Solutions fortes et comportement asymptotique pour un modèle de convection naturelle en milieu poreux

Résumé Nous étudions ici un système d'équations aux dérivées partielles qui gouverne la convection naturelle dans un milieu poreux soumis à un gradient de température T. Sous leur forme la plus générale, ces équations s'écrivent:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacqGHiiIZdaWcaaqaaiab% gkGi2kabeg8aYbqaaiabgkGi2kaadshaaaGaey4kaSIaaeizaiaabM% gacaqG2bGaaeiiaiabeg8aYjaadAfacqGH9aqpcaaIWaaabaWaaSaa% aeaacqaHbpGCaeaacqGHiiIZaaWaaSaaaeaacqGHciITaeaacqGHci% ITcaWG0baaaiabgUcaRiaabEgacaqGYbGaaeyyaiaabsgacaqGGaGa% amiCaiabgkHiTiabeg8aYjaadEgacqGHRaWkcqaH8oqBcaWGlbWaaW% baaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaadAfacqGH9aqpcaaI% WaaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiaacIcacq% aHbpGCcaWGJbGaaiykamaaCaaaleqabaGaaiOkaaaakiaadsfacqGH% sislcaqGGaGaaeizaiaabMgacaqG2bGaaeiiaiaabU5adaahaaWcbe% qaaiaabQcaaaGccaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfa% cqGHRaWkcaGGOaGaeqyWdiNaam4yaiaacMcadaWgaaWcbaGaamOzaa% qabaGccaWGwbGaaeiiaiabgwSixlabgEGirlaadsfacqGH9aqpcaaI% WaaabaGaeqyWdiNaeyypa0JaeqyWdi3aaSbaaSqaaiaadkhaaeqaaO% GaaiikaiaaigdacqGHsislcqaHXoqycaGGOaGaamivaiabgkHiTiaa% dsfadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiykaaaacaGL7baaaa% a!9527!\[P_1 \left\{ \begin{gathered} \in \frac{{\partial \rho }}{{\partial t}} + {\text{div }}\rho V = 0 \hfill \\ \frac{\rho }{ \in }\frac{\partial }{{\partial t}} + {\text{grad }}p - \rho g + \mu K^{ - 1} {\text{ }}V = 0 \hfill \\ \frac{\partial }{{\partial t}}(\rho c)^* T - {\text{ div \Lambda }}^{\text{*}} {\text{grad }}T + (\rho c)_f V{\text{ }} \cdot \nabla T = 0 \hfill \\ \rho = \rho _r (1 - \alpha (T - T_r )) \hfill \\ \end{gathered} \right.\] désigne la porosité, la masse volumique du fluide, V la vitesse, p la pression, T la température du fluide, la viscosité, K et % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4MdmaaCa% aaleqabaGaaeOkaaaaaaa!37E8!\[{\text{\Lambda }}^{\text{*}} \] sont les tenseurs respectifs de perméabilité et de conductivité thermique. La chaleur volumique du fluide est notée (c) _f , celle du solide (c) _s , et on définit alors la chaleur volumique équivalente par la relation: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiabeg% 8aYjaadogacaqGPaWaaWbaaSqabeaacaqGQaaaaOGaeyypa0Jaeyic% I4Saaiikaiabeg8aYjaadogacaGGPaWaaSbaaSqaaiaadAgaaeqaaO% Gaey4kaSIaaiikaiaaigdacaqGGaGaeyOeI0IaeyicI4Saaiykaiaa% cIcacqaHbpGCcaWGJbGaaiykaaaa!4C87!\[{\text{(}}\rho c{\text{)}}^{\text{*}} = \in (\rho c)_f + (1{\text{ }} - \in )(\rho c)\].De façon très classique, dans les problèmes de convection, on simplifie ce modèle en faisant l'approximation de Boussinesq qui consiste à négliger les variations de la masse volumique sauf dans le terme g, voir par exemple [6]. Ce modèle connu depuis longemps a été très étudié par de nombreux physiciens et numériciens depuis une dizaine d'années (voir par exemple [3–5, 7, 8, 18, 24]) mais à notre connaissance accune étude théorique n'a été entreprise jusqu'à aujourd'hui.On se limitera ici au cas d'un milieu homogène isotrope remplissant une cavité parallélépipédique dont l'un des axes a même direction que l'accélération de la pesanteur g. Sous forme adimensionnelle le système P ₂ s'écrit:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacqGHiiIZdaWcaaqaaiab% gkGi2kabeg8aYbqaaiabgkGi2kaadshaaaGaey4kaSIaaeizaiaabM% gacaqG2bGaaeiiaiabeg8aYjaadAfacqGH9aqpcaaIWaaabaWaaSaa% aeaacqaHbpGCaeaacqGHiiIZaaWaaSaaaeaacqGHciITaeaacqGHci% ITcaWG0baaaiabgUcaRiaabEgacaqGYbGaaeyyaiaabsgacaqGGaGa% amiCaiabgkHiTiabeg8aYjaadEgacqGHRaWkcqaH8oqBcaWGlbWaaW% baaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaadAfacqGH9aqpcaaI% WaaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiaacIcacq% aHbpGCcaWGJbGaaiykamaaCaaaleqabaGaaiOkaaaakiaadsfacqGH% sislcaqGGaGaaeizaiaabMgacaqG2bGaaeiiaiaabU5adaahaaWcbe% qaaiaabQcaaaGccaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfa% cqGHRaWkcaGGOaGaeqyWdiNaam4yaiaacMcadaWgaaWcbaGaamOzaa% qabaGccaWGwbGaaeiiaiabgwSixlabgEGirlaadsfacqGH9aqpcaaI% WaaabaGaeqyWdiNaeyypa0JaeqyWdi3aaSbaaSqaaiaadkhaaeqaaO% GaaiikaiaaigdacqGHsislcqaHXoqycaGGOaGaamivaiabgkHiTiaa% dsfadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiykaaaacaGL7baaaa% a!9527!\[P_1 \left\{ \begin{gathered} \in \frac{{\partial \rho }}{{\partial t}} + {\text{div }}\rho V = 0 \hfill \\ \frac{\rho }{ \in }\frac{\partial }{{\partial t}} + {\text{grad }}p - \rho g + \mu K^{ - 1} {\text{ }}V = 0 \hfill \\ \frac{\partial }{{\partial t}}(\rho c)^* T - {\text{ div \Lambda }}^{\text{*}} {\text{grad }}T + (\rho c)_f V{\text{ }} \cdot \nabla T = 0 \hfill \\ \rho = \rho _r (1 - \alpha (T - T_r )) \hfill \\ \end{gathered} \right.\]Dans % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey% ypa0Jaai4EaiaacIcacaWG4bGaaiilaiaabccacaWG5bGaaeilaiaa% bccacaWG6bGaaiykaiabgIGiolaac2facaaIWaGaaiilaiaabccaca% WGmbGaai4waerbbjxAHXgaiuaacaWFfrGaaiyxaiaaicdacaGGSaGa% aeiiaiaadYgacaGGBbGaa8xreiaac2facaaIWaGaaiilaiaabccaca% WGObGaai4waiaac2haaaa!54B3!\[\Omega = \{ (x,{\text{ }}y{\text{, }}z) \in ]0,{\text{ }}L[]0,{\text{ }}l[]0,{\text{ }}h[\} \]: de frontière les conditions aux limites sont:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGub% GaaiikaiaadIhacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiaiaabcda% caGGPaGaeyypa0JaaGymaiaacYcacaqGGaGaaeiiaiaabccacaqGGa% GaaeiiaiaadsfacaGGOaGaamiEaiaacYcacaqGGaGaamyEaiaacYca% caqGGaGaamiAaiaacMcacqGH9aqpcaaIWaGaaiilaaqaamacmc4caa% qaiWiGcWaJaAOaIyRaiWiGdsfaaeacmcOamWiGgkGi2kacmc4G4baa% aiaacIcacaaIWaGaaiilaiaabccacaWG5bGaaiilaiaabccacaWG6b% Gaaiykaiabg2da9maalaaabaGaeyOaIyRaamivaaqaaiabgkGi2kaa% dIhaaaGaaiikaiaadYeacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcaWGubaabaGaeyOa% IyRaamiEaaaacaGGOaGaamiEaiaacYcacaqGGaGaamiBaiaacYcaca% qGGaGaamOEaiaacMcacqGH9aqpcaaIWaGaaiilaaqaaiaadAfacqGH% flY1caqGGaGaamOBamaaBaaaleaaruqqYLwySbacfaGaa8hFaiabgk% Gi2kabfM6axbqabaGccqGH9aqpcaaIWaaaaaa!8886!\[\begin{gathered} T(x,{\text{ }}y,{\text{ 0}}) = 1,{\text{ }}T(x,{\text{ }}y,{\text{ }}h) = 0, \hfill \\ \frac{{\partial T}}{{\partial x}}(0,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(L,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(x,{\text{ }}l,{\text{ }}z) = 0, \hfill \\ V \cdot {\text{ }}n_{|\partial \Omega } = 0 \hfill \\ \end{gathered} \], où n est la normale unitaire sortante à .Le vecteur adimensionnel k est pris égal à-e _z, Ra ^* est un paramètre proportionnel à la contrainte exercée au milieu et S un paramètre très petit [Smin(10^-6, 10^-6 Ra ^*)] que l'on fera tendre par la suite vers zéro.Dans [10, 11] nous avons étudié le problème bidimensionnel aussi bien d'évolution que stationnaire et nous avons montré, outre un théorème d'existence, d'unicité et de régularité, la présence de plusieurs solutions stationnaires. Le phénomène nous a incité à étudier le comportement asymptotique des