We consider, for a bounded open domain Ω in <em>R<sup>n</sup></em> and a function <em>u</em> : Ω → <em>R<sup>m</sup></em>, the quasilinear elliptic system...We consider, for a bounded open domain Ω in <em>R<sup>n</sup></em> and a function <em>u</em> : Ω → <em>R<sup>m</sup></em>, the quasilinear elliptic system: <img src="Edit_8a3d3105-dccb-405b-bbbc-2084b80b6def.bmp" alt="" /> (1). We generalize the system (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> in considering a right hand side depending on the jacobian matrix <em>Du</em>. Here, the star in (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> indicates that <em>f </em>may depend on <em>Du</em>. In the right hand side, <em>v</em> belongs to the dual space <em>W</em><sup>-1,<em>P</em>’</sup>(Ω, <span style="white-space:nowrap;"><em>ω</em></span><sup>*</sup>,<em> R<sup>m</sup></em>), <img src="Edit_d584a286-6ceb-420c-b91f-d67f3d06d289.bmp" alt="" />, <em>f </em>and <em>g</em> satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for <em>σ</em>, but with only very mild monotonicity assumptions.展开更多
Following & recent paper of the authors in Communications in Partial Differential Equa- tions, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed c...Following & recent paper of the authors in Communications in Partial Differential Equa- tions, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.展开更多
Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system (denoted by CNSP), which converges up to extraction of a subsequence, the limit system may not be the same...Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system (denoted by CNSP), which converges up to extraction of a subsequence, the limit system may not be the same system. By introducing Young measures as in [6, 15], the authors deduce the system (HCNSP) which the limit functions must satisfy. Then they solve this system in a subclass where Young measures are convex combinations of Dirac measures, to give the information on the propagation of density-oscillations. The results for strong solutions to (CNSP) (see Corollary 6.1) are also obtained.展开更多
文摘We consider, for a bounded open domain Ω in <em>R<sup>n</sup></em> and a function <em>u</em> : Ω → <em>R<sup>m</sup></em>, the quasilinear elliptic system: <img src="Edit_8a3d3105-dccb-405b-bbbc-2084b80b6def.bmp" alt="" /> (1). We generalize the system (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> in considering a right hand side depending on the jacobian matrix <em>Du</em>. Here, the star in (<em>QES</em>)<sub>(<em>f</em>,<em>g</em>)</sub> indicates that <em>f </em>may depend on <em>Du</em>. In the right hand side, <em>v</em> belongs to the dual space <em>W</em><sup>-1,<em>P</em>’</sup>(Ω, <span style="white-space:nowrap;"><em>ω</em></span><sup>*</sup>,<em> R<sup>m</sup></em>), <img src="Edit_d584a286-6ceb-420c-b91f-d67f3d06d289.bmp" alt="" />, <em>f </em>and <em>g</em> satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for <em>σ</em>, but with only very mild monotonicity assumptions.
基金The first author is supported by the Chinese Youth Foundation and the innovation Funds of theChinese Academy of Sciences. The
文摘Following & recent paper of the authors in Communications in Partial Differential Equa- tions, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.
基金the National Natural Science Foundation of China (No. 10531020)the Program of 985Innovation Engineering on Information in Xiamen University (2004–2007)the New Century ExcellentTalents in Xiamen University
文摘Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system (denoted by CNSP), which converges up to extraction of a subsequence, the limit system may not be the same system. By introducing Young measures as in [6, 15], the authors deduce the system (HCNSP) which the limit functions must satisfy. Then they solve this system in a subclass where Young measures are convex combinations of Dirac measures, to give the information on the propagation of density-oscillations. The results for strong solutions to (CNSP) (see Corollary 6.1) are also obtained.
