Lectures on Nonlinear Wave Equations

By Christopher D. Sogge. 2005

330 citations recorded by the source corpus.

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Citations in the corpus

Ill-posedness for nonlinear Schrodinger and wave equations (2003, methodology): “Again, the energy method (see [18]) shows that this problem has a local solution inHk+1×Hk, which exists as long as the Hk+1×Hk norm stays bounded.”
Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, background): “The Strichartz estimates then read as follows; for a proof, see [8, 13, 37].”
Almost global wellposedness of the 2-D full water wave problem (2009, unknown): “lies that |g(reiθ,t)|2 . 1 r2 Z ∞ r s(|L0g|2 + X2 1 |L0∂αj |D| −1Hg|2)ds + t2 r4 X2 j=1 Z ∞ r s(|Ω0jg|2 +|Ω0j∂α j |D| −1Hg|2)ds. (2.23) Now in (2.23) we let g= Υk∂α l f. Using Lemma 1.2 on page 40 of [32], and (2.17), (2.3), we obtain |∂α l f(reiθ0,t)|2 . X k≤2 Z 2π 0 |Υk∂ αlf(re iθ,t)|2 dθ. 1 r2 X2”
The Global Nonlinear Stability of Minkowski Space for Self-gravitating Massive Fields (2015, methodology): “, for instance, [32]) while for w2 we apply (3.”
Strichartz estimates for the wave equation on manifolds with boundary (2008, background): “If this is combined with the aforementioned boundary estimates in [5] and [6] one can prove the global existence results in these papers by using the now-standard arguments that are found in [22] for convex obstacles, and [19] and [25] for the case where Ω = R3.”
On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles (2008, background): “orem 1.1 for the Dirichlet-wave equation outside of nontrapping obstacles under a somewhat more restrictive smallness assumption was obtained in [8]. Also, when Ω = R3, ∆ g = ∆, it was shown in Sogge [35] that, for the spherically symmetric case, the variant of the condition (1.7) saying that the H˙”
Weighted Strichartz estimates with angular regularity and their applications (2008, background): “ata (f,g) ∈ H˙ sc × H˙ sc−1, where s c is the minimal regularity assumption. When the small radial data (f,g) ∈ H˙ sc ×H˙ c−1, there are results dealing with either n ≤ 4 or p > ph, see e.g. Sogge [34], LindbladSogge [24] and Hidano [14]. We can apply the above estimates to this problem for the s”
Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions (2009, unknown): “ince the above scaling leaves the energy invariant, the (NLW) problem is referred to as “energy critical”. Local well-posedness for the Cauchy problem (NLW) has been studied in many papers (see, e.g. [25, 9, 22, 28, 29, 30, 13, 17]). Here we recall a version of the local well-posedness result as pre”
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation (2006, unknown): “Introduction In this paper we consider the energy critical non-linear wave equation ∂2t u− ∆u = ± |u| 4 N−2 u (x, t) ∈ RN × R u ∣∣ t=0 = u0 ∈ Ḣ1(RN ) ∂tu ∣∣ t=0 = u1 ∈ L2(RN ) Here the − sign corresponds to the defocusing problem, while the + sign corresponds to the focusing problem.”
Nonlinear dispersive equations : local and global analysis (2006, methodology): “See [Sog] for a more detailed and general treatment of this vector fields approach.”
Global wellposedness of the 3-D full water wave problem (2011, unknown): “40 of [22], and (2.”
Global existence of null-form wave equations in exterior domains (2006, background): “of (2.28) yields the desired estimate. 3. Decay estimates Classically, the necessary decay to prove long-time existence is aﬀorded to us by the Klainerman-Sobolev inequalities (see [14]; see also [7],[30]). These inequalities, however, require the use of the Lorentz rotations which does not seem per”
The Glassey conjecture with radially symmetric data (2011, background): “Besides the critical regularity sc = n/2−2/(p−1), there is one more Sobolev regularity, namely sd = 1/2−1/p (see Sogge [20] Section IV.”