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Institute
We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.
This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.
Formal poincare lemma
(2007)
Let X be a smooth n-dimensional manifold and D be an open connected set in X with smooth boundary OD. Perturbing the Cauchy problem for an elliptic system Au = f in D with data on a closed set Gamma subset of partial derivativeD, we obtain a family of mixed problems depending on a small parameter epsilon > 0. Although the mixed problems are subjected to a noncoercive boundary condition on partial derivativeDF in general, each of them is uniquely solvable in an appropriate Hilbert space D-T and the corresponding family {u(epsilon)} of solutions approximates the solution of the Cauchy problem in D-T whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in D-T is equivalent to the boundedness of the family {u(epsilon)}. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing the solution. Examples for Dirac operators in the Euclidean space R-n are treated. In this case, we obtain a family of mixed boundary problems for the Helmholtz equation
Let X be a smooth n -dimensional manifold and D be an open connected set in X with smooth boundary ∂D. Perturbing the Cauchy problem for an elliptic system Au = f in D with data on a closed set Γ ⊂ ∂D we obtain a family of mixed problems depending on a small parameter ε > 0. Although the mixed problems are subject to a non-coercive boundary condition on ∂D\Γ in general, each of them is uniquely solvable in an appropriate Hilbert space DT and the corresponding family {uε} of solutions approximates the solution of the Cauchy problem in DT whenever the solution exists. We also prove that the existence of a solution to the Cauchy problem in DT is equivalent to the boundedness of the family {uε}. We thus derive a solvability condition for the Cauchy problem and an effective method of constructing its solution. Examples for Dirac operators in the Euclidean space Rn are considered. In the latter case we obtain a family of mixed boundary problems for the Helmholtz equation.
This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.
We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
We prove the existence of Hp(D)-limit of iterations of double layer potentials constructed with the use of Hodge parametrix on a smooth compact manifold X, D being an open connected subset of X. This limit gives us an orthogonal projection from Sobolev space Hp(D) to a closed subspace of Hp(D)-solutions of an elliptic operator P of order p ≥ 1. Using this result we obtain formulae for Sobolev solutions to the equation Pu = f in D whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of double layer potentials. Similar regularization is constructed also for a P-Neumann problem in D.
Let A be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write Ssub(A)(D) for the space of solutions to thesystem Au = 0 in a domain D ⊂ X. Using reproducing kernels related to various Hilbert structures on subspaces of Ssub(A)(D) we show explicit identifications of the dual spaces. To prove the "regularity" of reproducing kernels up to the boundary of D we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the ∂-Neumann problem. The duality itself takes place only for those domains D which possess certain convexity properties with respect to A.