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Inhomogeneous Strichartz Estimates for the Schrödinger Equation
Abstract
We study Strichartz estimates for the solution of the Cauchy problem associated with the inhomogeneous free Schrödinger equation in the case when the initial data is equal to zero, proving some new estimates for certain exponents and giving counterexamples for some others.
... A starting point for this result is provided by the endpoint Strichartz estimates of Keel-Tao [11]. However, in addition we also use the larger class of inhomogeneous Strichartz estimates developed [2,17,18,34]. The latter play a key role in lowering the regularity assumptions for the initial data in our theorem. ...
... Here we consider the Schrödinger equation (2.24), and prove the Strichartz bounds in Proposition 1.3. First, we introduce the endpoint Strichartz estimates of Keel-Tao [11] and the inhomogeneous Strichartz estimates developed by [2,17,18,34]. Then we use these to bound the linear and nonlinear part, respectively. ...
... Here we will use the endpoint pair (q, r) = (2, 2d d−2 ). Next, we state the inhomogenous Strichartz estimates, which summarize several known results, see [2,17,18,34]. Definition 6.1. We say that the pair (q, r) is Schrödinger-acceptable if ...
The skew mean curvature flow is an evolution equation for a d dimensional manifold immersed into , and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions . This extends the local well-posedness result in \cite{HT}.
... This indicates the possibility of (1.4) being true for non-admissible pairs. In fact, for non-admissible pairs, various authors including Cazenave, Weissler [7] in 1992, Kato [14] in 1994, Foschi [9] in 2005, Vilela [32] in 2007, Koh [22] in 2011, proved the inequality (1.4) for q, r,q,r satisfying (1.7) and other restrictions. But the problem of finding all possible exponents satisfying the estimate (1.4), is still open. ...
... To achieve these estimates, first, we improve the result of Vilela [32,Theorem 2.4]. We would like to point out that the author in [32] proved the estimate (1.4) in the zero potential case, whereas we in the following result establish the stronger estimate (1.11): Theorem 1.1. ...
... To achieve these estimates, first, we improve the result of Vilela [32,Theorem 2.4]. We would like to point out that the author in [32] proved the estimate (1.4) in the zero potential case, whereas we in the following result establish the stronger estimate (1.11): Theorem 1.1. Let V = 0 and (q, r), (q,r) satisfy (1.7), r,r > 2 along with ...
We establish inhomogeneous Strichartz estimates for the Schrödinger equation with singular and time-dependent potentials for some non-admissible pairs. Our work extends the results of Vilela (Trans Am Math Soc 359:2123–2136, 2007) and Foschi (J Hyperbolic Differ Equ 2:1–24, 2005), where they proved the results in the absence of potential. It also extends the works of Pierfelice (Asymptot Anal 47:1–18, 2006) and Burq et al. (J Funct Anal 203:519–549, 2003), who proved the estimates for admissible pairs. We also extend the recent work of Mizutani et al. (J Funct Anal 278:108350, 2020), and as an application of it, we improve the stability result of Kenig–Merle (Invent Math 166:645–675, 2006), which in turn establishes a proof (alternative to Yang in Commun Pure Appl Anal 20:77, 2020) of the existence of scattering solution for the energy-critical focusing NLS with inverse-square potentials.
... Here, · p,r denotes the norm of the Lorentz space L p,r (R d ) (for example, see [54]). To extend the range of uniform boundedness we devise a kind of T T * argument which combines the typical T T * argument ( [28,20,61]) with a representation formula for the projection operator Π λ (see (2.3)). The extension of the range of (1.5) in Theorem 1.2 to the off-diagonal p, q is in analogue with the enlargement of the admissible mixed norm spaces on which the inhomogeneous Strichartz estimate holds. ...
... The extension of the range of (1.5) in Theorem 1.2 to the off-diagonal p, q is in analogue with the enlargement of the admissible mixed norm spaces on which the inhomogeneous Strichartz estimate holds. In particular, Foschi [20] and Vilela [61] extended the admissible range of the inhomogeneous Strichartz estimate for the Schrödinger equation beyond the range given by the admissible pairs of the homogeneous Strichartz estimate (see Keel and Tao [28]). ...
... It seems natural to expect the uniform boundedness range in Theorem 1.2 is optimal except some endpoints but there is a gap between the range in Theorem 1.2 and the current necessary condition. In fact, (1.5) can be true only if (1/p, 1/q) ∈ P := {(a, b) ∈ [1/2, 1] × [0, 1/2] : a − b ≤ 2/d, (d − 1)/d ≤ a + b ≤ (d + 1)/d} as can be easily seen from duality and the lower bounds (3.10) and (3.12) in Section 3. So, uniform boundedness of Π λ p→q remains open for p, q such that (1/p, 1/q) ∈ P \ P when d ≥ 3. The current situation has similarity to that of the inhomogeneous Strichartz estimate for the Schrödinger equation whose optimal range of boundedness remains open for d ≥ 3 (see for example [20,61]). ...