solutions du problème d'évolution. Afin de rendre cette étude plus complète nous avons décidé de travailler en dimension 3 d'espace.Ce papier donne les résultats préliminaires à une étude un peu fine du comportement asymptotique. Nous allons en particulier établir un théorème de régularité et donner une majoration uniforme des dérivées secondes en espace des solutions dans le cas où S=0. Ces propriétés sont similaires à celles connues pour les équations de Navier-Stokes dans le cas bidimensionnel [13, 26] et généralisent à la dimension trois ceux que nous avons obtenus dans [10].La clef de le preuve du théorème d'existence et d'unicité est une estimation L en temps et en espace de la température T obtenue en découplant l'équation de l'énergie (0.3) et l'équation de Darcy (0.2). Ensuite on applique une méthode de point fixe. La régularité en espace est liée à la structure particulière de l'ouvert ainsi qu'à la nature des conditions limites. Cela étant acquis, les majorations uniformes en temps sont obtenues de façon assez classique. Nous étendons enfin à notre système les résultats obtenus par Foias et Temam [15] pour les équations de Navier-Stokes en dimension deux d'espace. Rappelons qu'il s'agit alors de montrer que la solution est parfaitement déterminée par ses valeurs prises sur un ensemble fini de points.Avant d'aller plus avant dans ce travail, signalons que l'on se ramène à des conditions aux limites homogènes en posant % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2% da9iabeI7aXjabgUcaRiaaigdacqGHsislcaGGOaGaamOEaiaac+ca% caWGObGaaiykaaaa!4004!\[T = \theta + 1 - (z/h)\]. Le système devient:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaadaWcaaqaaiabgkGi2kab% eI7aXbqaaiabgkGi2kaadshaaaGaeyOeI0IaeyiLdqKaeqiUdeNaey% 4kaSIaamOvaiabgwSixlabgEGirlabeI7aXjabgkHiTmaalaaabaGa% aGymaaqaaiaadIgaaaGaeqyXdu3aaSbaaSqaaiaaiodaaeqaaOGaey% ypa0JaaGimaaqaaiaadofadaWcaaqaaiabgkGi2kaadAfaaeaacqGH% ciITcaWG0baaaiabgUcaRiaadAfacqGHRaWkcqGHhis0cqaHapaCcq% GHRaWkcaWGsbGaamyyamaaCaaaleqabaGaaiOkaaaakiaadUgacqaH% 4oqCcqGH9aqpcaaIWaaabaGaaeizaiaabMgacaqG2bGaaeiiaiaadA% facqGH9aqpcaaIWaaabaGaamOvaiabgwSixlaad6gadaWgaaWcbaqe% feKCPfgBaGqbaiaa-XhacqqHtoWraeqaaOGaeyypa0JaaGimaaqaai% abeI7aXjaacIcacaWG4bGaaiilaiaadMhacaGGSaGaaGimaiaacMca% cqGH9aqpcqaH4oqCcaGGOaGaamiEaiaacYcacaWG5bGaaiilaiaadI% gacaGGPaGaeyypa0JaaGimaaqaamaalaaabaGaeyOaIyRaeqiUdeha% baGaeyOaIyRaamiEaaaacaGGOaGaaGimaiaacYcacaWG5bGaaiilai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcqaH4oqCaeaacqGH% ciITcaWG4baaaiaacIcacaWGmbGaaiilaiaadMhacaGGSaGaamOEai% aacMcacqGH9aqpcaaIWaaabaWaaSaaaeaacqGHciITcqaH4oqCaeaa% cqGHciITcaWG5baaaiaacIcacaWG4bGaaiilaiaaicdacaGGSaGaam% OEaiaacMcacqGH9aqpdaWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi% 2kaadMhaaaGaaiikaiaadIhacaGGSaGaamiBaiaacYcacaWG6bGaai% ykaiabg2da9iaaicdaaaGaay5Eaaaaaa!B7C4!\[P_1 \left\{ \begin{gathered} \frac{{\partial \theta }}{{\partial t}} - \Delta \theta + V \cdot \nabla \theta - \frac{1}{h}\upsilon _3 = 0 \hfill \\ S\frac{{\partial V}}{{\partial t}} + V + \nabla \pi + Ra^* k\theta = 0 \hfill \\ {\text{div }}V = 0 \hfill \\ V \cdot n_{|\Gamma } = 0 \hfill \\ \theta (x,y,0) = \theta (x,y,h) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial x}}(0,y,z) = \frac{{\partial \theta }}{{\partial x}}(L,y,z) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial y}}(x,0,z) = \frac{{\partial \theta }}{{\partial y}}(x,l,z) = 0 \hfill \\ \end{gathered} \right.\]

