arXiv subjects

Alexander Shlapunov

Publications and source records attributed to Alexander Shlapunov.

21 records · Page 2Linked to original sources

arXiv2021

On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.

math.AP↗

arXiv2020

On the stability phenomenon of the Navier-Stokes type Equations for Elliptic Complexes

Let ${\mathcal X}$ be a Riemannian $n$-dimensional smooth compact closed manifold, $n\geq 2$, $E^i$ be smooth vector bundles over $\mathcal X$ and $\{A^i,E^i\}$ be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with $\{A^i,E^i\}$ on the scale of anisotropic Hölder spaces over the layer ${\mathcal X} \times [0,T]$ with finite time $T > 0$. Using the properties of the differentials $A^i$ and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator. It appears that the Fréchet derivative $(I+K)'$ is continuously invertible at every point of each Banach space under the consideration and the map $(I+K)$ is open and injective in the space.

math.AP↗

arXiv2019

On a Singular Perturbation of the Navier-Stokes Equations

The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and uniqueness is established for the weak solutions of the Cauchy problem. The solution of the regularised system is shown to converge to the solution of the conventional Navier-Stokes equations provided it is uniformly bounded in parameter.

math.AP↗