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Zbl 0763.35014
DiPerna, R.J.; Lions, P.L.; Meyer, Y.
$L\sp p$ regularity of velocity averages.
(English)
[J] Ann. Inst. Henri PoincarĂ©, Anal. Non LinĂ©aire 8, No.3-4, 271-287 (1991). ISSN 0294-1449

The paper is concerned with the regularity of velocity averages for solutions of transport equations: $$v.\nabla\sb xf=g\quad\text{for } x\in\bbfR\sp N,\quad v\in\bbfR\sp N,\quad\text{or} \tag 1$$ $$\partial f/\partial t+v.\nabla\sb xf=g\quad\text{for } x\in\bbfR\sp N,\quad v\in\bbfR\sp N,\quad t\in\bbfR. \tag 2$$ In the time-independent case (equation (1)), the authors prove that, if $f\in L\sp p(\bbfR\sp N\times\bbfR\sp N)$ and $g\in L\sp p(\bbfR\sp N\times\bbfR\sp N)$, $1<p\le 2$, then for every $\psi\in D(\bbfR\sp N)$, the velocity average $\bar f(x)=\int\sb{\bbfR\sp N}f(x,v)\psi(v)dv$ belongs to the Besov space $B\sb 2\sp{s,p}(\bbfR\sp N)$ where $s=1/p'$.\par In the time dependent case (equation (2)), they prove similar results when $g$ admits the following decomposition: $g=(I-\Delta\sb x)\sp{\tau/2}(I-\Delta\sb v)\sp{m/2}G$, $\ \tau\in[0,1)$, $\ m\ge 0$, $\ G\in L\sp p(\bbfR\sp N\times\bbfR\sp N\times\bbfR)$.\par Some applications to Vlasov-Maxwell systems and to other models are also given. These results extend many previous results [see for example {\it F. Golse}, {\it P. L. Lions}, {\it B. Perthame}, {\it R. Sentis}, J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)].
[J.-P.Raymond (Toulouse)]
MSC 2000:
*35B65 Smoothness of solutions of PDE
35F05 General theory of first order linear PDE
35Q40 PDE from quantum mechanics
42B25 Maximal functions
42B30 Hp-spaces (Fourier analysis)
46E35 Sobolev spaces and generalizations
42B15 Multipliers, several variables
82B40 Kinetic theory of gases
42B20 Singular integrals, several variables

Keywords: Littlewood-Paley type decompositions; interpolating arguments; spectral decomposition; Sobolev; Besov spaces; $L(\sup p)$-multipliers; transport equations; time-independent case; time dependent case; Vlasov-Maxwell systems

Citations: Zbl 0652.47031

Cited in: Zbl 0798.35025

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