Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross-Piatevskii equations in 1D and 2D
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- 作者：Gordon Blower g.blower@lancaster.ac.uk" class="auth_mail" title="E-mail the corresponding author
- 关键词：Gibbs measure ; Logarithmic Sobolev inequality ; Transportation
- 刊名：Journal of Mathematical Analysis and Applications
- 出版年：2016
- 出版时间：1 June 2016
- 年：2016
- 卷：438
- 期：1
- 页码：240-266
- 全文大小：577 K
文摘
This paper concerns Gibbs measures m>ν m> for some nonlinear PDE over the m>D m>-torus mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=c6cbb7f4effbf2ab80a84f9c02ea813b" title="Click to view the MathML source">TDmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msup><mrow><mi mathvariant="bold">Tmi>mrow><mrow><mi>Dmi>mrow>msup>math>. The Hamiltonian mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si2.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=5fecca1e6a8226931d0f0dcc57c850f0" title="Click to view the MathML source">H=∫TD‖∇u‖2−∫TD|u|pmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mi>Hmi><mo>=mo><msub><mrow><mo>∫mo>mrow><mrow><msup><mrow><mi mathvariant="bold">Tmi>mrow><mrow><mi>Dmi>mrow>msup>mrow>msub><msup><mrow><mo stretchy="false">‖mo><mi mathvariant="normal">∇mi><mi>umi><mo stretchy="false">‖mo>mrow><mrow><mn>2mn>mrow>msup><mo>−mo><msub><mrow><mo>∫mo>mrow><mrow><msup><mrow><mi mathvariant="bold">Tmi>mrow><mrow><mi>Dmi>mrow>msup>mrow>msub><mo stretchy="false">|mo><mi>umi><msup><mrow><mo stretchy="false">|mo>mrow><mrow><mi>pmi>mrow>msup>math> has canonical equations with solutions in mmlsi3" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si3.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=2d366a5848b6897d9fe082fac975e79b" title="Click to view the MathML source">ΩN={u∈L2(TD):∫|u|2≤N}mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi mathvariant="normal">Ωmi>mrow><mrow><mi>Nmi>mrow>msub><mo>=mo><mo stretchy="false">{mo><mi>umi><mo>∈mo><msup><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msup><mo stretchy="false">(mo><msup><mrow><mi mathvariant="bold">Tmi>mrow><mrow><mi>Dmi>mrow>msup><mo stretchy="false">)mo><mo>:mo><mo>∫mo><mo stretchy="false">|mo><mi>umi><msup><mrow><mo stretchy="false">|mo>mrow><mrow><mn>2mn>mrow>msup><mo>≤mo><mi>Nmi><mo stretchy="false">}mo>math>; this m>N m> is a parameter in quantum field theory analogous to the number of particles in a classical system. For mmlsi287" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si287.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=951d4096942df7d12f0a02d52cabc5e3" title="Click to view the MathML source">D=1mathContainer hidden">mathCode"><math altimg="si287.gif" overflow="scroll"><mi>Dmi><mo>=mo><mn>1mn>math> and mmlsi5" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si5.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=4cbb7d74b290ae8b946197da7097b4dd" title="Click to view the MathML source">2≤p<6mathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mn>2mn><mo>≤mo><mi>pmi><mo><mo><mn>6mn>math>, mmlsi207" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si207.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=0a16b5f9ac3ce491766e4f74903d1630" title="Click to view the MathML source">ΩNmathContainer hidden">mathCode"><math altimg="si207.gif" overflow="scroll"><msub><mrow><mi mathvariant="normal">Ωmi>mrow><mrow><mi>Nmi>mrow>msub>math> supports the Gibbs measure mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si7.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=6fd26255797d49a8961cda4f0879db4c" title="Click to view the MathML source">ν(du)=Z−1e−H(u)∏x∈Tdu(x)mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mi>νmi><mo stretchy="false">(mo><mi>dmi><mi>umi><mo stretchy="false">)mo><mo>=mo><msup><mrow><mi>Zmi>mrow><mrow><mo>−mo><mn>1mn>mrow>msup><msup><mrow><mi>emi>mrow><mrow><mo>−mo><mi>Hmi><mo stretchy="false">(mo><mi>umi><mo stretchy="false">)mo>mrow>msup><msub><mrow><mo>∏mo>mrow><mrow><mi>xmi><mo>∈mo><mi mathvariant="bold">Tmi>mrow>msub><mi>dmi><mi>umi><mo stretchy="false">(mo><mi>xmi><mo stretchy="false">)mo>math> which is normalized and formally invariant under the flow generated by the PDE. The paper proves that mmlsi8" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si8.