Inverse problems are prevalent in astrophysics and many methods were developed for the reconstruction of these ill-posed equations. We present a new method for inverting Abel's integral equation by explicit diagonalization of the re lated Volterra operator. This method is particularly suitable for determining the velocity laws of stellar winds. We test the method for different frequently used p...

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L(R) for positive times is entirely determined by the trace of the vorticity at t = 0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa, and Osada, and by Kato, this uniqueness property implies that the Cauchy problem f...

The time-dependent Maxwell system is supplemented with the constitutive relations of linear bianisotropic media and is treated as a neutral integro-differential equation in a Hilbert space. By using the theory of abstract Volterra equations and strongly continuous semigroups we obtain general well-posedness results for the corresponding mathematical problem.

In this study, nonlinear vibration of a composite cable is investigated by considering nonlinear stress-strain relations. The composite cable is composed of an aluminum wire as reinforcement and a rubber coating as matrix. The nonlinear governing equations of motion are derived about to an initial curve and based on the fundamentals of continuum mechanics and the nonlinear Green-Lagrangian stra...

We propose time regularizations for ill posed evolution equations of the type of the Perona-Malik equation of image processing, prove that they are well posed, and give numerical evidence for their superiority to the widely used space regularizations.

Solving large-scale algebraic Riccati equations (AREs) is one of the central tasks in solving optimal control problems for linear and, using receding-horizon techniques, also nonlinear instationary partial differential equations. Large-scale AREs also occur in several model reduction methods for dynamical systems. Due to sparsity and large dimensions of the resulting coefficient matrices, stand...

We obtain Calderón-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity u0 ∈ H (R) for s > 0 in 2-D, or u0 ∈ H (R) satisfying ‖u0‖L2‖∇u0‖L2 being sufficiently small in 3-D. This in particular improves t...

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in R with N ≥ 2. We address the question of well-posedness for large data having critical Besov regularity and we aim at stating well-posedness in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon...

By using the continuous induction method, we prove that the initial value problem of the three dimensional Navier-Stokes equations is globally well-posed in L(R)∩L(R) for any 3 < p < ∞. The proof is rather simple.