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Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. (English) Zbl 0990.35027
For Parts I and II see [Am. J. Math. 122, 349-376 (2000; Zbl 0959.35125) and ibid. 123, 385-423 (2001; Zbl 0988.35037)].
From the introduction: We are interested in Strichartz estimates for a second order operator of the form $$P(x,D)=\partial_i g^{ij}(x) \partial_j$$, which is strongly hyperbolic with respect to time.
In Part II we have shown that the full estimates hold in all dimensions for operators with $$C^2$$ coefficients, and we also obtained appropriate weaker estimates for operators with $$C^s$$ coefficients for $$0<s<2$$. The main goal of this article is to prove that the estimates are still true if the coefficients have two derivatives in $$L^1(L^\infty)$$, and then to explore some consequences of this result.
It is assumed that the matrices $$(g^{ij}(x))$$, $$(g^{ij} (x))^{-1}$$ are uniformly bounded and of signature $$(1,n)$$, furthermore, the surfaces $$x_0=$$const are space-like uniformly in $$x$$, i.e. that $$g^{00}> c>0$$.

##### MSC:
 35B45 A priori estimates in context of PDEs 35L10 Second-order hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations
##### Keywords:
nonlinear wave equation
Full Text:
##### References:
 [1] Hajer Bahouri and Jean-Yves Chemin, Équations d’ondes quasilinéaires et effet dispersif, Internat. Math. Res. Notices 21 (1999), 1141 – 1178 (French). · Zbl 0938.35106 [2] Hajer Bahouri and Jean-Yves Chemin, Équations d’ondes quasilinéaires et estimations de Strichartz, Amer. J. Math. 121 (1999), no. 6, 1337 – 1377 (French, with French summary). · Zbl 0952.35073 [3] Philip Brenner, On \?_{\?}-\?_{\?$$^{\prime}$$} estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251 – 254. · Zbl 0321.35052 [4] Jean-Marc Delort, F.B.I. transformation, Lecture Notes in Mathematics, vol. 1522, Springer-Verlag, Berlin, 1992. Second microlocalization and semilinear caustics. · Zbl 0760.35004 [5] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50 – 68. · Zbl 0849.35064 [6] Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273 – 294 (1977). · Zbl 0361.35046 [7] L. V. Kapitanskiĭ, Estimates for norms in Besov and Lizorkin-Triebel spaces for solutions of second-order linear hyperbolic equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 171 (1989), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 20, 106 – 162, 185 – 186 (Russian, with English summary); English transl., J. Soviet Math. 56 (1991), no. 2, 2348 – 2389. · Zbl 0759.35014 [8] Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891 – 907. · Zbl 0671.35066 [9] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955 – 980. · Zbl 0922.35028 [10] Sergiu Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982), no. 1, 73 – 98. · Zbl 0502.35015 [11] Hans Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math. 118 (1996), no. 1, 1 – 16. · Zbl 0855.35080 [12] Hans Lindblad, Counterexamples to local existence for quasilinear wave equations, Math. Res. Lett. 5 (1998), no. 5, 605 – 622. · Zbl 0932.35149 [13] Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65 – 130. · Zbl 0776.58037 [14] Hart F. Smith, A parametrix construction for wave equations with \?^{1,1} coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797 – 835 (English, with English and French summaries). · Zbl 0974.35068 [15] Hart F. Smith and Christopher D. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett. 1 (1994), no. 6, 729 – 737. · Zbl 0832.35018 [16] Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705 – 714. · Zbl 0372.35001 [17] Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), no. 2, 349 – 376. · Zbl 0959.35125 [18] Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients II. Amer. J. Math., 123(3):385-423, 2001. · Zbl 0988.35037 [19] Michael E. Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. · Zbl 0746.35062
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