Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers: The two-dimensional case

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• 作者：John W. Barrett^a ; ^{jwb@imperial.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Endre Sü ; li^b ; ^{endre.suli@maths.ox.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35Q30 ; 76N10 ; 82D60
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 July 2016
• 年：2016
• 卷：261
• 期：1
• 页码：592-626
• 全文大小：539 K

文摘

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead–spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain Ω in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si1.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=874098cfcadef22d7129cf2ece60f150" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si2.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=09dfad758fd1250f298dda08d78848bf" title="Click to view the MathML source">d=2class="mathContainer hidden">class="mathCode">$d=2$, for the density ρ  , the velocity class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si3.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=002d2e6c598742aa90b5b273adb9c3da">class="imgLazyJSB inlineImage" height="11" width="10" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616001236-si3.gif">class="mathContainer hidden">class="mathCode">$\underset{˜}{u}$ and the pressure p   of the fluid, with an equation of state of the form class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si4.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=d49ea0ae810bbe112bf95c88a50f4377" title="Click to view the MathML source">p(ρ)=c_pρ^γclass="mathContainer hidden">class="mathCode">$p(\rho )=c_p{\rho }^{\gamma }$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si5.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=054889cad5fd4bda5f3e052a0e7b357e" title="Click to view the MathML source">c_pclass="mathContainer hidden">class="mathCode">$c_p$ is a positive constant and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si6.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=a375ede06d19777a901b6407e083d224" title="Click to view the MathML source">γ>1class="mathContainer hidden">class="mathCode">$\gamma >1$. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This extends the result in our paper J.W. Barrett and E. Süli (2016) [9], which established the existence of global-in-time weak solutions to the system for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si7.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=5eabca414b48cccc7d39656d6614ecb6" title="Click to view the MathML source">d∈{2,3}class="mathContainer hidden">class="mathCode">$d\in \{2,3\}$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si8.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=9a1f8578c5176950818c4436635759f8">class="imgLazyJSB inlineImage" height="22" width="40" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616001236-si8.gif">class="mathContainer hidden">class="mathCode">$\gamma >\frac{3}{2}$, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si2.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=09dfad758fd1250f298dda08d78848bf" title="Click to view the MathML source">d=2class="mathContainer hidden">class="mathCode">$d=2$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si6.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=a375ede06d19777a901b6407e083d224" title="Click to view the MathML source">γ>1class="mathContainer hidden">class="mathCode">$\gamma >1$ the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si9.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=6de0fb6971a91e0184ed237e7e54fb03" title="Click to view the MathML source">ρ₀∈L^∞(Ω)class="mathContainer hidden">class="mathCode">${\rho }_0\in L^{\infty }(\mathrm{\Omega })$ for the continuity equation; a square-integrable initial velocity datum class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si10.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=b2f8c1ce5ece164ef397ce64b187709f">class="imgLazyJSB inlineImage" height="11" width="18" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616001236-si10.gif">class="mathContainer hidden">class="mathCode">${\underset{˜}{u}}_0$ for the Navier–Stokes momentum equation; and a nonnegative initial probability density function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si11.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=1863401e63ce3348e2c44444cf32e5b3" title="Click to view the MathML source">ψ₀class="mathContainer hidden">class="mathCode">${\psi }_0$ for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via   a limiting procedure on a pressure regularization parameter, the existence of a global-in-time bounded-energy weak solution class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si12.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=784e5a2c05cd2f7cb2e08fe7cfbf3a90">class="imgLazyJSB inlineImage" height="15" width="144" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616001236-si12.gif">class="mathContainer hidden">class="mathCode">$t\mapsto (\rho (t),\underset{˜}{u}(t),\psi (t))$ to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616001236&_mathId=si13.gif&_user=111111111&_pii=S0022039616001236&_rdoc=1&_issn=00220396&md5=efa70f2080881628cb4d90b84d21aab8">class="imgLazyJSB inlineImage" height="15" width="215" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616001236-si13.gif">class="mathContainer hidden">class="mathCode">$(\rho (0),\underset{˜}{u}(0),\psi (0))=({\rho }_0,{\underset{˜}{u}}_0,{\psi }_0)$.