We study - estimate for the spectral projection operator defined by the normalized Hermite functions in which are the eigenfucntions with the eigenvalue of the Hermite operator . Such estimates were previously available only for , equivalently with p=2 or q=2 (by argument) except for the estimates which are straightforward consequences of interpolation between those estimates. As shown in the works of Karadzhov, Thangavelu, and Koch and Tataru, the local and global estimates for are of different nature. Especially, exhibits complicated behavior near the set . Compared with the spectral projection operator associated with the Laplacian, the - boundedness of is not so well understood up to now for general p,q. In this paper we consider the - estimate for in a general framework including the local and global estimates with and undertake the work of characterizing the sharp bounds on . We establish various new sharp estimates in an extended range of p,q. First of all, we provide a complete characterization of the local estimate for which was first considered by Thangavelu. Secondly, for , we prove the endpoint - estimate for which has been left open since the work of Koch and Tataru. Thirdly, we extend the range of p,q for which the operator is uniformly bounded from to . As an application, we obtain new - resolvent estimates for the Hermite operator H and the Carleman estimate for the heat operator. This allows us to prove the strong unique continuation property of the heat operator for the potentials contained in .
... We establish inhomogeneous Strichartz Estimates for the Schrödinger equation with inverse square and time dependent potentials for non-admissible pairs. Our work extends the results provided by Vilela [21] and Foschi [6] where they proved the results in the absence of potential. It also extends the works of Pierfelice [19] and Burq, Planchon, Stalker, Tahvildar-Zadeh [3], who proved the estimates for admissible pairs. ...
... In 1998, Keel and Tao [10] proved the end point case. For non-admissible pairs, various authors including in 1992, Kato [9] in 1994 and recently Foschi [6] in 2005, Vilela [21] in 2006, Koh [15] in 2011, proved the inequality (1.4) for q, r,q,r satisfying (1.6) and other restrictions. Let us state the result of Vilela: ...
... Proof of Theorem 3.1. We use the techniques from Keel-Tao [10] and Vilela [21]. Note that by Duhamel's formula ...
We establish inhomogeneous Strichartz Estimates for the Schr{\"o}dinger equation with inverse square and time dependent potentials for non-admissible pairs. Our work extends the results provided by Vilela [21] and Foschi [6] where they proved the results in the absence of potential. It also extends the works of Pierfelice [19] and Burq, Planchon, Stalker, Tahvildar-Zadeh [3], who proved the estimates for admissible pairs. We also extend the recent work of Mizutani, Zhang, Zheng [17] and as an application of it, we improve the stability result of Kenig-Merle [12], which in turn establishes a proof (alternative to [23]) of existence of scattering solution for the energy critical focusing NLS with inverse square potential.
... Furthermore, it is known that there exist the exponent pairs (q, r) and (q,r) which do not satisfy the admissible condition, but the inhomogeneous estimate can be still valid; we refer the reader to T. Cazenave and F. Weissler [8] and T. Kato [26] for Schrödinger and to Harmse [20] and Oberlin [33] for wave with q = r. After that, D. Foschi [13] and M. Vilela [43] independently and greatly extended the range of the exponent pairs (q, r) and (q,r) for which the inhomogeneous Strichartz estimate holds. R. J. Taggart [40] generalized the inhomogeneous Strichartz estimate in an abstract mechanism. ...
... Remark 1.5. The set A ν 0 is an intersection of two sets, the first set is related to the known result of the inhomogeneous Strichartz estimates in [13,30,43,38] when V = 0 and the second set R ν 0 is from Theorem 1.1. The picture of inhomogeneous Strichartz estimate is far to be completed even though in the case without potential. ...
... we obtain . By the inhomogeneous Strichartz estimate in [13,30,43,38] for the Schrödinger without potential , we have Recall V = r −2 V 0 (θ) with V 0 ∈ C ∞ (S n−1 ), then one has V ∈ L n 2 ,∞ . Thus we obtain from the Strichartz estimate (3.5) , we need two lemmas. . ...
We study the uniform resolvent estimates for the Schr\"odinger operator with a Hardy-type singular potential. Let where is the usual Laplacian on and where and is a real function such that the operator is a strictly positive operator on . We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator .
... The problem of determining all possible inhomogeneous Strichartz estimates (1.4) for the wave and Schrödinger equations is still open. Foschi [7] and Vilela [20], following the scheme of Keel and Tao [11], independently obtained a wider range of Lebesgue exponents than those given in Theorem A. Foschi stated his results in the abstract framework above, while Vilela's statement is particular to the Schrödinger equation. ...
... s easily seen to be necessary. Also, in [7,20], several further conditions were shown to be necessary for (1.4) to hold, including ...
... In [7,20] it was shown that (1.4) holds under (1.7) and (1.8), and further assumptions which differ depending on whether d " 1, d " 2 or d ě 3; we refer the reader to these papers for the precise statement of their result (see also [12,13] for certain improvements to the range of exponents). Although (1.4) fails when pq, rq are such that (1.8) marginally fails, that is 1 q " dp 1 2´1 r q, the third author and Seo proved in [14] that certain weak-type estimates of the form ...
Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.
... The problem of determining all possible inhomogeneous Strichartz estimates (1.4) for the wave and Schrödinger equations is still open. Foschi [7] and Vilela [20], following the scheme of Keel and Tao [11], independently obtained a wider range of Lebesgue exponents than those given in Theorem A. Foschi stated his results in the abstract framework above, while Vilela's statement is particular to the Schrödinger equation. ...