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Strong solutions and asymptotic behaviour for a natural convection problem in porous media

We discuss a system of partial differential equations which describes natural convection in a porous medium under a temperature gradient T. In their most general form these equations can be written% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacqGHiiIZdaWcaaqaaiab% gkGi2kabeg8aYbqaaiabgkGi2kaadshaaaGaey4kaSIaaeizaiaabM% gacaqG2bGaaeiiaiabeg8aYjaadAfacqGH9aqpcaaIWaaabaWaaSaa% aeaacqaHbpGCaeaacqGHiiIZaaWaaSaaaeaacqGHciITaeaacqGHci% ITcaWG0baaaiabgUcaRiaabEgacaqGYbGaaeyyaiaabsgacaqGGaGa% amiCaiabgkHiTiabeg8aYjaadEgacqGHRaWkcqaH8oqBcaWGlbWaaW% baaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaadAfacqGH9aqpcaaI% WaaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiaacIcacq% aHbpGCcaWGJbGaaiykamaaCaaaleqabaGaaiOkaaaakiaadsfacqGH% sislcaqGGaGaaeizaiaabMgacaqG2bGaaeiiaiaabU5adaahaaWcbe% qaaiaabQcaaaGccaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfa% cqGHRaWkcaGGOaGaeqyWdiNaam4yaiaacMcadaWgaaWcbaGaamOzaa% qabaGccaWGwbGaaeiiaiabgwSixlabgEGirlaadsfacqGH9aqpcaaI% WaaabaGaeqyWdiNaeyypa0JaeqyWdi3aaSbaaSqaaiaadkhaaeqaaO% GaaiikaiaaigdacqGHsislcqaHXoqycaGGOaGaamivaiabgkHiTiaa% dsfadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiykaaaacaGL7baaaa% a!9527!\[P_1 \left\{ \begin{gathered} \in \frac{{\partial \rho }}{{\partial t}} + {\text{div }}\rho V = 0 \hfill \\ \frac{\rho }{ \in }\frac{\partial }{{\partial t}} + {\text{grad }}p - \rho g + \mu K^{ - 1} {\text{ }}V = 0 \hfill \\ \frac{\partial }{{\partial t}}(\rho c)^* T - {\text{ div \Lambda }}^{\text{*}} {\text{grad }}T + (\rho c)_f V{\text{ }} \cdot \nabla T = 0 \hfill \\ \rho = \rho _r (1 - \alpha (T - T_r )) \hfill \\ \end{gathered} \right.\]where represents the porosity, is the fluid density, T is the temperature, is the dynamic viscosity, K and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4MdmaaCa% aaleqabaGaaeOkaaaaaaa!37E8!\[{\text{\Lambda }}^{\text{*}} \] are, respectively, the tensor of permeability and of thermal conductivity. The heat capacity of fluid (resp., solid) is denoted by (c) _f (resp., (c) _s ). Thus, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiabeg% 8aYjaadogacaqGPaWaaWbaaSqabeaacaqGQaaaaOGaeyypa0Jaeyic% I4Saaiikaiabeg8aYjaadogacaGGPaWaaSbaaSqaaiaadAgaaeqaaO% Gaey4kaSIaaiikaiaaigdacaqGGaGaeyOeI0IaeyicI4Saaiykaiaa% cIcacqaHbpGCcaWGJbGaaiykaaaa!4C87!\[{\text{(}}\rho c{\text{)}}^{\text{*}} = \in (\rho c)_f + (1{\text{ }} - \in )(\rho c)\] represents the equivalent heat capacity.As is usual in convection problems, we simplify the model by adopting the Boussinesq approximation which consists of neglecting the density variations except in the g term, (cf., for instance, [6]). This well-known model has often been studied by physicists and numerical analysts, but ([3–5, 7, 8, 18, 24]), as far as we know, it seems that a theoretical approach has not yet been developed. We shall restrict our study to the case of a homogeneous isotropic medium filling a parallelepipedic cavity, one of the axis of which is colinear to the gravitational acceleration g. In dimensionless form, the system P ₁ can be written% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacaqGWaGaaeOlaiaabgda% caqGGaGaaeiiaiaabccacaqGKbGaaeyAaiaabAhacaqGGaGaamOvai% abg2da9iaaicdaaeaacaaIWaGaaiOlaiaaikdacaqGGaGaaeiiaiaa% bccacaWGtbWaaSaaaeaacqGHciITcaWGwbaabaGaeyOaIyRaamiDaa% aacqGHRaWkcaWGwbGaey4kaSIaae4zaiaabkhacaqGHbGaaeizaiaa% bccacaWGWbGaey4kaSIaamOuaiaadggadaahaaWcbeqaaiaacQcaaa% GccaWGRbGaamivaiabg2da9iaaicdaaeaacaaIWaGaaiOlaiaaioda% caqGGaGaaeiiaiaabccadaWcaaqaaiabgkGi2kaadsfaaeaacqGHci% ITcaWG0baaaiabgkHiTiabgs5aejaadsfacqGHRaWkcaqGGaGaamOv% aiaabccacaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfacqGH9a% qpcaaIWaGaaiOlaaaacaGL7baaaaa!71EF!\[P_1 \left\{ \begin{gathered} {\text{0}}{\text{.1 div }}V = 0 \hfill \\ 0.2{\text{ }}S\frac{{\partial V}}{{\partial t}} + V + {\text{grad }}p + Ra^* kT = 0 \hfill \\ 0.3{\text{ }}\frac{{\partial T}}{{\partial t}} - \Delta T + {\text{ }}V{\text{ grad }}T = 0. \hfill \\ \end{gathered} \right.\]With boundary conditions in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey% ypa0Jaai4EaiaacIcacaWG4bGaaiilaiaabccacaWG5bGaaeilaiaa% bccacaWG6bGaaiykaiabgIGiolaac2facaaIWaGaaiilaiaabccaca% WGmbGaai4waerbbjxAHXgaiuaacaWFfrGaaiyxaiaaicdacaGGSaGa% aeiiaiaadYgacaGGBbGaa8xreiaac2facaaIWaGaaiilaiaabccaca% WGObGaai4waiaac2haaaa!54B3!\[\Omega = \{ (x,{\text{ }}y{\text{, }}z) \in ]0,{\text{ }}L[]0,{\text{ }}l[]0,{\text{ }}h[\} \]:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGub% GaaiikaiaadIhacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiaiaabcda% caGGPaGaeyypa0JaaGymaiaacYcacaqGGaGaaeiiaiaabccacaqGGa% GaaeiiaiaadsfacaGGOaGaamiEaiaacYcacaqGGaGaamyEaiaacYca% caqGGaGaamiAaiaacMcacqGH9aqpcaaIWaGaaiilaaqaamacmc4caa% qaiWiGcWaJaAOaIyRaiWiGdsfaaeacmcOamWiGgkGi2kacmc4G4baa% aiaacIcacaaIWaGaaiilaiaabccacaWG5bGaaiilaiaabccacaWG6b% Gaaiykaiabg2da9maalaaabaGaeyOaIyRaamivaaqaaiabgkGi2kaa% dIhaaaGaaiikaiaadYeacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcaWGubaabaGaeyOa% IyRaamiEaaaacaGGOaGaamiEaiaacYcacaqGGaGaamiBaiaacYcaca% qGGaGaamOEaiaacMcacqGH9aqpcaaIWaGaaiilaaqaaiaadAfacqGH% flY1caqGGaGaamOBamaaBaaaleaaruqqYLwySbacfaGaa8hFaiabgk% Gi2kabfM6axbqabaGccqGH9aqpcaaIWaaaaaa!8886!\[\begin{gathered} T(x,{\text{ }}y,{\text{ 0}}) = 1,{\text{ }}T(x,{\text{ }}y,{\text{ }}h) = 0, \hfill \\ \frac{{\partial T}}{{\partial x}}(0,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(L,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(x,{\text{ }}l,{\text{ }}z) = 0, \hfill \\ V \cdot {\text{ }}n_{|\partial \Omega } = 0 \hfill \\ \end{gathered} \], where n is the outward normal unit sector to .The dimensionless vector k stands for the unit gravitational acceleration vector and Ra ^* is a parameter which is proportional to the constraint acting on the medium. S is a small parameter (Smin{(10⁶, 10^-6 Ra ^*)}) which will eventually vanish to zero.In an earlier work [10, 11], we studied the two-dimensional case for both the evolution and stationary problem and showed the existence uniqueness and regularity of the evolution problem. However, we did show that several stationary solutions exist.We were then led to study the asymptotic behaviour of the solution of the evolution problem. To make this study more general we