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| Softcover ISBN: | 978-1-4704-4299-6 |
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| Softcover ISBN: | 978-1-4704-4299-6 |
| Product Code: | MEMO/267/1301 |
| List Price: | $85.00 |
| MAA Member Price: | $76.50 |
| AMS Member Price: | $68.00 |
| eBook ISBN: | 978-1-4704-6401-1 |
| Product Code: | MEMO/267/1301.E |
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| MAA Member Price: | $76.50 |
| AMS Member Price: | $68.00 |
| Softcover ISBN: | 978-1-4704-4299-6 |
| eBook ISBN: | 978-1-4704-6401-1 |
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| List Price: | $170.00 $127.50 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 267; 2020; 129 ppMSC: Primary 35
The author shows that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on \(\mathbb R^3+1\) constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter \(\lambda (t) = t^-1-\nu \) is sufficiently close to the self-similar rate, i. e. \(\nu >0\) is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ -\partial _t^2 + \partial _r^2 + \frac 2r\partial _r +V(\lambda (t)r) \] for suitable monotone scaling parameters \(\lambda (t)\) and potentials \(V(r)\) with a resonance at zero.
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Table of Contents
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Chapters
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1. Introduction
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2. Recall of the construction of $u_{\nu }$ and the distorted Fourier transform
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3. Growth properties of the forward linear parametrix
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4. Setting up the perturbative problem
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5. Nonlinear estimates
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6. Outline of the iterative scheme
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7. Control of the first iterate; contribution of the linear terms $\mathcal {R}(\tau , \underline {x}^{(0)})$
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8. Control of the first iterate; contribution of the nonlinear terms
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9. Iterative step; a priori control of higher iterates
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10. Preparations for the proof of convergence; refined estimates
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11. Improvements upon re-iteration
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12. Convergence of the iterative scheme
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13. Proof of the main technical theorem
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14. Appendix
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Additional Material
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The author shows that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on \(\mathbb R^3+1\) constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter \(\lambda (t) = t^-1-\nu \) is sufficiently close to the self-similar rate, i. e. \(\nu >0\) is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ -\partial _t^2 + \partial _r^2 + \frac 2r\partial _r +V(\lambda (t)r) \] for suitable monotone scaling parameters \(\lambda (t)\) and potentials \(V(r)\) with a resonance at zero.
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Chapters
-
1. Introduction
-
2. Recall of the construction of $u_{\nu }$ and the distorted Fourier transform
-
3. Growth properties of the forward linear parametrix
-
4. Setting up the perturbative problem
-
5. Nonlinear estimates
-
6. Outline of the iterative scheme
-
7. Control of the first iterate; contribution of the linear terms $\mathcal {R}(\tau , \underline {x}^{(0)})$
-
8. Control of the first iterate; contribution of the nonlinear terms
-
9. Iterative step; a priori control of higher iterates
-
10. Preparations for the proof of convergence; refined estimates
-
11. Improvements upon re-iteration
-
12. Convergence of the iterative scheme
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13. Proof of the main technical theorem
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14. Appendix
