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In this chapter, we explore continuous dependence on initial data for the Cauchy problem. We start with Hadamard’s example, which shows that small errors in initial data for the Laplace equation can lead to large effects on the solution. We then prove the Lax-Mizohata Theorem, establishing the connection between solvability and continuous dependence on initial data.

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The Hadamard Example: Solvability of the Cauchy Problem and Continuous Dependence by the Data

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  • 发表日期：

    2025年

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  • Sergio Vessella

摘要

In this chapter, we explore continuous dependence on initial data for the Cauchy problem. We start with Hadamard’s example, which shows that small errors in initial data for the Laplace equation can lead to large effects on the solution. We then prove the Lax-Mizohata Theorem, establishing the connection between solvability and continuous dependence on initial data.