Quasi-Linear Equations with a Small Diffusion Term and the Evolution of Hierarchies of Cycles

We study the long-time behavior (at times of order \(\exp (\lambda /\varepsilon ^2\))) of solutions to quasi-linear parabolic equations with a small parameter \(\varepsilon ^2\) at the diffusion term. The solution to a PDE can be expressed in terms of diffusion processes, whose coefficients, in turn, depend on the unknown solution. The notion of a hierarchy of cycles for diffusion processes was introduced by Freidlin and Wentzell and applied to the study of the corresponding linear equations. In the quasi-linear case, it is not a single hierarchy that corresponds to an equation, but rather a family of hierarchies that depend on the timescale \(\lambda \). We describe the evolution of the hierarchies with respect to \(\lambda \) in order to gain information on the limiting behavior of the solution of the PDE.     

On the shape of solution sets of systems of (functional) equations

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations.     

On a Modification of Olver’s Method: A Special Case

We consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter \(\Lambda \): \(x^my''-\Lambda ^2y=g(x)y\), with \(m\in \mathbb {Z}\) and g continuous. Olver studies in detail the cases \(m\ne 2\), especially the cases \(m=0, \pm 1\), giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case \(m=2\) is different, as the behavior of the solutions for large \(\Lambda \) is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case \(m=2\). We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.     

Convergence Analysis of Weighted Difference Approximations on Piecewise Uniform Grids to a Class of Singularly Perturbed Functional Differential Equations

We consider the numerical approximation of a singularly perturbed time delayed convection diffusion problem on a rectangular domain. Assuming that the coefficients of the differential equation be smooth, we construct and analyze a higher order accurate finite difference method that converges uniformly with respect to the singular perturbation parameter. The method presented is a combination of the central difference spatial discretization on a Shishkin mesh and a weighted difference time discretization on a uniform mesh. A?priori explicit bounds on the solution of the problem are established. These bounds on the solution and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. It is shown that the proposed method is $L_{2}^{h}$ -stable. The analysis done permits its extension to the case of adaptive meshes which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practical satisfies the theoretical predictions.     

A Method to Study the Cauchy Problem for a Singularly Perturbed Homogeneous Linear Differential Equation of Arbitrary Order

We construct a sequence converging to the solution to the Cauchy problem for a singularly perturbed linear homogeneous differential equation of any order. This sequence is asymptotic in the following sense: the distance (with respect to the norm of the space of continuous functions) between its nth element and the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter.     

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt $(0<q<1)$ . This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to ${m} (m \leqslant 2)$ , the global accuracy of k level corrected approximation is $O(N^{-(2m(k+1)-\varepsilon)})$ , where N is the number of the nodes, and $\varepsilon$ is an arbitrary small positive number.     

On the sequences of zeros of holomorphic solutions of linear second-order differential equations

We find necessary and sufficient conditions under which a finite or infinite sequence of complex numbers is the sequence of zeros of a holomorphic solution of the linear differential equation f″ + a ₀ f = 0 with a meromorphic coefficient a ₀ that has second-order poles. In addition, we present a criterion for all solutions of second-order linear equations to be meromorphic.     

Asymptotic Stability of Solutions to Perturbed Linear Difference Equations with Periodic Coefficients

We consider perturbed linear systems of difference equations with periodic coefficients. The zero solution of a nonperturbed system is assumed asymptotically stable, i.e., all eigenvalues of the monodromy matrix belong to the unit disk {||<1}. We obtain conditions on the perturbation of this system under which the zero solution of the system is asymptotically stable and also establish continuous dependence of one class of numeric characteristics of asymptotic stability of solutions on the coefficients of the system.     

Generalized Ginzburg–Landau equations in high dimensions

In this work, we study critical points of the generalized Ginzburg–Landau equations in dimensions \(n\ge 3\) which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres, and the convergence is strong away from a finite set consisting (1) of the infinite energy singularities of the limiting map, and (2) of points where bubbling off of finite energy n-harmonic maps could take place. The latter case is specific to dimensions \({>}2\). We also exhibit a criticality condition satisfied by the limiting n-harmonic maps which constrains the location of the infinite energy singularities. Finally we construct an example of non-minimizing solutions to the generalized Ginzburg–Landau equations satisfying our assumptions.     

