On evolution equations for moving domains.
On evolution Galerkin methods for the Maxwell and the linearized Euler equations
The subject of the paper is the derivation and analysis of evolution Galerkin schemes for the two dimensional Maxwell and linearized Euler equations. The aim is to construct a method which takes into account better the infinitely many directions of propagation of waves. To do this the initial function is evolved using the characteristic cone and then projected onto a finite element space. We derive the divergence-free property and estimate the dispersion relation as well. We present some numerical...
On evolution inequalities of a modified Navier-Stokes type. II
The present part of the paper continues the study of the abstract evolution inequality from the first part. Theorem 1 states the existence and uniqueness of a weak solution to the evolution inequality under consideration. The proof is based on the method of approximation of the weak solution by a sequence of strong solutions. Theorem 2 yields two regularity results for the strong solution.
On evolution inequalities of a modified Navier-Stokes type. I
The paper present an existence theorem for a strong solution to an abstract evolution inequality where the properties of the operators involved are motivated by a type of modified Navier-Stokes equations under certain unilateral boundary conditions. The method of proof rests upon a Galerkin type argument combined with the regularization of the functional.
On evolution inequalities of a modified Navier-Stokes type. III
This is the last from a series of three papers dealing with variational equations of Navier-Stokes type. It is shown that the theoretical results from the preceding parts (existence and regularity of solutions) can be applied to the problem of motion of a fluid through a tube.
On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions
In this paper, we establish the large-data and long-time existence of a suitable weak solution to an initial and boundary value problem driven by a system of partial differential equations consisting of the Navier-Stokes equations with the viscosity polynomially increasing with a scalar quantity that evolves according to an evolutionary convection diffusion equation with the right hand side that is merely -integrable over space and time. We also formulate a conjecture concerning regularity...
On exact controllability for the Navier-Stokes equations
On exact controllability for the Navier-Stokes equations
We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain with control distributed in an arbitrary fixed subdomain. The result that we obtain in this paper is as follows. Suppose that we have a given stationary point of the Navier-Stokes equations and our initial condition is sufficiently close to it. Then there exists a locally distributed control such that in a given moment of time the solution of the Navier-Stokes...
On exact results in the finite element method
We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution . We show that the Galerkin approximation of based on the so-called biharmonic finite elements is independent of the values of in the interior of any subelement.
On exact solutions of a degenerate quasilinear wave equation with source term.
On exact solutions of one-dimensional diffusion problems with free boundaries for parabolic equations.
On existence and nonexistence results for nonlinear Schrödinger equations
On existence and regularity of solutions to a class of generalized stationary Stokes problem
We investigate the existence of weak solutions and their smoothness properties for a generalized Stokes problem. The generalization is twofold: the Laplace operator is replaced by a general second order linear elliptic operator in divergence form and the “pressure” gradient is replaced by a linear operator of first order.
On existence and Schauder regularity of solutions to a class of generalized stationary Stokes problems
On existence and uniqueness of solutions of nonlocal problems for hyperbolic differential-functional equations in two independent variables
We seek for classical solutions to hyperbolic nonlinear partial differential-functional equations of the second order. We give two theorems on existence and uniqueness for problems with nonlocal conditions in bounded and unbounded domains.
On existence in the Cauchy problem for the biharmonic equation
On existence of a boundary layer near a three-phase contact point.
On existence of a unique generalized solution to systems of elliptic PDEs at resonance
The Dirichlet boundary value problem for systems of elliptic partial differential equations at resonance is studied. The existence of a unique generalized solution is proved using a new min-max principle and a global inversion theorem.
On existence of harmonic maps from a surface to the spcae of it
On existence of positive solutions for a class of Caffarelli-Kohn-Nirenberg type equations
We investigate the solvability of a singular equation of Caffarelli-Kohn-Nirenberg type having a critical-like nonlinearity with a sign-changing weight function. We shall examine how the properties of the Nehari manifold and the fibering maps affect the question of existence of positive solutions.