... s easily seen to be necessary. Also, in [7,20], several further conditions were shown to be necessary for (1.4) to hold, including ...
... In [7,20] it was shown that (1.4) holds under (1.7) and (1.8), and further assumptions which differ depending on whether d " 1, d " 2 or d ě 3; we refer the reader to these papers for the precise statement of their result (see also [12,13] for certain improvements to the range of exponents). Although (1.4) fails when pq, rq are such that (1.8) marginally fails, that is 1 q " dp 1 2´1 r q, the third author and Seo proved in [14] that certain weak-type estimates of the form ...
Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.
... x L ∞ t was considered. (3) Inhomogeneous Strichartz estimates for non-admissible pairs for the free Schrödinger equation i∂ t u + ∆u = F with u| t=0 = 0 has been studied by several authors [10,39,41,15,58,44,45] under suitable condition on (p, q) (see [44]). The estimates (1.16) correspond to the endpoint cases for this condition. ...
... Compared with the case under the admissible condition, inhomogeneous estimates for nonadmissible pairs are less understood. By virtue of the abstract method established by [15,58], the dispersive (L 1 → L ∞ ) estimate for e −itH P ac implies inhomogeneous Strichartz estimates for non-admissible pairs satisfying a suitable condition which is wider than in Theorem 1.9. However, to obtain the dispersive estimate, much stronger condition on the potential V than that in this paper is usually required (see [22,25]). ...
... (4.1) is due to [56,18,41]. (4.2) for n 2(n−1) < s < 3n−4 2(n−1) was proved independently by [15] and [58]. The case s = n 2(n−1) or 3n−4 2(n−1) was settled recently by [45]. ...
This paper is concerned with global estimates for the Schr\"odinger operator with a real-valued potential which belongs to the scaling-critical Lebesgue space. Assuming that zero energy is neither an eigenvalue nor a resonance in a suitable sense, we show uniform Sobolev estimates for the resolvent and the same range of Lebesgue exponents as in the free case. As applications, we also prove (i) global-in-time inhomogeneous Strichartz estimates for the absolutely continuous part of the Schr\"odinger evolution group for all admissible and some non-admissible pairs; (ii) Keller type eigenvalue bounds for non-self-adjoint Schr\"odinger operators with complex-valued potentials. In the proof of above results, we study in details the space of zero resonant states which is defined as a subspace of the scaling-critical homogeneous Sobolev space.
... Indeed, (3) and the unitary property of the flow on L 2 .R d / are sufficient to obtain all known Strichartz estimates; endpoint cases are more delicate (see [8,17,30]). On any boundaryless Riemann manifold . ...
... Again, at 2;c , Remark 4 implies @ p 0 Á c D @ q 0 Á c and @ p 0˛c D @ q 0˛c . In turn, the functions ‚ 1;2 in Lemma 5 coincide as well, hence the functions E 1;2 defined in (29), (30) coincide also at 2;c . We abuse notation with E 1;2 as functions of .p ...
We consider an anisotropic model case for a strictly convex domain \Omega\subset\mathbb{R}^d of dimension d\geq 2 with smooth boundary \partial\Omega\neq\emptyset and we describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semiclassical flow: a loss of (\frac ht)^{1/4} occurs with respect to the boundaryless case due to repeated swallowtail-type singularities, and is proven optimal. Corresponding Strichartz estimates allow us to solve the cubic nonlinear Schrödinger equation on such a three-dimensional model convex domain, hence matching known results on generic compact boundaryless manifolds.
... , for all t = 0. Indeed, (3) and the unitary of the propagator on L 2 (R d ) are all that is required to obtain all known Strichartz estimates ; the endpoint cases are more delicate (see [21], [11], [35]). ...
... Proof. As t 2 √ γ ≥ 8 m 0 it follows, using (35), that only V N,h,γ (t, ·) with N ≥ 2 may provide significant contributions in G h,γ (t, ·). Let therefore N ≥ 2. We will show that the usual stationary phase applies in σ, s : the critical points satisfy 17). ...
We consider a model case for a strictly convex domain of dimension with smooth boundary and we describe dispersion for the semi-classical Schr{\"o}dinger equation with Dirichlet boundary condition. More specifically, we obtain the optimal fixed time decay rate for the linear semi-classical flow : a loss of occurs with respect to the boundary less case due to repeated swallowtail type singularities. The result is optimal and implies corresponding Strichartz estimates.
... The next one is an inhomogeneous estimate for non-admissible pairs. Proposition 2.6 ( [21,12,28,49]). Let 1 p 1 , p 2 , q 1 , q 2 ∞. The estimate ...
... We remark that the condition (2.5) comes from the acceptability. It is known that the condition (2.3) can be relaxed slightly (see [12,28,49]). However, we do not recall it since, under the diagonal assumption, the condition (2.3) is already weaker than (2.5). ...
In this paper, we introduce two minimization problems on non-scattering solutions to nonlinear Schr\"odinger equation. One gives us a sharp scattering criterion, the other is concerned with minimal size of blowup profiles. We first reformulate several previous results in terms of these two minimizations. Then, the main result of the paper is existence of minimizers to the both minimization problems for mass-subcritical nonlinear Schr\"odinger equations. To consider the latter minimization, we consider the equation in a Fourier transform of generalized Morrey space. It turns out that the minimizer to the latter problem possesses a compactness property, which is so-called almost periodicity modulo symmetry.
