By Ta-Tsien Li
This worthy quantity is a suite of articles in reminiscence of Jacques-Louis Lions, a number one mathematician and the founding father of the modern French utilized arithmetic university. The contributions were written through his pals, colleagues and scholars, together with C Bardos, A Bensoussan, S S Chern, P G Ciarlet, R Glowinski, Gu Chaohao, B Malgrange, G Marchuk, O Pironneau, W Strauss, R Temam, and so on.
The ebook matters many very important leads to research, geometry, numerical tools, fluid mechanics, keep watch over conception, and so forth.
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Extra resources for Frontiers In Mathematical Analysis And Numerical Methods: In Memory Of Jacques-Louis Lions
3(1963), 1032-1066. REGULARITY THEORY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS A . BENSOUSSAN University Paris Dauphine and CNES (Centre National d’Etudes Spatiales), 2, Place Maurice Quentin, 75001, Paris, France J. FREHSE Institut fur Angewandte Mathematik, Universitat Bonn, Bonn, Germany Abstract The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case.
GUO W. 4. 3. 23) holds. Proof. 24) where any element v E M * is a solution of the mean field equation with a temperature p ( v ) < -87r. Observe by contradiction that the L2(s2)distance d between p and M * is strictly positive. 27) This completes the proof. The previous stability results involve, as in most of the contributions in the subject (cf. for instance [l, 13, 19]), the L2 norm of the vorticity. Finally we conclude this section by proving, using the notion of entropic convergence, a weaker result under a weaker hypothesis.
1, let i;l solve the linear equation (at + v T h e n for any 1 5 p * 0 ) G = 0. 1. Hence since the Jacobian equals one. This proves the desired estimate on the first derivatives. C. BARDOS 40 Y. GUO W. STRAUSS Now we prove that the eigenfunctions are smooth. 3. A n y eigenfunction $A E L2(s2)corresponding t o a n eigenvalue X with Re X > o belongs t o W',P(n)for all p < 00. Proof. v (curl-l$A - V W ~ )d - ~ . 21) By assumption, 4~ is in L2(s2) so that curl-'$x - V w o is in H1(R). V w ' , P (+ ~> w',P(~) with norm O(e("+'It) for any E > 0.