Asymptotic solutions of the one-dimensional linearized Korteweg–de Vries equation with localized initial data

The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exactGreen function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.     

Non-Gaussian limit distributions of solutions of the many-dimensional Bürgers equation with random initial data

Limit distributions of solutions of the multidimensional Bürgers equation are found in the case where an initial condition is a random field of type ² of degreek with a long-range dependence.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 330–336, March, 1995.     

Limiting distributions of the solutions of the many-dimensional Bürgers equation with random initial data. II

We find non-Gaussian limiting distributions of the solutions of the many-dimensional Burgers equation with the initial condition given by a homogeneous isotropic Gaussian random ²-type field with strong dependence.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1003–1010, August, 1994.     

Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations

Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several conditions on u ₀ to prove the existence of a unique strong solution with u(0) = u ₀ in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given u ₀, q, s as above and the Stokes operator A ₂, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions.     

Non-uniform dependence on initial data for the Benjamin–Ono equation

In the paper, we consider the initial value problem to the Benjamin–Ono equation in the line. We show that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces.     

Long-time behavior of solutions to the derivative nonlinear Schrödinger equation for soliton-free initial data

The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.     

On optimal initial value conditions for local strong solutions of the Navier–Stokes equations

Consider a smooth bounded domain , and the Navier–Stokes system in with initial value and external force f =  div F, where , are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition in order to obtain a unique strong solution in some initial interval [0, T), . Up to now several sufficient conditions on u ₀ are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition , A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if , , then any weak solution u in the usual sense does not satisfy Serrin’s condition for each 0 < T ≤ ∞.      

Existence of solutions for the 3D-micropolar fluid system with initial data in Besov–Morrey spaces

In this paper, we show a local-in-time existence result for the 3D micropolar fluid system in the framework of Besov–Morrey spaces. The initial data class is larger than the previous ones and contains strongly singular functions and measures.     

Reaction–Diffusion Equations in Immunology

Computational Mathematics and Mathematical Physics - The paper is devoted to the recent works on reaction–diffusion models of virus infection dynamics in human and animal organisms. Various...     

On Error Control in the Numerical Solution of Reaction–Diffusion Equation

Computational Mathematics and Mathematical Physics - A novel method for deriving a posteriori error bounds for approximate solutions of reaction–diffusion equations is proposed. As a model...     

On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces

This paper is concerned with the Cauchy problem for the Dullin–Gottwald–Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.     

Stability of solutions of control problems for the convection–diffusion–reaction equation with a strong nonlinearity

We consider a boundary control problem for the stationary convection–diffusion–reaction equation in which the reaction constant depends on the concentration of matter in such a way that the equation has a fifth-order nonlinearity. We prove the solvability of the boundary value problem and an extremal problem, derive an optimality system, and analyze it to derive estimates for the local stability of the solution of the extremal problem under small perturbations of both the performance functional and one of the given functions.     

Existence of weak solutions of the g-Kelvin–Voight equation

The 2D $g$-Navier–Stokes equations have the form $\frac{?u}{?t}?\nu \Delta u+u.?u+?p=f\text{in }\Omega$ with the continuity equation $?.\left(gu\right)=0\text{in }\Omega$ in a bounded domain $\Omega ?{\mathbb{R}}^2$ where $g=g\left(x_{1,}{x}_2\right)$ is a smooth real valued function defined on $\Omega$. We use the method described by Roh [J. Roh, $g$-Navier Stokes equations, Ph.D. Thesis, University of Minnesota, 2001] for the derivation of $g$-Kelvin–Voight equations represented by $\frac{?u}{?t}?\nu {\Delta }_{g}u+\frac{\nu }{g}(?g??)u?\alpha {\Delta }_g{u}_t+\frac{\alpha }{g}(?g??)u_t+u??u+?p=f\left(x\right)\text{ in }\Omega$$?.\left(gu\right)=0\text{in }\Omega$ We discuss the existence and uniqueness of weak solutions of $g$-Kelvin–Voight equations by the use of the well known Feado–Galerkin method.     

Exact solutions of the generalized Sinh–Gordon equation

In this paper, we successfully derive a new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.