The jump problem for mixed-type equations with defects on the type change line.
The Laplace method of the linear partial differential equations of the -th order
The Lie-algebraic discrete approximation scheme for evolution equations with Dirichlet/Neumann data.
The lifespan of spherically symmetric solutions of the compressible Euler equations outside an impermeable sphere
The logarithmic gradient of the kernel of the heat equation with drift on a Riemannian manifold.
The multidimensional -transform and its use in solution of partial difference equations
The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes
We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region .
The positive radial solutions of a class of semilinear elliptic equations.
The problem of regular isometric embeddings and the Monge-Ampère equation of hyperbolic type
The quasi-stationary approximation for the Stefan problem with a convective boundary condition.
The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation
Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip . We prove a representation theorem for an arbitrary function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of...
The restriction of the Fourier transform to some curves and surfaces
The Riccati differential equation and a diffusion-type equation.
The role of symmetry and separation in surface evolution and curve shortening.
The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics
The existence of a periodic solution of a nonlinear equation is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.
The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques
Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions...
The solution of an axially symmetric boundary value problem of the generalized heat equation and its application in the technology of microschemes
The space of exponentially decreasing entire functions and its application to solvability
The spread of the potential on a weighted graph.
The stack of microlocal perverse sheaves
In this paper we construct the abelian stack of microlocal perverse sheaves on the projective cotangent bundle of a complex manifold. Following ideas of Andronikof we first consider microlocal perverse sheaves at a point using classical tools from microlocal sheaf theory. Then we will use Kashiwara-Schapira’s theory of analytic ind-sheaves to globalize our construction. This presentation allows us to formulate explicitly a global microlocal Riemann-Hilbert correspondence.