and temperature are described by a nonlinear parabolic system. We prove the stability of boundary layers and rigorously justify the asymptotic behavior of solutions in the L∞ space for the small viscosities and heat-conduction limit in the Lagrangian coordinates, as long as the strength of the boundary layers is suitably small. Finally, we show that the similar asymptotic behavior of the small viscosities and heat conduction limit holds in the Eulerian coordinates for the compressible nonisentropic viscous flow. [ABSTRACT FROM AUTHOR]

In this paper we consider the multiscale analysis of a Steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure. There are mainly two new results in the present paper. First, we obtain the convergence rate with ε1/2 for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the Steklov eigenvalue problem. Second, the boundary layer solution is defined. Numerical simulations are then carried out to validate the above theoretical results. [ABSTRACT FROM AUTHOR]

We consider incompressible viscous flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier--Stokes equations in the limit when both the amplitude of the vibration and the thickness of the oscillatory boundary layers (the Stokes layers) are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions for the averaged flow. At leading order, the averaged flow is described by the stationary Navier--Stokes equations with an additional term which contains the leading-order Stokes drift velocity. The same equations had been derived by Craik and Leibovich in 1976 when they proposed their model of Langmuir circulations in the ocean [A. D. D. Craik and S. Leibovich, J. Fluid Mech., 73 (1976), pp. 401-426]. In the context of flows induced by an oscillating conservative body force, these equations had also been derived by Riley [Ann. Rev. Fluid Mech., 33 (2001), pp. 43-65]. The general theory is applied to two particular examples of steady streaming induced by transverse vibrations of the walls in the form of standing and traveling plane waves. In particular, in the case of waves traveling in the same direction, the induced flow is plane-parallel and the Lagrangian velocity profile can be computed analytically. This example may be viewed as an extension of the theory of peristaltic pumping to the case of high Reynolds numbers. [ABSTRACT FROM AUTHOR]

A fast deterministic vortex-grid method for the solution of the two-dimensional unsteady boundary layer equations is presented. The new method uses a Lagrangian particle formulation for the convective step of the momentum equation, as in Chorin's random vortex sheet method, while the diffusion step is solved with finite differences on an adaptive Eulerian grid. Two algorithms for choosing the grid are proposed and compared. A numerical convergence proof is given on a standard model problem and the rate of convergence is estimated. Finally the scheme is applied to the impulsively started circular cylinder. Our results show that the present scheme is able to describe the formation of the singularity, in agreement with the results of Van Dommelen and Shen [J. Comput. Phys., 38 (1980), pp. 125–140]. [ABSTRACT FROM AUTHOR]