Abstract
The focus of this article is on the geometric mechanism for the blow-up of solutions to the initial value problem for scalar conservation laws. We prove that the sufficient and necessary condition of blow-up is the formation of characteristics envelope. Whether the solution blows up or not relates to the topology structure of a set dominated by initial data. At last we take Burger’s equation as an example to verify our main theorem.
Highlights
In this short article we consider the blow-up phenomena to scalar conservation laws
The focus of this article is on the geometric mechanism for the blow-up of solutions to the initial value problem for scalar conservation laws
We prove that the sufficient and necessary condition of blow-up is the formation of characteristics envelope
Summary
In this short article we consider the blow-up phenomena to scalar conservation laws. We are interested in the blow-up mechanism of the following initial value problem ut u. The solution to (1.1) blows up once the characteristics intersect (see [1,2,3,4]). What is needed for the blow-up to happen is the formation of the envelopes. In order to give an explicit description to the geometric blow-up mechanics for scalar conservation laws, we write this article. In the last section we take the Burger’s equation as an example to test and verify our main theorems by constructing certain initial data
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