Displaying similar documents to “Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation”

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti, Veronica Felli, Andrea Malchiodi (2005)

Journal of the European Mathematical Society

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V ( x ) | x | α , 0 < α < 2 , and K ( x ) | x | β , β > 0 . Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1 , 2 ( N ) is proved under the assumption that σ < p < ( N + 2 ) / ( N 2 ) for some σ = σ N , α , β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of 𝒜 = V θ K 2 / ( p 1 ) , where θ = ( p + 1 ) / ( p 1 ) 1 / 2 .

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (2021)

Czechoslovak Mathematical Journal

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data u 0 X , where X { M 2 , q s ( ) , H σ ( 𝕋 ) , H s 1 ( ) + H s 2 ( 𝕋 ) } and q [ 1 , 2 ] , s 0 , or σ 0 , or s 2 s 1 0 . Moreover, if M 2 , q s ( ) L 3 ( ) , or if σ 1 6 , or if s 1 1 6 and s 2 > 1 2 we show that the Cauchy problem is unconditionally wellposed in X . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ...

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

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We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in N , u H 1 ( N ) , where N 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k O ( 2 ) denotes the dihedral...

H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

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Let A = -Δ + V be a Schrödinger operator on d , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of H A p if the maximal function s u p t > 0 | T t f ( x ) | belongs to L p ( d ) , where T t t > 0 is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space H A p admits a special atomic decomposition.

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...

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.

Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov (2016)

Czechoslovak Mathematical Journal

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Let G be a graph of order n and λ ( G ) the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in G . One of the main results of the paper is the following theorem: Let k 2 , n k 3 + k + 4 , and let G be a graph of order n , with minimum degree δ ( G ) k . If λ ( G ) n - k - 1 , then G has a Hamiltonian cycle, unless G = K 1 ( K n - k - 1 + K k ) or G = K k ( K n - 2 k + K ¯ k ) .

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...

On Schrödinger maps from T 1 to  S 2

Robert L. Jerrard, Didier Smets (2012)

Annales scientifiques de l'École Normale Supérieure

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We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from T 1 to  S 2 . This estimate yields some continuity properties of the flow map for the topology of  L 2 ( T 1 , S 2 ) , provided one takes its quotient by the continuous group action of  T 1 given by translations. We also prove that without taking this quotient, for any t &gt; 0 the flow map at time t is discontinuous as a map from 𝒞 ( T 1 , S 2 ) , equipped with the weak topology of  H 1 / 2 , to the space of distributions ( 𝒞 ( T 1 , 3 ) ) * . The argument relies...

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...

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 ) ...............................................................

A radial estimate for the maximal operator associated with the free Schrödinger equation

Sichun Wang (2006)

Studia Mathematica

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Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator S d and its associated global maximal operator S * * d by ( S d f ) ( x , t ) = 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, ( S * * d f ) ( x ) = s u p t | 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ | , f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, S d f is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values u ( η 1 , t ) , , u ( η q , t ) with 0 η 1 < η 2 < < η q < 1 . By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case ( P q ) of (P) in which the nonlinear term contains the sum S q [ u 2 ] ( t ) = q - 1 i = 1 q u 2 ( ( i - 1 ) q , t ) . Under suitable conditions, we prove that the solution of ( P q ) converges to the solution...

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang (2016)

Czechoslovak Mathematical Journal

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Let 1 = - Δ + V be a Schrödinger operator and let 2 = ( - Δ ) 2 + V 2 be a Schrödinger type operator on n ( n 5 ) , where V 0 is a nonnegative potential belonging to certain reverse Hölder class B s for s n / 2 . The Hardy type space H 2 1 is defined in terms of the maximal function with respect to the semigroup { e - t 2 } and it is identical to the Hardy space H 1 1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator b = b f - ( b f ) generated by the Riesz transform = 2 2 - 1 / 2 , where b BMO θ ( ρ ) , which is larger...

Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

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We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H k , p ( G ) , where k * , 1 p and G is either the open unit disk 𝔻 or the annular domain G s , 0 < s < 1 of the complex space . More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H k , p ( G ) from its behavior on any open connected subset I of the boundary G of G with respect to the L 1 -norm. Our results can be viewed as an improvement and generalization of...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

Jaeyoung Byeon, Kazunaga Tanaka (2013)

Journal of the European Mathematical Society

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We consider a singularly perturbed elliptic equation ϵ 2 Δ u - V ( x ) u + f ( u ) = 0 , u ( x ) > 0 on N , 𝚕𝚒𝚖 x u ( x ) = 0 , where V ( x ) > 0 for any x N . The singularly perturbed problem has corresponding limiting problems Δ U - c U + f ( U ) = 0 , U ( x ) > 0 on N , 𝚕𝚒𝚖 x U ( x ) = 0 , c > 0 . Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential V under possibly general conditions...