Global existence and boundedness in a Keller-Segel-(Navier-)Stokes system with signal-dependent sensitivity

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• 作者：Ji Liu^a ; Yifu Wang^a ; ^b ; ^{wangyifu@bit.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Keller&ndash ; Segel ; Navier&ndash ; Stokes ; Stokes ; Boundedness ; Global existence
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：499-528
• 全文大小：564 K

文摘

In this paper, we consider the following Keller–Segel(–Navier)–Stokes systemiv class="formula" id="fm0010">iv class="label">equationia-hidden="true">(⋆ia-hidden="true">)iv>iv class="mathml">id="mmlsi1" class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=0ba0a78ea7cb23535f78240bb93c5ce8"><img class="imgLazyJSB inlineImage" height="87" width="372" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306242-si1.gif">ipt><img height="87" border="0" style="vertical-align:bottom" width="372" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306242-si1.gif">ipt>iner hidden">i1.gif" overflow="scroll">{ing="0em">ign="left">i>ni>i>ti>+i>ui>⋅i mathvariant="normal">∇i>i>ni>=i mathvariant="normal">Δi>i>ni>&minus;i mathvariant="normal">∇i>⋅(i>ni>i>χi>(i>ci>)i mathvariant="normal">∇i>i>ci>),idth="1em">ign="left">i>xi>&isin;i mathvariant="normal">Ωi>, i>ti>>0,ign="left">i>ci>i>ti>+i>ui>⋅i mathvariant="normal">∇i>i>ci>=i mathvariant="normal">Δi>i>ci>&minus;i>ci>+i>ni>,idth="1em">ign="left">i>xi>&isin;i mathvariant="normal">Ωi>, i>ti>>0,ign="left">i>ui>i>ti>+i>κi>(i>ui>⋅i mathvariant="normal">∇i>)i>ui>=i mathvariant="normal">Δi>i>ui>+i mathvariant="normal">∇i>i>Pi>+i>ni>i mathvariant="normal">∇i>i>ϕi>,idth="1em">ign="left">i>xi>&isin;i mathvariant="normal">Ωi>, i>ti>>0,ign="left">i mathvariant="normal">∇i>⋅i>ui>=0,idth="1em">ign="left">i>xi>&isin;i mathvariant="normal">Ωi>, i>ti>>0,<img class="temp" src="/sd/blank.gif">iv>iv> where id="mmlsi2" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=e4abc320761a742afee2b5e3fc8f0488" title="Click to view the MathML source">Ω⊂R^N (N=2,3)iner hidden">i2.gif" overflow="scroll">i mathvariant="normal">Ωi>⊂i mathvariant="double-struck">Ri>i>Ni> (i>Ni>=2,3) is a bounded domain with smooth boundary id="mmlsi3" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si3.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=5769d6ee8ba7d17d8edc99ce5166762e" title="Click to view the MathML source">∂Ω, κ&isin;Riner hidden">i3.gif" overflow="scroll">∂i mathvariant="normal">Ωi>, i>κi>&isin;i mathvariant="double-struck">Ri> and id="mmlsi4" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=a07b2039b64b35da5153f5b1a9c7ca84" title="Click to view the MathML source">χ(c)iner hidden">i4.gif" overflow="scroll">i>χi>(i>ci>) is assumed to generalize the prototypeiv class="formula" id="fm0020">iv class="mathml">id="mmlsi5" class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=94f617645bf54dc3ecabe600da68a931"><img class="imgLazyJSB inlineImage" height="33" width="173" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306242-si5.gif">ipt><img height="33" border="0" style="vertical-align:bottom" width="173" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16306242-si5.gif">ipt>iner hidden">i5.gif" overflow="scroll">i>χi>(i>ci>)=i>χi>0(1+i>μi>i>ci>)2, i>ci>≥0.<img class="temp" src="/sd/blank.gif">iv>iv> It is proved that i) for id="mmlsi10" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=bedaf6d382f6eef443a821816fe4381b" title="Click to view the MathML source">κ≠0iner hidden">i10.gif" overflow="scroll">i>κi>≠0 and id="mmlsi45" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si45.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=feda4009a227987ca5f5c17d669483be" title="Click to view the MathML source">N=2iner hidden">i45.gif" overflow="scroll">i>Ni>=2 or id="mmlsi46" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si46.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=060951556131ab072c96af03968e43d1" title="Click to view the MathML source">κ=0iner hidden">i46.gif" overflow="scroll">i>κi>=0 and id="mmlsi47" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si47.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=9d80ac4109ad23728c26f3359f4d2dd3" title="Click to view the MathML source">N&isin;{2,3}iner hidden">i47.gif" overflow="scroll">i>Ni>&isin;{2,3}, the corresponding initial–boundary problem admits a unique global classical solution which is bounded; ii) for id="mmlsi10" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si10.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=bedaf6d382f6eef443a821816fe4381b" title="Click to view the MathML source">κ≠0iner hidden">i10.gif" overflow="scroll">i>κi>≠0 and id="mmlsi11" class="mathmlsrc">ixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306242&_mathId=si11.gif&_user=111111111&_pii=S0022247X16306242&_rdoc=1&_issn=0022247X&md5=94f1c460f83c1c3027d833f8f6a67007" title="Click to view the MathML source">N=3iner hidden">i11.gif" overflow="scroll">i>Ni>=3, the corresponding initial–boundary problem possesses at least one global weak solution.