A local existence theorem for the quasilinear wave equation with initial values of bounded variation

1985 ◽  
pp. 9-24
Author(s):  
H. D. Alber
Keyword(s):  
Wave Equation ◽  
Existence Theorem ◽  
Bounded Variation ◽  
Local Existence ◽  
Quasilinear Wave Equation ◽  
Local Existence Theorem ◽  
Initial Values

scholarly journals The Einstein-Vlasov system in maximal areal coordinates---Local existence and continuation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sebastian Günther ◽  
Gerhard Rein
Keyword(s):  
Nonlinear System ◽  
Existence Theorem ◽  
Local Existence ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Asymptotically Flat ◽  
Numerical Investigations ◽  
Local Existence Theorem ◽  
Complicated Form ◽  
Derivatives Of

<p style='text-indent:20px;'>We consider the spherically symmetric, asymptotically flat Einstein-Vlasov system in maximal areal coordinates. The latter coordinates have been used both in analytical and numerical investigations of the Einstein-Vlasov system [<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>], but neither a local existence theorem nor a suitable continuation criterion has so far been established for the corresponding nonlinear system of PDEs. We close this gap. Although the analysis follows lines similar to the corresponding result in Schwarzschild coordinates, essential new difficulties arise from to the much more complicated form which the field equations take, while at the same time it becomes easier to control the necessary, highest order derivatives of the solution. The latter observation may be useful in subsequent investigations.</p>

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