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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
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