decided to work in three-dimensional space.This article contains the preliminary results to a somewhat fine study to an asymptotic behaviour. More precisely, we establish a regularity theorem and give a uniform estimation in time of second-order space derivatives of the solutions in the case S=0. These properties are similar to those found in two-dimensional Navier-Stokes equations and extend the solutions obtained in [10] to three dimensions.The key to the proof of the existence and uniqueness theorem is an L estimation in space and time of temperature T obtained by rendering the energy equation (0.3) and the Darcy equation (0.2) independent. Then a fixed point method is applied. Space regularity is related to a particular structure of the domain and also to the type of boundary conditions. Uniform time estimates can thus be obtained by a fairly classical method.In the spirit of the Foias and Temam paper [15], we extend some of their results to our system and show that the solution is completely determined by its nodal values on a finite set.Before proceding further, it should be pointed out that the change of the unknown % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2% da9iabeI7aXjabgUcaRiaaigdacqGHsislcaGGOaGaamOEaiaac+ca% caWGObGaaiykaaaa!4004!\[T = \theta + 1 - (z/h)\] leads to homogeneous boundary conditions. The system can then be written% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaadaWcaaqaaiabgkGi2kab% eI7aXbqaaiabgkGi2kaadshaaaGaeyOeI0IaeyiLdqKaeqiUdeNaey% 4kaSIaamOvaiabgwSixlabgEGirlabeI7aXjabgkHiTmaalaaabaGa% aGymaaqaaiaadIgaaaGaeqyXdu3aaSbaaSqaaiaaiodaaeqaaOGaey% ypa0JaaGimaaqaaiaadofadaWcaaqaaiabgkGi2kaadAfaaeaacqGH% ciITcaWG0baaaiabgUcaRiaadAfacqGHRaWkcqGHhis0cqaHapaCcq% GHRaWkcaWGsbGaamyyamaaCaaaleqabaGaaiOkaaaakiaadUgacqaH% 4oqCcqGH9aqpcaaIWaaabaGaaeizaiaabMgacaqG2bGaaeiiaiaadA% facqGH9aqpcaaIWaaabaGaamOvaiabgwSixlaad6gadaWgaaWcbaqe% feKCPfgBaGqbaiaa-XhacqqHtoWraeqaaOGaeyypa0JaaGimaaqaai% abeI7aXjaacIcacaWG4bGaaiilaiaadMhacaGGSaGaaGimaiaacMca% cqGH9aqpcqaH4oqCcaGGOaGaamiEaiaacYcacaWG5bGaaiilaiaadI% gacaGGPaGaeyypa0JaaGimaaqaamaalaaabaGaeyOaIyRaeqiUdeha% baGaeyOaIyRaamiEaaaacaGGOaGaaGimaiaacYcacaWG5bGaaiilai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcqaH4oqCaeaacqGH% ciITcaWG4baaaiaacIcacaWGmbGaaiilaiaadMhacaGGSaGaamOEai% aacMcacqGH9aqpcaaIWaaabaWaaSaaaeaacqGHciITcqaH4oqCaeaa% cqGHciITcaWG5baaaiaacIcacaWG4bGaaiilaiaaicdacaGGSaGaam% OEaiaacMcacqGH9aqpdaWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi% 2kaadMhaaaGaaiikaiaadIhacaGGSaGaamiBaiaacYcacaWG6bGaai% ykaiabg2da9iaaicdaaaGaay5Eaaaaaa!B7C4!\[P_1 \left\{ \begin{gathered} \frac{{\partial \theta }}{{\partial t}} - \Delta \theta + V \cdot \nabla \theta - \frac{1}{h}\upsilon _3 = 0 \hfill \\ S\frac{{\partial V}}{{\partial t}} + V + \nabla \pi + Ra^* k\theta = 0 \hfill \\ {\text{div }}V = 0 \hfill \\ V \cdot n_{|\Gamma } = 0 \hfill \\ \theta (x,y,0) = \theta (x,y,h) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial x}}(0,y,z) = \frac{{\partial \theta }}{{\partial x}}(L,y,z) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial y}}(x,0,z) = \frac{{\partial \theta }}{{\partial y}}(x,l,z) = 0 \hfill \\ \end{gathered} \right.\]

