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Counterexamples to Strichartz estimates for the wave equation in domains. (English) Zbl 1201.35060
For the solutions to the wave equation in the flat space \(\mathbb{R}^n\)
\[ (\partial_t^2-\Delta)u(t,x)=0, \quad u(0,x)=f, \quad \partial_t u(0,x)=g,\quad (t,x)\in \mathbb R\times\mathbb R^n, \]
the Strichartz estimates are a family of space time integrability estimates and state that
\[ \|u\|_{L^q_t L^r_x}\leq C \big(\|f\|_{\dot H^s}+\|g\|_{\dot H^{s-1}}\big) \]
for
\[ \frac{1}{q}\leq \frac{n-1}{2} \bigg(\frac{1}{2}-\frac{1}{r}\bigg), \quad q\geq 2, \quad (q,r)\neq \Bigg(\max \bigg\{2,\frac{4}{n-1}\bigg\},\infty\Bigg) \]
and \(s=\frac{n}{2}-\frac{n}{r}-\frac{1}{q}\) [see D. Fang and C. Wang, Nonlinear Anal., Theory Methods Appl. 65, No. 3 (A), 697–706 (2006; Zbl 1096.35026)].
There have been many works in proving the Strichartz estimates in various settings. If the domain is the exterior in \(\mathbb{R}^n\) of a compact set with smooth nontrapping boundary, the estimates were established for the full range of exponents \((q,r,s)\) with \(q>2\) and \(r<\infty\) [see K. Hidano, J. Metcalfe, H. F. Smith, C. D. Sogge and Y. Zhou, Trans. Am. Math. Soc. 362, No. 5, 2789–2809 (2010; Zbl 1193.35100)]. If the domain is a compact manifold without boundary, the local in time estimates have also been established for the full range of exponents \((q,r,s)\) with \(r<\infty\) [see D. Tataru, J. Am. Math. Soc. 15, No. 2, 419–442 (2002; Zbl 0990.35027)]. For a manifold with boundary, together with either Dirichlet or Neumann homogeneous boundary conditions, there have been some works showing that the local in time Strichartz estimates hold for certain subset of the range of indices [see M. D. Blair, H. F. Smith and C. D. Sogge, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 5, 1817–1829 (2009; Zbl 1198.58012)].
In this work, for a sample 2-dimensional strictly convex domain with Dirichlet boundary condition, the author provide a counterexample to the local Strichartz estimates for
\[ \frac{3}{q}+\frac{1}{r}>\frac{15}{24}. \]
More precisely, for \((q,r)\) with \(\frac{2}{q}=\frac{1}{2}-\frac{1}{r}\) and \(4<r<\infty\), the author proves that to have the local Strichartz estimates hold, we need to add at least \(\frac{1}{6}(\frac{1}{4}-\frac{1}{r})\) on \(s\).

MSC:
35B45 A priori estimates in context of PDEs
35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
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[1] Blair, M.D., Smith, H.F., Sogge, C.D.: Strichartz estimates for the wave equation on manifolds with boundary. To appear in Ann. Inst. H. Poincaré, Anal. Non Liréaire · Zbl 1198.58012
[2] Burq N., Lebeau G., Planchon F.: Global existence for energy critical waves in 3-D domains. J. Am. Math. Soc. 21(3), 831–845 (2008) · Zbl 1204.35119
[3] Eskin G.: Parametrix and propagation of singularities for the interior mixed hyperbolic problem. J. Analyse Math. 32, 17–62 (1977) · Zbl 0375.35037
[4] Ginibre J., Velo G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 2(4), 309–327 (1985) · Zbl 0586.35042
[5] Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. In: Partial differential operators and mathematical physics (Holzhau, 1994). Oper. Theory Adv. Appl., vol. 78, pp. 153–160. Birkhäuser, Basel (1995) · Zbl 0839.35016
[6] Hörmander, L.: The analysis of linear partial differential operators III. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274. Springer, Berlin, Pseudodifferential operators (1985) · Zbl 0601.35001
[7] Kapitanskiĭ L.V.: Some generalizations of the Strichartz-Brenner inequality. Algebra i Analiz 1(3), 127–159 (1989)
[8] Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998) · Zbl 0922.35028
[9] Koch H., Smith H.F., Tataru D.: Subcritical L p bounds on spectral clusters for Lipschitz metrics. Math. Res. Lett. 15(5), 993–1002 (2008) · Zbl 1159.58017
[10] Lebeau, G.: Estimation de dispersion pour les ondes dans un convexe. In: Journées ”Équations aux Dérivées Partielles” (Evian, 2006). See http://www.numdam.org/numdam-bin/fitem?id=JEDP_2006____A7_0
[11] Lindblad H., Sogge C.D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130(2), 357–426 (1995) · Zbl 0846.35085
[12] Martinez A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext. Springer, New York (2002) · Zbl 0994.35003
[13] Sjöstrand, J.: Singularités analytiques microlocales. In: Astérisque, 95, Astérisque, vol. 95, pp. 1–166. Soc. Math. France, Paris (1982)
[14] Smith H.F., Sogge C.D.: On the critical semilinear wave equation outside convex obstacles. J. Am. Math. Soc. 8(4), 879–916 (1995) · Zbl 0860.35081
[15] Smith H.F., Sogge C.D.: On the L p norm of spectral clusters for compact manifolds with boundary. Acta Math. 198(1), 107–153 (2007) · Zbl 1189.58017
[16] Strichartz R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977) · Zbl 0372.35001
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