On stability of solutions of the coefficient inverse extremal problems for the stationary convection-diffusion equation

The coefficient inverse extremal problems are studied for the stationary convectiondiffusion equation in a bounded domain under mixed boundary conditions on the boundary of the domain. The role of control is played by the velocity vector of a medium and the functions that are involved in the boundary conditions for temperature. The solvability of the extremal problems is proven both for an arbitrary weakly lower semicontinuous quality functional and for the particular quality functionals. On the basis of analysis of the optimality system some sufficient conditions are established on the initial data providing the uniqueness and stability of optimal solutions under sufficiently small perturbations of both the quality functional and one of the functions involved in the original boundary value problem.     

Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

Existence and uniqueness of solutions for the Navier-Stokes equations with hyperdissipation

The Navier-Stokes equations with hyperdissipation are used to numerically simulate turbulent flows. While numerical aspects of this model are well-understood, its analytic properties have not been thoroughly explored. In this article, we prove the existence and uniqueness of strong solutions to these equations on with periodic boundary conditions. Our proof extends the methods of Cannone (1995) and Cannone and Meyer (1995) to the discrete setting and relies on Littlewood-Paley theory and techniques from multiple Fourier series.     

Stability of solutions to extremal problems of boundary control for stationary heat convection equations

We study extremal problems of boundary control for stationary heat convection equations with Dirichlet boundary conditions on velocity and temperature. As the cost functional we choose the mean square integral deviation of the required temperature field from a given temperature field measured in some part of the flow region. The controls are functions appearing in the Dirichlet conditions on velocity and temperature. We prove the stability of solutions to these problems with respect to certain perturbations of both the quality functional and one of the known functions appearing in the original equations of the model.     

On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations

We study regularity properties of weak solutions of the eqns. of Navier-Stokes which are in L((O,T),Lⁿ()) or in LP((O,T),Lⁿ()) for some p2. We prove also that L((O,T), Lⁿ()) is a uniqueness class for weak solutions. Moreover we give a generalization of Serrin's uniqueness result.     

On the stationary solutions of the full compressible Navier-Stokes equations and its stability with respect to initial disturbance

This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier-Stokes equations effected by external force of general form in R³.     

On the regularity and uniqueness of conically self-similar free-vortex solutions to the Navier-Stokes equations

In this paper the real analyticity of all conically self-similar free-vortex solutions to the Navier-Stokes equations is proven. Furthermore, it is mathematically established that such solutions are uniquely determined by the values of three derivatives on the symmetry axis, and hence a numerical method, invented and successfully used by Shtern & Hussain (1993,1996), is justified mathematically. In addition, it is proven that these results imply that for any conically self-similar free-vortex solution to the Navier--Stokes equations there exists a second order non-swirling correction term. For this term it is also shown that the second order contribution to the total axial flow force vanishes in the cases of the entire space and a half-space, but that it need not vanish for general conical domains. In doing so an old claim by Burggraf & Foster (1977) is established mathematically, however not for Long's problem but for Shtern & Hussain's (1996) extension of this problem to the full Navier-Stokes equations and the entire space.     

Remark on uniqueness of mild solutions to the Navier-Stokes equations

We investigate a limiting uniqueness criterion to the Navier-Stokes equations. We prove that the mild solution is unique under the class , where bmo^-1 is the “critical” space including Lⁿ. As an application of uniqueness theorem, we also consider the local well-posedness of Navier-Stokes equations in bmo^-1.     

On the Navier problem for the stationary Navier-Stokes equations

The Navier problem is to find a solution of the steady-state Navier-Stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary. If Ω is exterior it is required that the velocity converges to an assigned constant vector u₀ at infinity. We prove that a solution exists in a bounded domain provided ‖a_‖L²(∂Ω) is less than a computable positive constant and is unique if ‖a_‖W1/2,2(∂Ω)+‖s_‖L²(∂Ω) is suitably small. As far as exterior domains are concerned, we show that a solution exists if ‖a_‖L²(∂Ω)+‖a−u₀⋅n_‖L²(∂Ω) is small.     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the uniqueness in critical spaces for compressible Navier-Stokes equations

We show some new uniqueness results for compressible flows with data having critical regularity. In the barotropic case, uniqueness is stated whenever the space dimension N satisfies N ≥ 2, and in the polytropic case, whenever N ≥ 3. The endpoints N = 2 in the barotropic case and N = 3 in the polytropic case were left open in [4], [5] and [6].     

On free boundary problems with moving contact points for the stationary two-dimensional Navier-Stokes equations

Solvability of the problem of slow drying of a plane capillary in the classical setting (i. e., with the adherence condition on a rigid wall) is established. The proof is based on a detailed study of the asymptotics of the solution near a point of contact of the free boundary with a moving wall, including estimates of the coefficients in well known asymptotic formulas. It is shown that the only value of the contact angle admitting a solution of the problem with finite energy dissipation equals π. Bibliography: 18 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 179–205. Translated by E. V. Frolova.     

A uniqueness theorem for weak solutions of the stationary semiconductor equations

In this paper we prove a uniqueness theorem for weak solutions of a mixed boundary-value problem for the stationary semiconductor equations (van Roosbroeck's system) under the assumption that the deviation of the carrier potentials from an equilibrium solution is sufficiently small.     

Weak-strong uniqueness for the generalized Navier-Stokes equations

We give a weak-strong uniqueness result for the weak solutions of the generalized Navier-Stokes equations in Besov space.     

On the stationary and nonstationary Navier-Stokes equations inR n

Sunto Si dimostra che le soluzioni dette equazioni di Navier-Stokes stazionarie considerate in tutto lo spazio Rⁿ sono asintoticamente stabili rispetto a piccole perturbazioni dei dati iniziali nella norma di L^p.     

Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations

Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.