... x variable. To overcome this obstacle we make an use, similarly to [37], of a class of inhomogeneous Strichartz estimates with respect the x variable true in a range of Lebesgue exponents larger than the one given in the usual homogeneous estimates context (see [16] and [38]). Observe also that following our discussion above we are interested to work with a space which is an algebra w.r.t. to y variable, i.e. ...
... x,y -bounded. At this point we give the following definition borrowed by [16] (see also [38]) ...
We study the nonlinear Schr\"odinger equations posed on the product spaces
, for and , with any k-dimensional compact Riemaniann manifold. The main results concern
global well-posedness and scattering for small data solutions in non-isotropic
Sobolev fractional spaces. In the particular case of k=2, -scattering is
also obtained.
... have been known as inhomogeneous Strichartz estimates. The results obtained so far (see [3,6,7,10,11]) are not conclusive when it comes to determining the optimal values of the Lebesue exponents q, r,q andr for which the estimate (2) holds. Trying to further understand this problem, we [1] found new necessary conditions on these exponents values. ...
... Now, applying Lemma 1 to the oscillatory integral I N (t, x, y) in (10) yields ...
Let be the oscillatory
integral operators defined by where
is the unit ball in and We compare the
asymptotic behaviour as of the operator norms
for
all We prove that, except for the dimension
this asymptotic behaviour depends on the linearity or quadraticity of
the phase in s only. We are led to this problem by an observation on
inhomogeneous Strichartz estimates for the Schr\"{o}dinger equation.
... The following Strichartz estimates treat the case of non-admissible pairs. When the second index of the Lorentz space is equal to 2, such estimates are well known (see [38,41]). Here we consider the general case which will be used to prove the global existence. ...
In this paper, we consider the inhomogeneous nonlinear Schrödinger equation i∂tu+Δu=K(x)|u|αu,u(0)=u0∈H1(RN),N≥3,|K(x)|+|x||∇K(x)|≲|x|-b,0<b<min(2,N-2),0<α<(4-2b)/(N-2). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L2-space for a new range α0(b)<α<(4-2b)/N. The value α0(b) is the positive root of Nα2+(N-2+2b)α-4+2b=0, which extends the Strauss exponent known for b=0b=0. Our results improve the known ones for K(x)=μ|x|-b, μ∈C. For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of α. In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K.
... In higher dimensions d ≥ 3, uniform boundedness of Π λ p→q remains open for (1/p, 1/q) ∈ P \ P, where P : (1.10) holds true only if (1/p, 1/q) ∈ P as can be seen easily by duality and the lower bounds (5.2) and (5.3) in Section 5. The current situation seem similar to that of the inhomogeneous Strichartz estimate for the Schrödinger equation whose optimal range of boundedness remains open for d ≥ 3 (see, for example, [15,42]). ...
We study - bounds on the spectral projection operator associated to the Hermite operator in . We are mainly concerned with a localized operator for a subset and undertake the task of characterizing the sharp -- bounds. We obtain sharp bounds in extended ranges of p,q. First, we provide a complete characterization of the sharp -- bounds when E is away from . Secondly, we obtain the sharp bounds as the set E gets close to . Thirdly, we extend the range of p,q for which the operator is uniformly bounded from to .
... The proof is based on the argument in [31]. We first see that the estimate holds for the case V ≡ 0 by [7,30]. Then, by assumption (A4), we see that the wave operator W = lim t→∞ e itH e it∆ and its adjoint W * are bounded in L p , 2 p 6. Combining these facts, we obtain the estimate. ...
We consider the global dynamics of solutions to the 3d cubic nonlinear Schr\"odinger equation in the presence of an external potential, in the setting in which the equation admits both ground state solitons and excited solitons at small mass. We prove that small mass solutions with energy below that of the excited solitons either scatter to the ground states or grow their -norm in time. In particular, we give an extension of the result of Nakanishi [19] from the radial to the non-radial setting.
... More inhomogeneous estimates are available by the bilinear interpolation due to Keel-Tao [21]. The estimates were further refined by Foschi [13] (see also [41,35,28,24]). For the sake of brevity, we only record the estimates due to Keel-Tao, but remark that the estimates from [13] hold in a wider range. Theorem 1.6. ...
Sharp Strichartz estimates are proved for Schr\"odinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how dispersive properties are inherited from the constant coefficient case. Global Strichartz estimates follow provided that the derivatives of the coefficients are integrable. The estimates extend to structured coefficients of bounded variations. As applications we derive Strichartz estimates with additional derivative loss for wave equations with H\"older-continuous coefficients and solve nonlinear Schr\"odinger equations. Finally, we record spectral multiplier estimates, which follow from the Strichartz estimates by well-known means.
... The following Strichartz estimates treat the case of non-admissible pairs. When the second index of the Lorentz space is equal to 2 such estimates are well known (See [37,40]). Here we consider the general case which will be used to prove the global existence. ...
In this paper we consider the inhomogeneous nonlinear Schr\"odinger equation . We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted -space for a new range . The value is the positive root of which extends the Strauss exponent known for b=0. Our results improve the known ones for , and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of . Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential K.