     

Existence and Uniqueness of a Weak Solution to the Initial Mixed Boundary-Value Problem for Quasilinear Elliptic-Parabolic Equations

We prove the existence and the uniqueness of a weak solution to the mixed boundary problem for the elliptic-parabolic equation

Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion:

$$\left\{ {\begin{array}{*{20}{c}}{{u_t} + \left( {u \cdot \nabla } \right)u + v{\Lambda ^{2a}}u = -\nabla p + \theta {e_3},\;{e_3} = {{\left( {0,0,1} \right)}^T},} \\ {{\theta _t} + \left( {u \cdot \nabla } \right)t = 0,} \\ {Divu = 0.} \end{array}} \right.$$

With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α ≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.

     

Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

Blow-up,quenching, aggregation and collapse in a chemotaxis model with reproduction term

In this paper,we consider the following chemotaxis model with ratio-dependent logistic reaction term u/t=D▽(▽u-u▽ω/ω)+u(α-bu/ω),(x,t)∈QT,ω/t=βu-δω,(x,t)∈QT,u▽㏑(u/w)·=0,x ∈Ω,0tT,u(x,0)=u0(x)0,x ∈,w(x,0)=w0(x)0,x ∈,It is shown that the solution to the problem exists globally if b+β≥0 and will blow up or quench if b+β0 by means of function transformation and comparison method.Various asymptotic behavior related to different coefficients and initial data is also discussed.     

The umbral calculus and orthogonal polynomials

We determine all orthogonal polynomials having Boas-Buck generating functions g(t)(xf(t)), where% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHOo% qwcaGGOaGaamiDaiaacMcacqGH9aqpruqqYLwySbacfaGaa8hiamaa% BeaaleaacaaIWaaabeaakiaadAeacaqGGaWaaSbaaSqaaiaabgdaae% qaaOGaaeikaiaadggacaGGSaGaa8hiaiaadshacaqGPaGaaeilaiaa% bccacaqGGaGaaeiiaiaadggacqGHGjsUcaaIWaGaaiilaiaa-bcacq% GHsislcaaIXaGaaiilaiaa-bcacqGHsislcaaIYaGaaiilaiablAci% ljaacUdaaeaacqqHOoqwcaGGOaGaamiDaiaacMcacqGH9aqpcaWFGa% WaaSraaSqaaiaaicdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOm% aaqabaGccaGGOaWaaSqaaSqaaiaaigdaaeaacaaIZaaaaOGaaiilai% aa-bcadaWcbaWcbaGaaGOmaaqaaiaaiodaaaGccaGGSaGaa8hiaiaa% dshacaGGPaGaa8hiamaaBeaaleaacaaIWaaabeaakiaadAeacaqGGa% WaaSbaaSqaaiaabkdaaeqaaOGaaeikamaaleaaleaacaaIYaaabaGa% aG4maaaakiaacYcacaWFGaWaaSqaaSqaaiaaisdaaeaacaaIZaaaaO% Gaaiilaiaa-bcacaWG0bGaaiykaiaacYcacaWFGaWaaSraaSqaaiaa% icdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOmaaqabaGccaGGOa% WaaSqaaSqaaiaaisdaaeaacaaIZaaaaOGaaiilaiaa-bcadaWcbaWc% baGaaGynaaqaaiaaiodaaaGccaGGSaGaa8hiaiaadshacaGGPaGaai% 4oaaqaaiabfI6azjaacIcacaWG0bGaaiykaiabg2da9iaa-bcadaWg% baWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaacaqGZaaabe% aakiaacIcadaWcbaWcbaGaaGymaaqaaiaaisdaaaGccaGGSaGaa8hi% amaaleaaleaacaaIYaaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaS% qaaiaaiodaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGaaiykaiaa% -bcadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaaca% qGZaaabeaakiaabIcadaWcbaWcbaGaaGOmaaqaaiaaisdaaaGccaGG% SaGaa8hiamaaleaaleaacaaIZaaabaGaaGinaaaakiaacYcacaWFGa% WaaSqaaSqaaiaaiwdaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGa% aiykaiaacYcaaeaadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiam% aaBaaaleaacaqGZaaabeaakiaacIcadaWcbaWcbaGaaG4maaqaaiaa% isdaaaGccaGGSaGaa8hiamaaleaaleaacaaI1aaabaGaaGinaaaaki% aacYcacaWFGaWaaSqaaSqaaiaaiAdaaeaacaaI0aaaaOGaaiilaiaa% -bcacaWG0bGaaiykaiaacYcacaGGUaGaa8hiamaaBeaaleaacaaIWa% aabeaakiaadAeacaqGGaWaaSbaaSqaaiaabodaaeqaaOGaaeikamaa% leaaleaacaaI1aaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaSqaai% aaiAdaaeaacaaI0aaaaOGaaiilaiaa-bcadaWcbaWcbaGaaG4naaqa% aiaaisdaaaGccaGGSaGaa8hiaiaadshacaGGPaGaaiOlaaaaaa!C1F3!\[\begin{gathered}\Psi (t) = {}_0F{\text{ }}_{\text{1}} {\text{(}}a, t{\text{), }}a \ne 0, - 1, - 2, \ldots ; \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{2}} (\tfrac{1}{3}, \tfrac{2}{3}, t) {}_0F{\text{ }}_{\text{2}} {\text{(}}\tfrac{2}{3}, \tfrac{4}{3}, t), {}_0F{\text{ }}_{\text{2}} (\tfrac{4}{3}, \tfrac{5}{3}, t); \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{3}} (\tfrac{1}{4}, \tfrac{2}{4}, \tfrac{3}{4}, t) {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{2}{4}, \tfrac{3}{4}, \tfrac{5}{4}, t), \hfill \\{}_0F{\text{ }}_{\text{3}} (\tfrac{3}{4}, \tfrac{5}{4}, \tfrac{6}{4}, t),. {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{5}{4}, \tfrac{6}{4}, \tfrac{7}{4}, t). \hfill \\\end{gathered}\]We also determine all Sheffer polynomials which are orthogonal on the unit circle. The formula for the product of polynomials of the Boas-Buck type is obtained.     

