Local regularity of nonlinear wave equations in three space dimensions

Journal Article · Fri Jan 01 04:00:00 EST 1993 · Communications in Partial Differential Equations; (United States)

DOI:https://doi.org/10.1080/03605309308820925· OSTI ID:6372187

Ponce, G; Sideris, T C ^[1]

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1. Univ. of California, Santa Barbara (United States)

This note examines the question of the minimal Sobolev regularity required to construct local solutions to the Cauchy problem for three-dimensional nonlinear wave equations of the form [partial derivative][sub t][sup 2]u [minus] [Delta]u = G(u, Du), (1), u(0) = f, [partial derivative][sub t]u(0) = g, (2), where Du = ([partial derivative][sub t]u, [del][sub x]u). For nonintegral s, this requires the use of an inequality. Thus it is crucial to gain control of the L[sup [ell][minus]1] norm in time of the maximum norm of the gradient of the solution in space. This is usually done by using the Sobolev imbedding theorem which leads to the restriction s > n/2+1 for the Sobolev exponent. The authors show that in three space dimensions (the case to which they restrict themselves throughout the paper), the lower bound for the Sobolev exponent can be reduced from 5/2 to s([ell]) [identical to] max[l brace]2, (5[ell] [minus] 7)/(2[ell] [minus] 2)[r brace] when the nonlinearity G in (1) grows no faster than order [ell] in Du. Thus, as [ell] [yields] [infinity] the classical result s > 5/2 is approached. They also show that this is sharp in the sense that the quantity [parallel]f[parallel][sub H[sup s([ell])]] + [parallel]g[parallel][sub H[sup s([ell])[minus]1]] is not sufficient, in general, to control the local existence time of solutions (for [ell] [ge] 3). The authors return to the spherically symmetric case at the end of the paper. For general nonlinearities, such a result just fails. It is shown how to apply instead a space-time estimate for the free solution due to Marshall and Pecher, as an extension of the work of Strichartz. 9 refs.

• OSTI ID: 6372187
• Journal Information: Communications in Partial Differential Equations; (United States), Journal Name: Communications in Partial Differential Equations; (United States) Vol. 18:1-2; ISSN 0360-5302; ISSN CPDIDZ
• Country of Publication: United States
• Language: English

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CAUCHY PROBLEM
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
SPACE-TIME
THREE-DIMENSIONAL CALCULATIONS
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