Endpoint Strichartz estimates
01 Oct 1998-American Journal of Mathematics (Johns Hopkins University Press)-Vol. 120, Iss: 5, pp 955-980
TL;DR: In this paper, an abstract Strichartz estimate for the wave equation (in dimension n ≥ 4) and for the Schrodinger equation (n ≥ 3) was proved.
Abstract: 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.
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TL;DR: In this paper, it was shown that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller compared to that of a standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5.
Abstract: We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.
945 citations
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...oblem (CP) ( i∂tu+∆u+|u| 4 N−2 u = 0 (x,t) ∈ RN ×R u|t=0 = u0 ∈ H˙ 1(RN) i.e., the H˙ 1 critical, focusing, Cauchy problem for NLS. We need two preliminary results. Lemma 2.1 (Strichartz estimate [7],[14]). We say that a pair of exponents (q,r) is admissible if 2 q + N r = 2 and 2 ≤ q, r ≤ ∞. Then, if 2 ≤ r ≤ 2 N−2 (N ≥ 3) (or 2 ≤ r < ∞, N = 2 and 2 ≤ r ≤ ∞, N = 1) we have i) eit∆h Lq tL r x ≤ C ||...
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...x)) ▽ v +{▽f(u(x))) −▽f(v(x))} ▽v, so |▽xf(u(x)) −▽xf(v(x))| ≤ C |u(x)| 4 N−2 |▽u−▽v| +C |▽v| n |u| 6−N N−2 +|v| 6−N N−2 o |u−v|. Remark 2.4. In the estimate ii) in Lemma 2.1, one can actually show: ([14]) ii’) ′ Z+∞ −∞ ei(t−τ)∆g(−,τ)dτ Lq t L r x ≤ C ||g||Lm t L n′ x , where (q,r), (m,n) are any pair of admissible indices as in i) of Lemma 2.1. Let us define S(I),W(I) norm for an interval I by ||v||S(...
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TL;DR: In this article, the authors obtained global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3, which is energy-critical.
Abstract: We obtain global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain (4) and Grillakis (20), which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain (4), but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant (first used in (12), (13)). The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of L 2 mass in fre- quency space, rules out the possibility of energy concentration.
485 citations
TL;DR: In this article, Strichartz estimates with fractional loss of derivatives for the Schrodinger equation on any Riemannian compact manifold were shown to have low regularity local well-posedness results in any dimension.
Abstract: kklWe prove Strichartz estimates with fractional loss of derivatives for the Schrodinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local well-posedness results in any dimension, as well as global existence results for the Cauchy problem of nonlinear Schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of cubic defocusing nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.
480 citations
Cites background or methods from "Endpoint Strichartz estimates"
...This forces us to use Keel-Tao’s endpoint estimate (1.6) with p = 2, q = 6 which corresponds exactly to a gain of 12 derivative....
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...(Keel-Tao [22]) Let (X,S,µ) be a σ-finite measured space, and U : R→ B(L2(X,S,µ)) be a weakly measurable map satisfying, for some A, σ > 0, (i) ‖U(t)‖L2→L2 ≤ A, t ∈ R, (ii) ‖U(t1)U(t2) f‖L∞ ≤ A |t1 − t2|σ ‖f‖L1 , t1, t2 ∈ R....
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...Finally, we apply the abstract Keel-Tao TT result and we obtain Strichartz inequalities with loss of derivatives by summing up on the time intervals....
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...Firstly we recall the nonhomogeneous version of Keel-Tao’s result [22]....
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...and satisfy moreover p ≥ 2, (p, q) = (2,∞) (for the case p = 2, we refer to the paper by Keel-Tao [22])....
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TL;DR: In this paper, the authors established dispersive estimates for solutions to the linear Schrodinger equation in three dimensions 0.1, 0.2 and 0.3, respectively.
Abstract: In this paper we establish dispersive estimates for solutions to the linear Schrodinger equation in three dimensions 0.1
$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$
where V(t,x) is a time-dependent potential that satisfies the conditions $$\sup_{t}\|V(t,\cdot)\|_{L^{\frac{3}{2}}(\mathbb{R}^3)} + \sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \int_{-\infty}^\infty\frac{|V(\hat{\tau},x)|}{|x-y|}\,d\tau\,dy < c_0.$$
Here c
0 is some small constant and $V(\hat{\tau},x$)
denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·)∈L
∞
t
(L
2
x
(ℝ3))∩L
2
t
(L
6
x
(ℝ3)) for any f∈L
2(ℝ3) satisfying the dispersive inequality 0.2
$$\|\psi(t)\|_{\infty} \le C|t-s|^{-\frac32}\,\|f\|_1 \text{\ \ for all times $t,s$.}$$
For the case of time independent potentials V(x), (0.2) remains true if $$\int_{\mathbb{R}^6} \frac{|V(x)|\;|V(y)|}{|x-y|^2} \, dxdy <(4\pi)^2\text{\ \ \ and\ \ \ }\|V\|_{\mathcal{K}}:=\sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|V(y)|}{|x-y|}\,dy<4\pi.$$
We also establish the dispersive estimate with an e-loss for large energies provided $\|V\|_{\mathcal{K}}+\|V\|_2<\infty$
. Finally, we prove Strichartz estimates for the Schrodinger equations with potentials that decay like |x|-2-e in dimensions n≥3, thus solving an open problem posed by Journe, Soffer, and Sogge.
