## Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients.(English)Zbl 0882.35026

The authors study the linear transport equation $u_t+\text{div}(a(t,x) u)=0 \text{ in }(0,\infty) \times \mathbb{R}^n,$ with initial data $$u(0,x)=u_0(x)$$. The coefficient $$a(t,x)$$ belongs to $$L^1_{loc}(0,\infty;L^{\infty}(\mathbb{R}^n))^n$$. The characteristics of such a system are defined by $$dX/dt=a(t,X(t;x))$$, with $$X(0;x)=x$$, and solution is meant in the sense of Fillipov. Under the assumption that the Fillipov characteristics are unique (for which a sufficient condition is given) it is proved that there is a unique measure solution of the transport equation given by $$u(t):=X(t)(u_0)$$, for any initial data $$u_0$$ taken from the space of bounded measures on $$\mathbb{R}^n$$. Precisely, this means that $$\langle u(t,\cdot),\varphi\rangle=\langle u_0,\varphi(X(t;\cdot))\rangle$$ for any function $$\varphi \in C^0_0(\mathbb{R}^n)$$.
Reviewer: E.Barron (Chicago)

### MSC:

 35D05 Existence of generalized solutions of PDE (MSC2000) 35F10 Initial value problems for linear first-order PDEs 35R05 PDEs with low regular coefficients and/or low regular data

### Keywords:

Fillipov solution
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### References:

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