Systems of differential equations with periodic coefficients

We consider linear systems of differential equations with periodic coefficients. We prove the solvability of nonhomogeneous systems in the Sobolev space W ₂ ¹ (R) and establish estimates for the solutions. This result implies a perturbation theorem for the exponential dichotomy of systems of differential equations with periodic coefficients.     

Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter

We provide existence results of multiple solutions for quasilinear elliptic equations depending on a parameter under the Neumann and Dirichlet boundary condition. Our main result shows the existence of two opposite constant sign solutions and a sign changing solution in the case where we do not impose the subcritical growth condition to the nonlinear term not including derivatives of the solution. The studied equations contain the \(p\) -Laplacian problems as a special case. Our approach is based on variational methods combining super- and sub-solution and the existence of critical points via descending flow.     

Selection problems of $${{\mathbb {Z}}}^2$$-periodic entropy solutions and viscosity solutions

\({{\mathbb {Z}}}^2\)-periodic entropy solutions of hyperbolic scalar conservation laws and \({{\mathbb {Z}}}^2\)-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate \({{\mathbb {Z}}}^2\)-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural \({{\mathbb {Z}}}^2\)-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.     

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of <Emphasis Type="Italic">p</Emphasis>-Laplacian type

We study general parabolic equations of the form \(u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,(|\mathbf {F}|^{p-2} \mathbf {F})+ f\) whose principal part depends on the solution itself. The vector field \(\mathbf {A}\) is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderón–Zygmund estimates for locally bounded weak solutions to the equations when \(p>2n/(n+2)\). This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285–320, 2007).     

Convergent and Asymptotic Methods for Second-order Difference Equations with a Large Parameter

We consider the second-order linear difference equation \(y(n+2)-2a y(n+1)-\Lambda ^2 y(n)=g(n)y(n)+f(n)y(n+1)\), where \(\Lambda \) is a large complex parameter, \(a\ge 0\) and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large \(\vert \Lambda \vert \) of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind.     

New approach to the solution of the classical sine-Gordon equation and its generalizations

We obtain new exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation. The three-dimensional solutions depend on an arbitrary function F(α) whose argument is a function α(x, y, z, t). The ansatz α is found from an equation linear in (x, y, z, t) whose coefficients are arbitrary functions of α that should satisfy a system of algebraic equations. By this method, we solve the classical and a generalized sine-Gordon equation; the latter additionally contains first derivatives with respect to (x, y, z, t). We separately consider an equation that contains only the first derivative with respect to time. We present approaches to the solution of the sine-Gordon equation with variable amplitude. The considered methods for solving the sine-Gordon equation admit a natural generalization to the case of integration of the same types of equations in a space of arbitrarily many dimensions.     

Geodesically convex energies and confinement of solutions for a multi-component system of nonlocal interaction equations

We consider a system of n nonlocal interaction evolution equations on \({\mathbb{R}^d}\) with a differentiable matrix-valued interaction potential W. Under suitable conditions on convexity, symmetry and growth of W, we prove \({\lambda}\)-geodesic convexity for some \({\lambda\in\mathbb{R}}\) of the associated interaction energy with respect to a weighted compound distance of Wasserstein type. In particular, this implies existence and uniqueness of solutions to the evolution system. In one spatial dimension, we further analyse the qualitative properties of this solution in the non-uniformly convex case. We obtain, if the interaction potential is sufficiently convex far away from the origin, that the support of the solution is uniformly bounded. Under a suitable Lipschitz condition for the potential, we can exclude finite-time blow-up and give a partial characterization of the long-time behaviour.     

Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if \(\partial _3 u\) belongs to the space \(L_t^{s_0}L_x^{r_0}(D)\) where \(2/s_0 + 3/r_0 \le 2 \) and \(9/4 \le r_0\le 5/2\), then the solution is Hölder continuous in D.     

Analysis of a FEM for a coupled system of singularly perturbed reaction–diffusion equations

We consider a system of ℓ ≥ 2 one-dimensional singularly perturbed reaction–diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We present a convergence theory for conforming linear finite elements on arbitrary meshes. As a result convergence independently of the perturbation parameters on a wide class of layer-adapted meshes is established.      

Blow-up solutions for a general class of the second-order differential equations on the half line

In this paper, we study the existence of positive blow-up solutions for a general class of the second-order differential equations and systems, which are positive radially symmetric solutions to many elliptic problems in ${\mathbb{R}}^N$. We explore fixed point arguments applied to suitable integral equations to get solutions.