433 citations
TL;DR: In this article, the authors consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrodinger equation (NLS) i∂tu + Δu + |u|2u = 0 scatter, i.e., approach the solution to a linear Schroffinger equation as t → ±∞.
Abstract: We consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrodinger equation (NLS) i∂tu + Δu + |u|2u = 0 scatter, i.e., approach the solution to a linear Schrodinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities \({\|u_0\|_{L^2}\|
abla u_0\|_{L^2}}\) and M[u]E[u], where u0 denotes the initial data, and M[u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution eitQ(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if M[u]E[u] \|Q\|_{L^2}\|
abla Q\|_{L^2}}\), then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS.
326 citations
References
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Book•
01 Feb 1971
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
Abstract: Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.
9,595 citations
Book•
18 Nov 2011
TL;DR: In this paper, the authors define the Riesz-Thorin Theorem as a necessary and sufficient condition for interpolation spaces, and apply it to approximate spaces in the context of vector spaces.
Abstract: 1. Some Classical Theorems.- 1.1. The Riesz-Thorin Theorem.- 1.2. Applications of the Riesz-Thorin Theorem.- 1.3. The Marcinkiewicz Theorem.- 1.4. An Application of the Marcinkiewicz Theorem.- 1.5. Two Classical Approximation Results.- 1.6. Exercises.- 1.7. Notes and Comment.- 2. General Properties of Interpolation Spaces.- 2.1. Categories and Functors.- 2.2. Normed Vector Spaces.- 2.3. Couples of Spaces.- 2.4. Definition of Interpolation Spaces.- 2.5. The Aronszajn-Gagliardo Theorem.- 2.6. A Necessary Condition for Interpolation.- 2.7. A Duality Theorem.- 2.8. Exercises.- 2.9. Notes and Comment.- 3. The Real Interpolation Method.- 3.1. The K-Method.- 3.2. The J-Method.- 3.3. The Equivalence Theorem.- 3.4. Simple Properties of ??, q.- 3.5. The Reiteration Theorem.- 3.6. A Formula for the K-Functional.- 3.7. The Duality Theorem.- 3.8. A Compactness Theorem.- 3.9. An Extremal Property of the Real Method.- 3.10. Quasi-Normed Abelian Groups.- 3.11. The Real Interpolation Method for Quasi-Normed Abelian Groups.- 3.12. Some Other Equivalent Real Interpolation Methods.- 3.13. Exercises.- 3.14. Notes and Comment.- 4. The Complex Interpolation Method.- 4.1. Definition of the Complex Method.- 4.2. Simple Properties of ?[?].- 4.3. The Equivalence Theorem.- 4.4. Multilinear Interpolation.- 4.5. The Duality Theorem.- 4.6. The Reiteration Theorem.- 4.7. On the Connection with the Real Method.- 4.8. Exercises.- 4.9. Notes and Comment.- 5. Interpolation of Lp-Spaces.- 5.1. Interpolation of Lp-Spaces: the Complex Method.- 5.2. Interpolation of Lp-Spaces: the Real Method.- 5.3. Interpolation of Lorentz Spaces.- 5.4. Interpolation of Lp-Spaces with Change of Measure: p0 = p1.- 5.5. Interpolation of Lp-Spaces with Change of Measure: p0 ? p1.- 5.6. Interpolation of Lp-Spaces of Vector-Valued Sequences.- 5.7. Exercises.- 5.8. Notes and Comment.- 6. Interpolation of Sobolev and Besov Spaces.- 6.1. Fourier Multipliers.- 6.2. Definition of the Sobolev and Besov Spaces.- 6.3. The Homogeneous Sobolev and Besov Spaces.- 6.4. Interpolation of Sobolev and Besov Spaces.- 6.5. An Embedding Theorem.- 6.6. A Trace Theorem.- 6.7. Interpolation of Semi-Groups of Operators.- 6.8. Exercises.- 6.9. Notes and Comment.- 7. Applications to Approximation Theory.- 7.1. Approximation Spaces.- 7.2. Approximation of Functions.- 7.3. Approximation of Operators.- 7.4. Approximation by Difference Operators.- 7.5. Exercises.- 7.6. Notes and Comment.- References.- List of Symbols.
4,025 citations
TL;DR: In this paper, the authors give a complete solution when S is a quadratic surface given by the duality argument for the special case S {(x, y) yZ xz I} and give the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic invariant norm.
Abstract: A simple duality argument shows these two problems are completely equivalent ifp and q are dual indices, (]/) + (I/q) ]. ]nteresl in Problem A when S is a sphere stems from the work of C. Fefferman [3], and in this case the answer is known (see [l I]). Interest in Problem B was recently signalled by 1. Segal [6] who studied the special case S {(x, y) yZ xz I} and gave the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic-invariant norm. In this paper we give a complete solution when S is a quadratic surface given by
1,351 citations
Additional excerpts
...These results extend a long line of investigation going back to a specific space-time estimate for the linear Klein-Gordon equation in [18] and the fundamental paper of Strichartz [24] drawing the connection to the restriction theorems of Tomas and Stein....
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1,309 citations