Small divisor problem in the theory of three-dimensional water gravity waves

We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta$ between them. \newline Denoting by $\mu =gL/c^{2}$ the dimensionless bifurcation parameter ($L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $\mu =\cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic expansion. Due to the \emph{occurence of small divisors}, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta$, the 3-dimensional travelling waves bifurcate for a set of "good" values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(\theta ,\mu ).$Comment: 119

Inhomogeneous boundary value problems for compressible Navier-Stokes equation: well-posedness and sensitivity analysis

In the paper compressible, stationary Navier-Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions, i.e., the solutions take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and in addition, the differentiability of solutions with respect to the coefficients of differential operators in shown. The results on the well-posedness of nonlinear problem are interesting on its own, and are used to obtain the shape differentiability of the drag functional for incompressible Navier-Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form, an appropriate adjoint state is introduced to this end. The shape derivatives of solutions to the Navier-Stokes equations are given by smooth functions, however the shape differentiability is shown in a weak norm. The method of analysis proposed in the paper is general, and can be used to establish the well-posedness for distributed and boundary control problems a well as for inverse problems in the case of the state equations in the form of compressible Navier-Stokes equations. The differentiability of solutions to the Navier-Stokes equations with respect to the date leads to the first order necessary conditions for a broad class of optimization problems

Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations

In the paper compressible, stationary Navier-Stokes eqautions are considered. A framework for analysis of such equations is established. The well-posedness for homogeneous boundary value problems of elliptic-hyperbolic type is shown

Reaction-diffusion equations with spatially distributed hysteresis

The paper deals with reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. In particular, such problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. We find sufficient conditions that guarantee the existence and uniqueness of solutions as well as their continuous dependence on initial data.Comment: 30 pages, 14 figure

Concentrations of solutions to time-discretizied compressible Navier-Stokes equations

The compactness properties of solutions to time-discretization of compressible Navier-Stokes equations are investigated in three dimensions. The existence of generalized solutions is established

Domain Dependence of Solutions to Compressible Navier-Stokes Equations

The minimization of the drag functional for the stationary, isentropic, compressible Navier\-Stokes equations (NSE) in three spatial dimensions is considered. In order to establish the existence of an optimal shape, the general result on compactness of families of generalized solutions to the NSE is established within in the framework of the modern theory of nonlinear PDEs [P. L. Lions, Mathematical Topics in Fluid Dynamics. Vol. $2$. Compressible Models, Oxford University Press, Clarendon Press, New York, 1998], [E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, UK, 2004]. The family of generalized solutions to the NSE is constructed over a family of admissible domains $\mathcal{U}_{ad}$. Any admissible domain $\Omega=B\setminus S$ contains an obstacle $S$, e.g., a wing profile. Compactness properties of the family of admissible domains in the form of the condition ($\mathcal{H}_\Omega$) is imposed. Roughly speaking, the condition ($\mathcal{H}_\Omega$) is satisfied, provided that for any sequence of admissible domains $\{\Omega_n\}\subset \mathcal{U}_{ad}$ there is a subsequence convergent both in Hausdorff metrics and in the sense of Kuratowski and Mosco. The analysis is performed for the adiabatic constant $\gamma>1$ in the pressure law $p(\rho)=\rho^\gamma$ and it is based on the technique used in [P. I. Plotnikov and J. Sokolowski, Comm. Math. Phys., 258 (2005), pp. 567-608] in the case of discretized NSE. The result is a generalization to the stationary equations with $\gamma>1$ of the results obtained in [E. Feireisl, A. H. Novotn‡, and H. Petzeltová, Math. Methods Appl. Sci., 25 (2002), pp. 1045-1073], [E. Feireisl, Appl. Math. Optim., 47 (2003), pp. 59-78] for evolution equations within the range $\gamma>3/2$ for the adiabatic ratio

On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations

The steady state system of isothermal Navier-Stokes equations is considered in two dimensional domain including an obstacle. The shape optimisation problem of minimisation of the drag with respect to the admissible shape of the obstacle is defined. The generalized solutions for the Navier-Stokes equations are introduced. The existence of an optimal shape is proved in the class of admissible domains. In general the solutions are not unique for the problem under considerations

Stationary solutions of Navier-Stokes equations for diatomic gases

in press in Russian Mathematical surveys, May-June 200