High energy estimates

Reinhard Racke
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Reinhard Racke: University of Konstanz, Department of Mathematics and Statistics

Chapter 6 in Lectures on Nonlinear Evolution Equations, 2015, pp 78-81 from Springer

Abstract: Abstract In this chapter we shall prove an energy estimate for the local solution $$ u = \left( {\partial _t y,\,\nabla _y } \right) $$ of the initial value problem (5.7), (5.8), where y solves (5.1), (5.2) with the assumptions (5.3) – (5.6) on the nonlinearity $$ \left( {case\,\alpha = 1} \right) $$ . Without loss of generality for further investigations we assume the constant $$ g_2^m $$ appearing in the local existence Theorem 5.8 to be sufficiently small as it will be needed in the proof of the next theorem.

Keywords: Partial Differential Equation; Positive Constant; Evolution Equation; Sobolev Space; Similar Argument (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-21873-1_7

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