Displaying similar documents to “Large data local solutions for the derivative NLS equation”

Divergent solutions to the 5D Hartree equations

Daomin Cao, Qing Guo (2011)

Colloquium Mathematicae

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We consider the Cauchy problem for the focusing Hartree equation i u t + Δ u + ( | · | - 3 | u | ² ) u = 0 in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of - Q + Δ Q + ( | · | - 3 | Q | ² ) Q = 0 in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂...

Seasonal time-series imputation of gap missing algorithm (STIGMA)

Eduardo Rangel-Heras, Pavel Zuniga, Alma Y. Alanis, Esteban A. Hernandez-Vargas, Oscar D. Sanchez (2023)

Kybernetika

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This work presents a new approach for the imputation of missing data in weather time-series from a seasonal pattern; the seasonal time-series imputation of gap missing algorithm (STIGMA). The algorithm takes advantage from a seasonal pattern for the imputation of unknown data by averaging available data. We test the algorithm using data measured every 10 minutes over a period of 365 days during the year 2010; the variables include global irradiance, diffuse irradiance, ultraviolet irradiance,...

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod, Gigliola Staffilani (2015)

Journal of the European Mathematical Society

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We also prove a long time existence result; more precisely we prove that for fixed T > 0 there exists a set Σ T , ( Σ T ) > 0 such that any data φ ω ( x ) H γ ( 𝕋 3 ) , γ < 1 , ω Σ T , evolves up to time T into a solution u ( t ) with u ( t ) - e i t Δ φ ω C ( [ 0 , T ] ; H s ( 𝕋 3 ) ) , s = s ( γ ) > 1 . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H 1 ( 𝕋 3 ) , that is in the supercritical scaling regime.

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation t u - ³ t x x u + 2 κ x u + x [ g ( u ) / 2 ] = γ ( 2 x u ² x x u + u ³ x x x u ) for the initial data u₀(x) in the Besov space B p , r s ( ) with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given C m -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

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Let  P be a long range metric perturbation of the Euclidean Laplacian on  d , d 2 . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to  P . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group e i t f ( P ) where f has a suitable development at zero (resp. infinity). ...

A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

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Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function f W k , p ( Ω ) possesses an L p derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space W k , p ( Ω ) . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

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For a principal type pseudodifferential operator, we prove that condition  ( ψ ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ + 3 / 2 for any ϵ &gt; 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesimply local solvability with a loss of 1...

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function f L l o c p ( ) the notion of p-mean variation of order 1, p ( f , ) is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space W 1 , p ( ) in terms of p ( f , ) is directly related to the characterisation of W 1 , p ( ) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u - χ · ( u v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω n , χ , r , μ > 0 , k > 1 and n 1 . It is shown with k ( 1 , 2 ) that the system possesses a global generalized solution for n 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1 .

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in 3

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

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We present a novel approach for bounding the resolvent of H = - Δ + i ( A · + · A ) + V = : - Δ + L 1 for large energies. It is shown here that there exist a large integer m and a large number λ 0 so that relative to the usual weighted L 2 -norm, ( L ( - Δ + ( λ + i 0 ) ) - 1 ) m < 1 2 2 for all λ > λ 0 . This requires suitable decay and smoothness conditions on A , V . The estimate (2) is trivial when A = 0 , but difficult for large A since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

On the persistence of decorrelation in the theory of wave turbulence

Anne-Sophie de Suzzoni (2013)

Journées Équations aux dérivées partielles

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We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable u 0 with values in the Sobolev space H s with s big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by e i θ for all θ . We investigate about the persistence of the decorrelation...

The number of minimum points of a positive quadratic form

G. L. Watson

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CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index p Φ ( 0 , 1 ] and ω A l o c ( ) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space h ω Φ ( ) via the local grand maximal function. Let ρ ( t ) t - 1 / Φ - 1 ( t - 1 ) for all t ∈ (0,∞). We also introduce the BMO-type space b m o ρ , ω ( ) and establish the duality between h ω Φ ( ) and b m o ρ , ω ( ) . Characterizations of h ω Φ ( ) , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional k -dimensional Jacobian of a map u in the Sobolev space W 1 , k 1 which takes values in the sphere S k 1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S k 1 . In case M is polyhedral, the...

Remarks on L B I -subalgebras of C ( X )

Mehdi Parsinia (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let A ( X ) denote a subalgebra of C ( X ) which is closed under local bounded inversion, briefly, an L B I -subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of A ( X ) , we generalize the notion of z A β -ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of C ( X ) ...

On the Schröder equation

M. Kuczma

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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. C p solutions in *................................................ 74. Change of variables..............................................................................

The centralizer of a classical group and Bruhat-Tits buildings

Daniel Skodlerack (2013)

Annales de l’institut Fourier

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Let G be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let H be the centralizer of a semisimple rational Lie algebra element of G . We prove that the Bruhat-Tits building 𝔅 1 ( H ) of H can be affinely and G -equivariantly embedded in the Bruhat-Tits building 𝔅 1 ( G ) of G so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let j and j be maps from 𝔅 1 ( H ) to 𝔅 1 ( G ) which preserve the Moy–Prasad filtrations....

Further characterizations of Sobolev spaces

Hoai-Minh Nguyen (2008)

Journal of the European Mathematical Society

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Let ( F n ) n be a sequence of non-decreasing functions from [ 0 , + ) into [ 0 , + ) . Under some suitable hypotheses of ( F n ) n , we will prove that if g L p ( N ) , 1 < p < + , satisfies lim inf n N N F n ( | g ( x ) - g ( y ) | ) / | x - y | N + p d x d y < + , then g W 1 , p ( N ) and moreover lim n N N F n ( | g ( x ) - g ( y ) | ) / | x - y | N + p d x d y = K N , p N | g ( x ) | p d x , where K N , p is a positive constant depending only on N and p . This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some...

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p q &lt; and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...