An existence principle for solutions to a singular boundary-value problems

The singular boundary-value problem

$ \left\{ {\begin{array}{*{20}{c}} {{u^{\prime\prime}} + g\left( {t,u,{u^{\prime}}} \right) = 0\quad {\text{for}}\quad t \in \left( {0,1} \right),} \hfill \\ {u(0) = u(1) = 0} \hfill \\ \end{array} } \right. $

is studied. The singularity may appear at u?=?0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions.

     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

This paper is concerned with the Cauchy problem for the nonlinear parabolic equation $${\partial _t}u| = \vartriangle u + F(x,t,u,\nabla u){\text{ in }}{{\text{R}}^N} \times (0,\infty ),{\text{ }}u(x,0) = \varphi (x){\text{ in }}{{\text{R}}^N},$$ , where $$\begin{gathered} N \geqslant 1, \hfill \\ F \in C(R^N \times (0,\infty ) \times R \times R^N ), \hfill \\ \phi \in L^\infty (R^N ) \cap L^1 (R^N ,(1 + |x|^K )dx)forsomeK \geqslant 0 \hfill \\ \end{gathered} $$ . We give a sufficient condition for the solution to behave like a multiple of the Gauss kernel as t → ∞ and obtain the higher order asymptotic expansions of the solution in W ^1,q (R ^N ) with 1 ≤ q ≤ ∞.     

Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).

     

Invertibility of an Operator Appearing in the Control Theory for Linear Systems

We give sufficient conditions for the existence of a bounded inverse operator for a linear operator appearing in the theory of optimal control of linear systems in Hilbert space and having a matrix representation of the form , where F₃, F₄ are nonnegative self-adjoint operators. The invertibility of the operator under study is used to prove the unique solvability of a certain two-point boundary-value problem that arises from conditions for optimal control.     

Nonexistence results and estimates for some nonlinear elliptic problems

Here we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type or . We give integral estimates and nonexistence results. They depend on properties of the supersolutions of the equationsL _A ^u=0,L _B ^v=0, which suppose weak coercivity conditions. Under stronger conditions, we give pointwise estimates in case of equalities, using Harnack properties.     

Solutions of a system of diffusion equations

We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on \mathbbR ×W{\mathbb{R}} \times \Omega :

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

Sums of Five, Seven and Nine Squares

Let r _k(n) denote the number of representations of an integer n as a sum of k squares. We prove that

Probabilistic approach to the Dirichlet problem of second order elliptic PDE

In this paper we provide a probabilistic approach to the following Dirichlet Problem{(∑x~4(α~(ij) x~j) ∑b~ix~i ξ)u=0, iD u=g, on D,without assuming that the eigenvalues of the operator∑x~i(α~(ij)x~j) ∑b~ix~i ξwith Dirichlet boundary conditions are all strictly negative. The results of this paper generalizedthose of Ma.     

The conductor of a cyclic quartic field using Gauss sums

Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q. It is known that there are unique integers A, B, C, D such that where A is squarefree and odd, D=B ²+C ² is squarefree, B $$ " align="middle" border="0"> 0 , C $$ " align="middle" border="0"> 0, GCD(A,D)=1. The conductor f(K) of K is f(K) = 2 ^l |A|D, where A simple proof of this formula for f(K) is given, which uses the basic properties of quartic Gauss sums.     

A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

The solvability of the nonlocal boundary value problem

Oscillatory properties of solutions of three-dimensional differential systems of neutral type

The purpose of this paper is to obtain oscillation criteria for the differential system