On wave equations with supercritical nonlinearities

Brenner, Philip; Kumlin, Peter

doi:10.1007/pl00000418

Cited by 10 publications

(13 citation statements)

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“…k = 1 or k = 2, or fails to be Lipschitz continuous. Among the recent works along these lines are [7], [24], 3 This (minimal) notion of local well-posedness is designed to provide meaning to rough solutions obtained through a limiting procedure of smooth functions. It differs subtly from what may be the "most natural" definition: For any R > 0 there exists T = T (R) > 0 such that the data-to-solution map S is uniformly continuous and uniquely defined from the ball {u0 ∈ H s x : u0 H s x < R} to the space C 0 ([−T, T ]; H s x ).…”

Section: Introductionmentioning

confidence: 99%

Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

Christ¹,

Colliander²,

Tao³

2003

ajm

374

465

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In a recent paper [18], Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities.In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces, in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the L 2 topology, despite the existence of global weak solutions at this regularity.Finally, we investigate the KdV equation at the endpoint regularity H −3/4 , and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval [−T, T ] and a locally Lipschitz continuous map taking the initial data in H −3/4 to a distributional solution u ∈ C 0 ([−T, T ]; H −3/4 ) which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.

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Section: Introductionmentioning

confidence: 99%

Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

Christ¹,

Colliander²,

Tao³

2003

ajm

374

465

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“…[10], [5] and [1]). Using constructions and ideas similar to those of Theorem 2.6 in the present paper, the authors proved in [6] that the solution operators for these nonlinear equations are not Lipschitz continuous as mappings from H 1 2 to L 2 for supercritical exponents ρ (ρ > 1 + 4 n−2 ). Related results for polynomial nonlinearities f have been obtained by Lebeau [14], [15].…”

Section: Definition 12 -A Family F Of Mappings Is Said To Admit Intmentioning

confidence: 73%

Nonlinear Maps between Besov and Sobolev spaces

Brenner¹,

Kumlin²

2010

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…In the case of Sobolev spaces X = X ′ = H (s) (R n ) and p = 2. The integral transform K allows us to avoid consideration in the phase space and to apply immediately the well-known decay estimates for the solution of the wave equation (operator EE) (see, e.g., [2,3,18]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%