Fourier Analysis Methods for PDEs

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1、FourierAnalysisMethodsforPDE’sR.DanchinNovember14,20052Contents1AnintroductiontoFourieranalysis71.1Notationsanddefinitions...............................71.2TheLittlewood-Paleydecomposition.........................81.2.1Bernsteininequalities...........................

2、..81.2.2ThenonhomogeneousLittlewood-Paleydecomposition...........91.2.3Abouttheperiodiccase............................101.3Littlewood-Paleydecompositionandfunctionalspaces...............111.3.1SobolevandH¨olderspaces...........................111.3.2Besovspaces.....

3、.............................131.3.3AfewpropertiesofBesovspaces.......................151.4Paradifferentialcalculus................................211.4.1Definitions...................................211.4.2Resultsofcontinuityfortheparaproductandtheremainder........221

4、.4.3Resultsofcontinuityfortheproduct....................231.4.4AresultofcompactnessinBesovspaces...................241.4.5Resultsofcontinuityforthecomposition..................271.5Calculusinhomogeneousfunctionalspaces.....................281.5.1HomogeneousLittlewood

5、-Paleydecomposition...............291.5.2HomogeneousBesovspaces..........................291.5.3Paradifferentialcalculusinhomogeneousspaces...............351.6Exercises........................................372Theheatequation392.1Generalities.........................

6、..............392.2AprioriestimatesinBesovspacesfortheheatequation..............402.2.1Spectrallocalization..............................412.2.2Estimatesfortheheatequation........................422.2.3Acounterexample...............................432.2.4Estimatesinn

7、onhomogeneousBesovspaces,andtheperiodiccase....432.3Optimalwell-posednessresultsforNavier-Stokesequations.............442.3.1Themodel...................................442.3.2Aboutscalingandcriticalspaces.......................462.3.3Globalwell-posednessforsmalldata.

8、....................472.3.4Furtherresults.................................502.4Exercises........................................513