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Local regularity of nonlinear wave equations in three space dimensions. (English) Zbl 0803.35096

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 \[ \partial^ 2_ tu - \Delta u = G(u,Du), \quad u(0) = f, \quad \partial_ tu (0) = g, \] where \(Du = (\partial_ tu, \nabla_ xu)\). If for example \(G = | Du |^ \ell\), then for any \(s \geq 0\), the classical energy method gives the bound \[ \bigl \| u(t) \bigr \|_{H^ s (\mathbb{R}^ n)} + \bigl \| \partial_ tu (t) \bigr \|_{H^{s-1} (\mathbb{R}^ n)} \leq \bigl( \| f \|_{H^ s (\mathbb{R}^ n)} + \| g \|_{H^{s-1} (\mathbb{R}^ n)} \bigr) \exp \left( C(t) \int^ t_ 0 \bigl \| Du (\tau) \bigr \|^{\ell-1}_{L^ \infty (\mathbb{R}^ n)} d \tau \right). \] This is usually done by using the Sobolev imbedding theorem which leads to the restriction \(s<n/2+1\) for the Sobolev exponent.
We show that in three space dimensions (the case to which we restrict ourselves throughout the paper), the lower bound for the Sobolev exponent can be reduced from 5/2 to \(s(\ell) \equiv \max \{2, (5 \ell - 7)/(2 \ell - 2)\}\) when the nonlinearity \(G\) in (1) grows not faster than order \(\ell\) in \(Du\). Thus, as \(\ell \to \infty\) the classical result \(s>5/2\) is approached. We also show that this is sharp in the sense that the quantity \(\| f \|_{H^{s (\ell)}} + \| g \|_{H^{s (\ell) - 1}}\) is not sufficient, in general, to control the local existence time of solutions (for \(\ell \geq 3)\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
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References:

[1] Coifman R., Au-delaádes operateurs pseudo-difféntiels, Asteriques 57 (1973)
[2] DOI: 10.1002/cpa.3160340103 · Zbl 0453.35060 · doi:10.1002/cpa.3160340103
[3] DOI: 10.1002/cpa.3160410704 · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[4] Marshall B., Proc.Conf. Harmonic Anal. pp 638– (1981)
[5] Pecher H., Math.Z. 185 pp 445– (1985)
[6] DOI: 10.1007/BF00280033 · Zbl 0564.35070 · doi:10.1007/BF00280033
[7] DOI: 10.1080/03605308308820304 · Zbl 0534.35069 · doi:10.1080/03605308308820304
[8] DOI: 10.1215/S0012-7094-77-04430-1 · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
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