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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.

##### MSC:
 35L65 Hyperbolic conservation laws 35F20 Nonlinear first-order PDEs
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