The Riemann method for equations with leading partial derivative in ℝn

We give a version of the Riemann method for solving the Cauchy problem for a linear equation with variable coefficients with leading partial derivative.     

On the directional derivative problem for an elliptic equation

Siberian Mathematical Journal -     

On a nonlinear equation with fractional derivative

We consider a nonlinear equation with fractional derivative in which the nonlinearity has the form of a linear combination of convective and nonconvective types. We prove the time-global existence of solutions of the Cauchy problem and find their asymptotics at large times uniformly with respect to the space variable. The proof method is based on a detailed investigation of the behavior of the Green function, which permits one to drop the restriction of smallness of the initial data in the case of a nonlinearity of special form.     

On the distribution of values of the derivative of the Riemann zeta function at its zeros. I

Let ζ′(s) be the derivative of the Riemann zeta function ζ(s). A study on the value distribution of ζ′(s) at the non-trivial zeros ρ of ζ(s) is presented. In particular, for a fixed positive number X, an asymptotic formula and a non-trivial upper bound for the sum Σ_{0<Im ρ≤T} ζ′(ρ)X ^ρ as T → ∞ are given. We clarify the dependence on the arithmetic nature of X.     

Model oblique derivative problem for the heat equation with a discontinuous boundary function

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H ₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H ₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.     

On an optimal control problem for a hyperbolic partial differential equation

Existence of solutions is proved for a minimum problem for a distributed-parameter control system described by a linear, hyperbolic partial differential equation. The cost function is an integral depending on boundary controls.This research was supported by the Consiglio Nazionale delle Ricerche, Rome, Italy.     

On the construction of the Lyapunov function with sign-definite derivative with the help of auxiliary functions with sign-constant derivatives

The short survey of studies of the asymptotic stability with the help of auxiliary functions (positive definite and nonnegative) whose derivatives are nonpositive by virtue of the equations of perturbed motion and the construction of positive definite Lyapunov functions with negative definite derivatives on the basis of these auxiliary functions is given. The example of the construction of the Lyapunov function with the use of an auxiliary nonnegative function with nonpositive derivative is presented.     

The Riemann problem for the Leray-Burgers equation

For Riemann data consisting of a single decreasing jump, we find that the Leray regularization captures the correct shock solution of the inviscid Burgers equation. However, for Riemann data consisting of a single increasing jump, the Leray regularization captures an unphysical shock. This behavior can be remedied by considering the behavior of the Leray regularization with initial data consisting of an arbitrary mollification of the Riemann data. As we show, for this case, the Leray regularization captures the correct rarefaction solution of the inviscid Burgers equation. Additionally, we prove the existence and uniqueness of solutions of the Leray-regularized equation for a large class of discontinuous initial data. All of our results make extensive use of a reformulation of the Leray-regularized equation in the Lagrangian reference frame. The results indicate that the regularization works by bending the characteristics of the inviscid Burgers equation and thereby preventing their finite-time crossing.     

On a Riemann–Hilbert problem for the Fokas–Lenells equation

The solution of the initial value problem (IVP) for the Fokas–Lenells equation (FLE) was constructed in terms of the solution $M(x,t,k)$ of a 2 $\times$ 2 matrix Riemann–Hilbert problem (RHP) as $k\to \infty$, and the one-soliton solution of the FLE was derived based on this Riemann–Hilbert problem, in Lenells and Fokas (2009). However, in fact, the derivative with respect to $x$ of the solution of the FLE ($u_x(x,t)$) was recovered from the RHP as $k\to \infty$. In this paper, we construct the solution of the FLE in terms of the RHP as $k\to 0$, because the Lax pair of the FLE contains the negative order of the spectral variable $k$. We show that the one-soliton solution of the FLE obtained in this paper is the same as Lenells and Fokas (2009), but avoiding a complex integral.     

On spectral theory of the Riemann zeta function

Science China Mathematics - Every nontrivial zero of the Riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a Hilbert-Pólya...     

On some expansions for the Euler Gamma function and the Riemann Zeta function

In the present paper we introduce some expansions which use the falling factorials for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.