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One-dimensional transport equations with discontinuous coefficients. (English) Zbl 0989.35130
From the introduction: This paper is devoted to one-dimensional homogeneous linear transport equations $\partial_t u+a(t,x) \partial_x u=0\quad \text{in } ]0,T[ \times\mathbb{R}, \tag{1}$ with $$T>0$$ and $$a$$ a given bounded coefficient. This equation will be referred to as the nonconservative problem. By differentiating (1) with respect to $$x$$, we obtain the conservative problem $\partial_t \mu+\partial_x \bigl(a(t,x)\mu\bigr)=0 \quad\text{in }]0, T [\times \mathbb{R},\tag{2}$ with $$\mu= \partial_xu$$.
In Section 2, we prove our main lemma which states that conservative and nonconservative equations are equivalent. We also prove two sharp results of uniqueness. Section 3 is concerned with the case of a piecewise continuous $$a$$. Finally, Section 4 is devoted to the case where $$a$$ satisfies the one-sided Lipschitz condition. In 4.1 we study the backward problem and Lipschitz solutions, in 4.2 we define duality solutions for the forward problem, 4.3 contains more sophisticated results and the relation with the generalized Filippov flow, and 4.4 is devoted to some comments about viscous problems.

##### MSC:
 35R05 PDEs with low regular coefficients and/or low regular data 35L65 Hyperbolic conservation laws
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