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000856402 019__ $$a1066179507$$a1067243474
000856402 020__ $$a9783319994864$$q(electronic book)
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000856402 020__ $$z9783319994857
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000856402 0247_ $$a10.1007/978-3-319-99486-4$$2doi
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000856402 1001_ $$aBerti, Massimiliano,$$eauthor.
000856402 24510 $$aAlmost global solutions of capillary-gravity water waves equations on the circle /$$cMassimiliano Berti, Jean-Marc Delort.
000856402 264_1 $$aCham, Switzerland :$$bSpringer,$$c2018.
000856402 300__ $$a1 online resource (x, 269 pages) :$$billustrations.
000856402 336__ $$atext$$btxt$$2rdacontent
000856402 337__ $$acomputer$$bc$$2rdamedia
000856402 338__ $$aonline resource$$bcr$$2rdacarrier
000856402 4901_ $$aLecture notes of the Unione Matematica Italiana,$$x1862-9113 ;$$v24
000856402 504__ $$aIncludes bibliographical references and index.
000856402 5050_ $$aIntro; Preface; Contents; 1 Introduction; 1.1 Main Theorem; 1.2 Introduction to the Proof; 1.3 Paradifferential Formulation and Good Unknown; 1.4 Reduction to Constant Coefficients; Step 1: Diagonalization of the System; Step 2: Reduction to Constant Coefficients at Principal Order; Step 3: Reduction to Constant Coefficients at Arbitrary Order; 1.5 Normal Forms; 2 Main Result; 2.1 The Periodic Capillarity-Gravity Equations; 2.2 Statement of the Main Theorem; 3 Paradifferential Calculus; 3.1 Classes of Symbols; 3.2 Quantization of Symbols; 3.3 Symbolic Calculus; 3.4 Composition Theorems
000856402 5058_ $$a3.5 Paracomposition4 Complex Formulation of the Equation and Diagonalization of the Matrix Symbol; 4.1 Reality, Parity and Reversibility Properties; 4.2 Complex Formulation of the Capillary-Gravity Water Waves Equations; 4.3 Diagonalization of the System; 5 Reduction to a Constant Coefficients Operator and Proof of the Main Theorem; 5.1 Reduction to Constant Coefficients of the Highest Order Part; 5.2 Reduction to Constant Coefficient Symbols; 5.3 Normal Forms; 5.4 Proof of Theorem 4.7; 6 The Dirichlet-Neumann Paradifferential Problem; 6.1 Paradifferential and Para-Poisson Operators
000856402 5058_ $$a6.2 Parametrix of Dirichlet-Neumann Problem6.3 Solving the Dirichlet-Neumann Problem; 7 Dirichlet-Neumann Operator and the Good Unknown; 7.1 The Good Unknown; 7.2 Paralinearization of the Water Waves System; 7.3 The Capillarity-Gravity Water Waves Equations in Complex Coordinates; 8 Proof of Some Auxiliary Results; 8.1 Non-Resonance Condition; 8.2 Precise Structure of the Dirichlet-Neumann Operator; References; Index
000856402 506__ $$aAccess limited to authorized users.
000856402 520__ $$aThe goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.--$$cProvided by publisher.
000856402 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed November 12, 2018).
000856402 650_0 $$aCauchy problem.
000856402 650_0 $$aSobolev spaces.
000856402 7001_ $$aDelort, Jean-Marc,$$d1961-$$eauthor.
000856402 77608 $$iPrint version:$$z3319994859$$z9783319994857$$w(OCoLC)1045466541
000856402 830_0 $$aLecture notes of the Unione Matematica Italiana ;$$v24.$$x1862-9113
000856402 852__ $$bebk
000856402 85640 $$3SpringerLink$$uhttps://univsouthin.idm.oclc.org/login?url=http://link.springer.com/10.1007/978-3-319-99486-4$$zOnline Access$$91397441.1
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