... For related results on the Schrödinger equation we refer to[32,19,20]. ...
We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lam\'e operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.
... Lemma 2.5 (Inhomogeneous Strichartz, [19,38]). Let I ⊂ R. Suppose that (q, r) satisfy ...
In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schr\"odinger equation. Recently, Dodson [arXiv:2004.09618] studied the global well-posedness in a critical Sobolev space . In this paper, we aim to show that if the initial data belongs to to guarantee the local existence, then some extra weak space which is subcritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to for , then the corresponding solution exists globally and scatters.
... which is not clear to hold for all 0 < s < 1. In fact, to our knowledge, the best known inhomogeneous Strichartz estimates were proved independently by Foschi [17] and Vilela [38]. According to their results, the estimate (1.12) holds true provided that (1.14) and ...
... For more details on this type of estimates, see e.g. [12], [24]. ...
An Lp-theory of local and global solutions for the one dimensional nonlinear Schrödinger equations with pure power like nonlinearities is developed. Firstly, twisted local well-posedness results in scaling subcritical Lp-spaces are established for p<2. This extends Zhou's earlier results for the gauge-invariant cubic NLS equation. Secondly, by a similar functional framework, the global well-posedness for small data in critical Lp-spaces is proved, and as an immediate consequence, Lp′-Lp type decay estimates for the global solutions are derived, which are well known for the global solutions to the corresponding linear Schrödinger equation. Finally, global well-posedness results for gauge-invariant equations with large Lp-data are proved, which improve earlier existence results, and from which it is shown that the global solution u has a smoothing effect in terms of spatial integrability at any large time. Various Strichartz type inequalities in the Lp-framework including linear weighted estimates and bi-linear estimates for Duhamel type operators play a central role in proving the main results.
... In fact, according to the best known inhomogeneous Strichartz estimates for Schrödinger-type equations (including the biharmonic NLS), which were proved independently by Foschi [15] and Vilela [35], we need to check the following conditions: ...
We consider the following class of focusing -supercritical fourth-order nonlinear Schr\"odinger equations
\[
i\partial_t u - \Delta^2 u + \mu \Delta u = - |u|^\alpha u, \quad (t,x) \in \R \times \R^N,
\]
where , , and with if and if . By using the localized Morawetz estimates and radial Sobolev embedding, we establish the energy scattering below the ground state threshold for the equation with radially symmetric initial data. We also address the existence of finite time blow-up radial solutions to the equation. In particular, we show a sharp threshold for scattering and blow-up for the equation with radial data. Our scattering result not only extends the one proved by Guo \cite{Guo}, where the scattering was proven for , but also provides an alternative simple proof that completely avoids the use of the concentration/compactness and rigidity argument. In the case , our blow-up result extends an earlier result proved by Boulenger-Lenzmann \cite{BL}, where the finite time blow-up was shown for initial data with negative energy.
... We expect our result can be extended to higher dimensions d ≥ 5, since the only obstacle is the stability theory. However, it seems difficult to control the masscritical term in the Sobolev spaces H 1 or the weaker space H sp even by using exotic Strichartz estimates in [22,52] when we are trying to establish the stability theory. ...
We consider the Cauchy problem for the nonlinear Schr\"odinger equation with double nonlinearities with opposite sign, with one term is mass-critical and the other term is mass-supercritical and energy-subcritical, which includes the famous two-dimensional cubic-quintic nonlinear Schr\"odinger equaton. We prove global wellposedness and scattering in below the threshold for non-radial data when .
... If the time interval I is not specified, we take I = R. We also refer to [13,22,23,33,40,41] for more discussion on the homogeneous and inhomogeneous type Strichartz estimates. ...
We mainly consider the focusing biharmonic Schr\"odinger equation with a large radial repulsive potential V(x): \begin{equation*} \left\{ \begin{aligned} iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If , \ (i.e. the -supercritical and -subcritical case ), and for some , then we firstly prove a global well-posedness and scattering result for the radial data which satisfies that where , and Q is the ground state of . We crucially establish full Strichartz estimates and smoothing estimates of linear flow with a large poetential V, which are fundamental to our scattering results. Finally, based on the method introduced in \cite[T. Boulenger, E. Lenzmann, Blow up for biharmonic NLS, Ann. Sci. c. Norm. Supr., 50(2017), 503-544]{B-Lenzmann}, we also prove a blow-up result for a class of potential V and the radial data satisfying that M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \ {\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}} u_{0}\|_{L^{2}}>\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}}.
... We will also use an extension for not admissible pairs for the inhomogeneous Strichartz estimates, see [13] and [27] for a general treatment. ...
We consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist. We prove that under the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The ingredients of the proof are variational characterization of the ground states energy, a suitable profile decomposition theorem and localized virial estimates, enabling to carry out a Concentration/Compactness and Rigidity scheme. As a byproduct we show that in the stable regime, where standing states do not exist, any initial data in the energy space scatters.