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=694b8f72ab2a19c3d8be742fc4a99f52" title="Click to view the MathML source">(ΩN,‖⋅‖L2,ν)mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll"><mo stretchy="false">(mo><msub><mrow><mi mathvariant="normal">Ωmi>mrow><mrow><mi>Nmi>mrow>msub><mo>,mo><msub><mrow><mo stretchy="false">‖mo><mo>⋅mo><mo stretchy="false">‖mo>mrow><mrow><msup><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msup>mrow>msub><mo>,mo><mi>νmi><mo stretchy="false">)mo>math> is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic m>KdV m>, the focussing cubic nonlinear Schrödinger equation and the periodic Zakharov system. For suitable subset of mmlsi207" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si207.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=0a16b5f9ac3ce491766e4f74903d1630" title="Click to view the MathML source">ΩNmathContainer hidden">mathCode"><math altimg="si207.gif" overflow="scroll"><msub><mrow><mi mathvariant="normal">Ωmi>mrow><mrow><mi>Nmi>mrow>msub>math>, a logarithmic Sobolev inequality also holds in the critical case mmlsi9" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si9.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=7e1471fefe57036a59553b5fc40b85f4" title="Click to view the MathML source">p=6mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><mi>pmi><mo>=mo><mn>6mn>math>. For mmlsi10" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si10.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=67f83629825e247718e5d32cb0c54e40" title="Click to view the MathML source">D=2mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll"><mi>Dmi><mo>=mo><mn>2mn>math>, the Gross–Piatevskii equation has mmlsi11" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si11.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=7fc84f0f1cb007e8a0558dd24d7a7157" title="Click to view the MathML source">H=∫T2‖∇u‖2−∫T2(V⁎|u|2)|u|2mathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll"><mi>Hmi><mo>=mo><msub><mrow><mo>∫mo>mrow><mrow><msup><mrow><mi mathvariant="bold">Tmi>mrow><mrow><mn>2mn>mrow>msup>mrow>msub><msup><mrow><mo stretchy="false">‖mo><mi mathvariant="normal">∇mi><mi>umi><mo stretchy="false">‖mo>mrow><mrow><mn>2mn>mrow>msup><mo>−mo><msub><mrow><mo>∫mo>mrow><mrow><msup><mrow><mi mathvariant="bold">Tmi>mrow><mrow><mn>2mn>mrow>msup>mrow>msub><mo stretchy="false">(mo><mi>Vmi><mo>⁎mo><mo stretchy="false">|mo><mi>umi><msup><mrow><mo stretchy="false">|mo>mrow><mrow><mn>2mn>mrow>msup><mo stretchy="false">)mo><mo stretchy="false">|mo><mi>umi><msup><mrow><mo stretchy="false">|mo>mrow><mrow><mn>2mn>mrow>msup>math>, for a suitable bounded interaction potential m>V m> and the Gibbs measure m>ν m> lies on a metric probability space mmlsi12" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si12.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=0250b719f0aefb0efe1f8a76e27c107b" title="Click to view the MathML source">(Ω,‖⋅‖H−s,ν)mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll"><mo stretchy="false">(mo><mi mathvariant="normal">Ωmi><mo>,mo><msub><mrow><mo stretchy="false">‖mo><mo>⋅mo><mo stretchy="false">‖mo>mrow><mrow><msup><mrow><mi>Hmi>mrow><mrow><mo>−mo><mi>smi>mrow>msup>mrow>msub><mo>,mo><mi>νmi><mo stretchy="false">)mo>math> which satisfies m>LSI m>. In the above cases, mmlsi13" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si13.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=3bd83ffe65aa946718406cb6736263c4" title="Click to view the MathML source">(Ω,d,ν)mathContainer hidden">mathCode"><math altimg="si13.gif" overflow="scroll"><mo stretchy="false">(mo><mi mathvariant="normal">Ωmi><mo>,mo><mi>dmi><mo>,mo><mi>νmi><mo stretchy="false">)mo>math> is the limit in mmlsi14" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si14.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=f0900b290a04d613a3c651e4b3e95355" title="Click to view the MathML source">L2mathContainer hidden">mathCode"><math altimg="si14.gif" overflow="scroll"><msup><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msup>math> transportation distance of finite-dimensional mmlsi15" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1600127X&_mathId=si15.gif&_user=111111111&_pii=S0022247X1600127X&_rdoc=1&_issn=0022247X&md5=09e60c78d363ef3e781190987bfa8ae6" title="Click to view the MathML source">(Ωn,‖⋅‖,νn)mathContainer hidden">mathCode"><math altimg="si15.gif" overflow="scroll"><mo stretchy="false">(mo><msub><mrow><mi mathvariant="normal">Ωmi>mrow><mrow><mi>nmi>mrow>msub><mo>,mo><mo stretchy="false">‖mo><mo>⋅mo><mo stretchy="false">‖mo><mo>,mo><msub><mrow><mi>νmi>mrow><mrow><mi>nmi>mrow>msub><mo stretchy="false">)mo>math> given by Fourier sums.