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Large time regular solutions to the MHD equations in cylindrical domains

Wisam Alame, Wojciech M. Zajączkowski (2011)

Applicationes Mathematicae

We prove the large time existence of solutions to the magnetohydrodynamics equations with slip boundary conditions in a cylindrical domain. Assuming smallness of the L₂-norms of the derivatives of the initial velocity and of the magnetic field with respect to the variable along the axis of the cylinder, we are able to obtain an estimate for the velocity and the magnetic field in W ₂ 2 , 1 without restriction on their magnitude. Then the existence follows from the Leray-Schauder fixed point theorem.

Levi's forms of higher codimensional submanifolds

Andrea D'Agnolo, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let X ≅ C n , let M be a C 2 hypersurface of X , S be a C 2 submanifold of M . Denote by L M the Levi form of M at z 0 ∈ S . In a previous paper [3] two numbers s ± ⁢ S , p , p ∈ T ˙ S * ⁢ X z 0 are defined; for S = M they are the numbers of positive and negative eigenvalues for L M . For S ⊂ M , p ∈ S × M T ˙ * S X ) , we show here that s ± ⁢ S , p are still the numbers of positive and negative eigenvalues for L M when restricted to T z 0 C ⁢ S . Applications to the concentration in degree for microfunctions at the boundary are given.

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

The estimate D k - 1 u L n / ( n - 1 ) ≤ A ( D ) u L 1 is shown to hold if and only if A ( D ) is elliptic and canceling. Here A ( D ) is a homogeneous linear differential operator A ( D ) of order k on ℝ n from a vector space V to a vector space E . The operator A ( D ) is defined to be canceling if ⋂ ξ ∈ ℝ n ∖ { 0 } A ( ξ ) [ V ] = { 0 } . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...

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