... Indeed, Kato [25] proved that inhomogeneous estimates (37) hold true when the pairs (q, r) and (q,r) are Schrödinger acceptable and satisfy the scaling condition 1/q + 1/q = d/2(1 − 1/r − 1/r) in the range 1/r, 1/r > (d − 2)/(2d). Afterwards, for d > 2, Foschi [18] improved this result by looking for the optimal range of Lebesgue exponents for which inhomogeneous Strichartz estimates hold (results almost equivalent have recently obtained by Vilela [43]). Actually, this range is larger than the one given by admissible exponents for homogeneous estimates, as was shown by the following result [18,Proposition 24]. ...
The objective of this paper is to report on recent progress on Strichartz estimates for the Schr\"odinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in Wiener amalgam and modulation spaces. We present and compare the different technicalities. Then, we illustrate applications to well-posedness.
... However, if we only consider the inhomogeneous Strichartz estimate, we can obtain this endpoint estimate (5.2) by following the argument of [38] and [35], although at this moment we only have the microlocalized dispersive estimates (4.8)-(4.10). For more inhomogeneous estimates, we refer the reader to [22,61] where the propagator satisfies the classical dispersive estimate. ...
Consider the metric cone with the metric where the cross section Y is a compact -dimensional Riemannian manifold (Y,h). Let be the Friedrich extension positive Laplacian on X and let be the positive Laplacian on Y, and consider the operator \LL_V=\Delta_g+V_0 r^{-2} where V_0\in\CC^\infty(Y) such that is a strictly positive operator on . In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator \LL_V. To this end, we show a Sobolev inequality and a boundedness of a generalized Riesz transform in this setting. As an application, we study the well-posed theory and scattering theory for energy-critical wave equation with small data on this setting of dimension .
... Remark 4. To prove the long time perturbation on S α (I) one may invoke the proof of perturbation of [26] based on the inhomogeneous Strichartz estimates with extended range of Schrödinger case (α = 2) as stated in [19,35]. Unfortunately the best known results for inhomogeneous estimates of the fractional case do not include the end point ( q, r) = (2, 2d d−α ) (see [11]). ...
We consider the fractional nonlinear Schrödinger equation (FNLS) with non-local dispersion |∇|α and focusing energy-critical Hartree type nonlinearity [-(|x|-2α * |u|²)u]. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].
... Remark 2.1. The estimate (2.1) may be extended to other sets of admisssible pairs: see [5] and [12]. However, the linear estimate (2.1) is not valid for any other pairs and for u 0 R L 2 pR d q. ...
We consider the Cauchy problem for the nonlinear Schr\"odinger equation on , where the initial data is in . We prove local well-posedness for large ranges of p and discuss some global well-posedness results.
... Then by Theorem 2.4 in [46], ...
In this paper, we prove the scattering of radial solutions to high dimensional energy-critical nonlinear Schrödinger equations with regular potentials in the defocusing case.
... The following estimates are due to Kato [10], Foschi [8] and Vilela [17]. ...
In this paper, we consider a Hamilton equation which is a Nonlinear Schr\" odinger equation (NLS) coupled to an ODE. This system is a simplified model of NLS around soliton solutions. Following Nakanishi, we show scattering of small radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on in time norms.
... Remark 2.2. Further extension is obtained in [4,18,28]. However, the above version is sufficient for our purpose. ...
This paper is concerned with time global behavior of solutions to nonlinear Schr\"odinger equation with a non-vanishing condition at the spatial infinity. Under a non-vanishing condition, it would be expected that the behavior is determined by the shape of the nonlinear term around the non-vanishing state. To observe this phenomenon, we introduce a generalized version of the Gross-Pitaevskii equation, which is a typical equation involving a non-vanishing condition, by modifying the shape of nonlinearity around the non-vanishing state. It turns out that, if the nonlinearity decays fast as a solution approaches to the non-vanishing state, then the equation admits a global solution which scatters to the non-vanishing element for both time directions.
... If P (D) = 1 4π 2 ∆, (1.2) is a particular case of the classical Hardy-Littlewood-Sobolev inequality. When P (D) is non-elliptic, (1.2) is closely related to the inhomogeneous Strichartz estimates ( [11,8,9,17]) for the dispersive equations such as the wave and the Klein-Gordon equations (see [24,18]). For these equations, estimates (1.2) were first shown by Strichartz [24] for some p, q. ...
We study uniform Sobolev inequalities for the second order differential
operators P(D) of non-elliptic type. For we prove that the Sobolev
type estimate
holds with C independent of the first order and the constant terms of P(D)
if and only if and .
We also obtain restricted weak type endpoint estimates for the critical
, . As consequence, the result extends the class of functions
for which the unique continuation for the inequality holds.
... The (CP) theory gives the rest. We will need 2 new ingredients: [6], 2003, Vilela [32], 2007) holds, provided 1 q ...
... Then by Theorem 2.4 in M.C. Vilela [30], ...
In this paper, we prove the scattering in energy space for nonlinear
Schr\"odinger equations with potentials for small initial data in a suitable
weighted Sobolev space. The nonlinearity we consider is the pure power case
namely , we prove scattering for all , where
n>2 is the dimension. The index 1+2/n is sharp for scattering by the result of
W. Strauss. Our result generalizes the one-dimensional work of S. Cuccagna, V.
Georgiev, N. Visciglia and our previous work for three dimension to all n>2.
