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Oscillatory solutions to transport equations. (English) Zbl 1098.35101
In many problems of the theory of transport equations it is desirable to have a function space \(X\subset L^\infty\) which has the following properties:
(1) \(X\) is compactly embedded in \(L_{\text{loc}}^1\);
(2) contains function with jump discontinuities, and;
(3) solutions to transport equation with data from \(X\) take values in \(X\).
The authors show that if one additionally requires \(X\) to contain functions of bounded variations, then the answer is negative if the dimension of the space is greater than 3.

35L65 Hyperbolic conservation laws
35F20 Nonlinear first-order PDEs
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