By V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)
Articles during this volume:
Square Integrability and specialty of the suggestions of the Kadomtsev–Petviashvili-I Equation
Soliton Asymptotics of options of the Sine-Gordon Equation
Werner Kirsch and Vladimir Kotlyarov
On the Davey–Stewartson and Ishimori Systems
Nakao Hayashi and Pavel I. Naumkin
Stochastic Isometries in Quantum Mechanics
Complex superstar Algebras
L. B. de Monvel
“Momentum” Tunneling among Tori and the Splitting of Eigenvalues of the Laplace–Beltrami Operator on Liouville Surfaces
S. Yu. Dobrokhotov and A. I. Shafarevich
Nonclassical Thermomechanics of Granular Materials
Random Operators and Crossed Products
Daniel H. Lenz
Schrödinger Operators with Empty Singularly non-stop Spectra
Michael Demuth and Kalyan B. Sinha
An Asymptotic growth for Bloch capabilities on Riemann Surfaces of endless Genus and nearly Periodicity of the Kadomcev–Petviashvilli Flow
Lifshitz Asymptotics through Linear Coupling of Disorder
Sharp Spectral Asymptotics and Weyl formulation for Elliptic Operators with Non-smooth Coefficients
Topological Invariants of Dynamical platforms and areas of Holomorphic Maps: I
Contents of quantity 2
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Additional resources for Mathematical Physics, Analysis and Geometry - Volume 2
In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic–hyperbolic Davey–Stewartson system can be reduced to Equation (A) with (1) (1) N1 = |u|2 u, Kx = ∂y (|u|2 ), Ky = ∂x (|u|2 ), and all the rest components of the vectors Kx and Ky are equal to zero. The elliptic–hyperbolic Ishimori system is involved in Equation (A), when N1 = (1 + |u|2 )−1 u(∇u)2 , and Kx = −Ky = (1 + |u|2 )−2 (ux uy − ux uy ). Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely if N (eiθ v) = eiθ N (v) for all θ ∈ R.
We consider the first term of the right-hand side of the √ above. 12) with g = φ, q = r = ∞, p = s = 2, we find Y∗ Su, Xφ∂x ψu φY∗ S |∂x |u + C u e3 ×e 2e4 ϕ ϕ ∞ ∞ ψ φ +C u e 2 6 ϕ ϕ ∞ φ 1,0,∞ (1 |∂x |u + C u e 3 ϕ S L∞ y S L∞ y |∂x |u ∞ 2 1,0,∞ φ 2 + + ψ ψ ∞ S L∞ y 2 1,0,∞ + ϕ ψ 1,0,∞ (1 |∂x |u 1+ ϕ 1,0,∞ ) × + ϕ 1,0,∞ ) 2 2 1,0,∞ . Thus the first estimate of the lemma is proved. 8) with p = q = 2 we have Su, S∂y−1 (φψ)∂x u = Y∗ Su, X∂y−1 (φψ)∂x u Y∗ Su, X∂y−1 (φ∂x ψ)u + Y∗ Su X∂y−1 (φψx )u Y∗ Su, X∂y−1 (φ∂x ψ)u + Ce2 φ × ψ 2 L∞ x Ly + ψx 2 L∞ x Ly ϕ ∞ 2 L∞ x Ly × 2 u .
1 we obtain Im(Su, Mu) Su, ωj2 S|∂j |u = 2 =2 j =1,2 ωj Su, ∂j ωj SHj u − [∂j , ωj S]Hj u j =1,2 64 N. HAYASHI AND P. I. NAUMKIN = −2 ωj S |∂j |u + |∂j |, ωj S u, j =1,2 − ωj S |∂j |u + |∂j |Hj , ωj S Hj u − 2 (ωj Su, [∂j , ωj S]Hj u) j =1,2 2 ωj S |∂j |u 2 − ωj S |∂j u |∂j |, ωj S u + j =1,2 |∂j |Hj , ωj S Hj u + − |∂j |, ωj S u |∂j |Hj , ωj S Hj u − − 2|(ωj Su, [∂j , ωj S]Hj u)| 2 ωj S |∂j |u −C u 2 exp(2 ϕ ∞) ω 4 ∞ + ω ω ∞ 1,0,∞ . 7). In the next lemma we prepare some estimates of different terms appearing in the nonlinearity.
Mathematical Physics, Analysis and Geometry - Volume 2 by V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)