... Strichartz estimates with nonadmissible pairs first appeared in [4, Lemma 2.1], but were not used there for low energy scattering. They have subsequently been developed in [11,6,16]. In this paper, we use Strichartz estimates with non-admissible pairs along with the low energy scattering argument of [4]. ...
In this paper we consider the nonlinear Schr\"o\-din\-ger equation . We prove that if and
, then every nontrivial -solution blows up in finite or
infinite time. In the case and , we improve the existing low energy scattering results in dimensions . More precisely, we prove that if , then small data give rise to global, scattering
solutions in .
... Proof. This set of estimates for the free Schrödinger equation was proved by Kato [11] for q j < 2 * , and by Vilela [25] for q j = 2 * . It is transferred to the time independent equation e itH P c by Yajima's argument of bounded wave operators [26]. ...
Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a
single negative eigenvalue. It has solitons with negative small energy, which
are asymptotically stable, and, if the nonlinearity is focusing, then also
solitons with positive large energy, which are unstable. In this paper we
classify the global dynamics below the second lowest energy of solitons under
small mass and radial symmetry constraints.
... We also refer to [6,16,17,32] for more precise discussion on the inhomogeneous Strichartz estimates. ...
We consider the focusing -critical biharmonic Schr\"odinger
equation, and prove a global wellposedness and scattering result for the radial
data satisfying and where
and Q is the ground state of .
... In the classical case a = 2, such an estimate was first obtained by Strichartz [41] in L q t,x (R n+1 ) norms. Since then, Strichartz's estimate has been studied by many authors [15,24,5,21,14,46,25,30] naturally in more general mixed norms L q t (R; L r x (R n )). (See also [12,36,31] and references therein for different related norms.) ...
We obtain weighted Strichartz estimates for Schr\"odinger equations
, u(x,0)=f(x), of general orders
with radial data f,F with respect to the spatial variable x, whenever the
weight is in a Morrey-Campanato type class. This is done by making use of a
useful property of maximal functions of the weights together with
frequency-localized estimates which follow from using bilinear interpolation
and some estimates of Bessel functions. As a consequence, we give an
affirmative answer to a question posed in \cite{BBCRV} concerning weighted
homogeneous Strichartz estimates. We also apply the weighted estimates to
the well-posedness theory for the Schr\"odinger equations with time-dependent
potentials in the class.
... Long range effects are described in [12], in the case n = d − 1 and σ = 1 (cubic nonlinearity, which is exactly the threshold to have long range scattering in one dimension). A technical difference with [19] is that for the Cauchy problem, we do not make use of inhomogeneous Strichartz for non-admissible pairs like established in [5,7,20], and for scattering theory, such estimates are not needed when d 4. ...
We consider the defocusing nonlinear Schr{\"o}dinger equation in several
space dimensions, in the presence of an external potential depending on only
one space vari-able. This potential is bounded from below, and may grow
arbitrarily fast at infinity. We prove existence and uniqueness in the
associated Cauchy problem, in a suitable functional framework, as well as the
existence of wave operators when the power of the nonlinearity is sufficiently
large. Asymptotic completeness then stems from at least two approaches, which
are briefly recalled.
... In the classical case α = 2, the Strichartz estimates originated by Strichartz [23] have been extensively studied by many authors ( [9,14,2,12,8,24,15,17,18,5,6,20]). Over the past several years, considerable attention has been paid to the fractional order where 1 < α < 2 in the radial case (see [21,11,13] and references therein). ...
In this paper we obtain some new inhomogeneous Strichartz estimates for the
fractional Schr\"odinger equation in the radial case. Then we apply them to the
well-posedness theory for the equation
, , with radial
initial data below and radial potentials under the scaling-critical range .
We study the nonlinear Schrödinger equation posed on product spaces
The objective of this paper is to report on recent progress on Strichartz estimates for the Schrödinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in Wiener amalgam and modulation spaces. We present and compare the different technicalities. Then, we illustrate applications to well-posedness.El objetivo de este trabajo es reportar los progresos recientes sobre estimativas de Strichartz para la ecuación de Schrödinger y presentar el estado de arte. Estas estimativas han sido obtenidas en espacios de Lebesgue, espacios de Sobolev, y recientemente, en espacios de Wiener amalgamados y de modulación. Presentamos y comparamos los diferentes aspectos técnicos envueltos. Ilustramos los resultados con aplicaciones a buena colocación.
Foschi and Vilela in their independent works [3,13] showed that the range of (1/r,1/r~) for which the inhomogeneous Strichartz estimate (norm of matrix)∫0tei(t-s)δF(.,s)ds(norm of matrix)LtqLxr≲(norm of matrix)F(norm of matrix)Ltq~'Lxr~' holds for some q, q~ is contained in the closed pentagon with vertices A, B, B', P, P' except for the points P, P' (see Fig. 1). We obtain the estimate for the corner points P, P'.
We consider the focusing Ḣsc-critical biharmonic Schrödinger equation, and prove a global wellposedness and scattering result for the radial data u0 ∈ H2(ℝN) satisfying and where sc ∈ (0, 2) and Q is the ground state of Δ2Q + (2 − sc)Q − |Q|p−1Q = 0.
In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson system {i∂tu + Δu = -a-u-p-1u + b1uvx1 ; (t; x) ∈ ℝ × ℝ3; -Δv = b2(-u-2)x1 ; where a > 0; b1b2 > 0, 4/3 + 1 < p < 5. We first use a variational approach to give a dichotomy of blow-up and scattering for the solution of mass supercritical equation with the initial data satisfying J(uo) < J(R), where J stands for the Lagrange functional. The basic strategy is the concentration-compactness arguments from Kenig and Merle [17]. We overcome the main difficulties coming from the lack of scaling invariance and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). Furthermore, we adapt the standard method from [9] to obtain the blow up criterion.
We obtain space-time estimates on the solution to the Cauchy problem of damped fractional wave equation. We mainly focus on the linear equation.
The almost everywhere convergence of the solution to linear equations as is also studied, with the initial data satisfying certain regularity conditions.
We extend previously known Carleman estimates for the (time-dependent) Schrödinger operator i∂ t +Δ to a wider range for which inhomogeneous Strichartz estimates are known to hold. Then we apply them to obtain new results on unique continuation for the Schrödinger equation which include more general classes of potentials. Also, we obtain a unique continuation result for nonlinear Schrödinger equations.
Recently, Cordero and Nicola [4] obtained inhomogeneous Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation. In this paper we extend these estimates to a wider range for which the usual estimate ([8, 22]) in Lebesgue spaces is known to hold. This enables us to establish new Carleman estimates which are closely related to the extended estimates. Then we apply them to obtain new results on unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces.
Let u(x, t) be the solution of the Schrödinger or wave equation with L 2 initial data. We provide counterexamples to plausible conjectures involving the decay in t of the BMO norm of u(t, ·). The proofs make use of random methods, in particular, Brownian motion.
. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrodinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation. 1. Introduction In this paper we shall prove a Strichartz estimate in the following abstract setting 1 . Let (X; dx) be a measure space and H a Hilbert space. We'll write the Lebesgue norm of a function f : X ! C by kfk p j kfk L p (X) j Gamma Z X jf(x)j p dx Delta 1 p : Suppose that for each time t 2 R we have an operator U (t) : H ! L 2 (X) which obeys the energy estimate: ffl For all t and all f 2 H we have kU (t)fk L 2 x . kfkH (1) and that for some oe ? 0, one of the following decay estimates: 1991 Mathematics Subject Classification. 35L05, 35J10, 42B15, 46B70. 1 In the ...
. Let T be a bounded linear, or sublinear, operator from L p (Y ) to L q (X). To any sequence of subsets Y j of Y is associated a maximal operator T f(x) = sup j jT (f Delta Y j )(x)j. Under the hypotheses that q ? p and the sets Y j are nested, we prove that T is also bounded. Classical theorems of Menshov and Zygmund are obtained as corollaries. Multilinear generalizations of this theorem are also established. These results are motivated by applications to the spectral analysis of Schrodinger operators. 1. Introduction Let (X; ); (Y; ) be arbitrary measure spaces. Denote by E the characteristic function of a set E. To any sequence of measurable subsets fY n : n 2 Zg of Y and any operator T defined on L p (Y ) can be associated a maximal operator T f(x) = sup n jT (f Delta Yn )(x)j: The operator norm of T : L p (Y ) 7! L q (X) is denoted by kTk p;q . By a filtration of Y we will mean any sequence of subsets Y n ae Y which are nested: Y n ae Y n+1 for ev...
The L6 norm in space-time of a solution of the Klein-Gordon equation in two space-time dimensions is bounded relative to the Lorentz-invariant Hilbert space norm; the Lp norms for p ≥ 6 are bounded relative to certain similar larger Hilbert space norms, including the energy norm.
The endpoint Strichartz estimates for the Schrödinger equation is known to be false in two dimensions[7]. However, if one averages the solution in L2 in the angular variable, we show that the homogeneous endpoint and the retarded halfendoint estimates hold, but the full retarded endpoint fails. In particular, the original verisions of these estimates hold for radial data
We consider the initial value problem for the nonlinear Schrdinger equation in H1(Rn). We establish local existence and uniqueness for a wide class of subcritical nonlinearities. The proofs make use of a truncation argument, space-time integrability properties of the linear equation, anda priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.
We consider global solutions of the nonlinear Schrödinger equation(NLS)
where λ∈R and
. In particular, for
, we show that for every φεH1 (RN) such thatxφ(x)∈L2(RN), the solution of (NLS) with initial value φ(x)ei(b|x|2/4) is global and rapidly decaying ast→∞ ifb is large enough. Furthermore, by applying the pseudo-conformal transformation and studying the resulting nonautonomous nonlinear Schrödinger equation, we obtain both new results and simpler proofs of some known results concerning the scattering theory. In particular, we construct the wave operators for
. Also, we establish a low energy scattering theory for the same range of α and show that, at least for λ0, we prove asymptotic completeness for
.
We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationi∂u/∂t=(−1/2)Δu+V(t,x)u,u(0)=u
0. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u
0εC
1(ℝ,L
2) ∩C
0(ℝH
2(ℝn
)) foru
0εH
2(ℝn
). The conditions are general enough to accommodate moving singularities of type ∣x∣−2+ɛ(n≧4) or ∣x∣−n/2+ɛ(n≧3).