Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term
Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term
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含“粘度”项的振动膜方程 BV 函数空间解的存在唯一性
DOI：
10.1155/2013/936915
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
K. Kikuchi
中科院分区：
文献类型：
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作者：
K. Kikuchi
研究了一个含有粘性项的振动膜非线性运动方程。通常，这一项被添加，并且众所周知，这个方程在函数空间中是适定的。本文将粘性项改为，并证明了若初始数据是稍微光滑的（但属于是充分的），则弱解在BV函数空间中唯一存在.
A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. Usually, the term <path id="x1D462" d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z" /> is added, and it is well known that this equation is well posed in the space of <path id="x1D44A" d="M1004 650l-6 -29q-54 -6 -71 -19.5t-51 -74.5l-271 -539h-33l-98 506h-3l-258 -506h-30l-78 532q-10 67 -21.5 80t-66.5 21l6 29h241l-8 -29l-26 -5q-34 -6 -41 -16t-3 -47l59 -425h4l251 510h31l102 -510h2q150 299 198 423q14 40 8.5 48.5t-47.5 16.5l-28 5l7 29h231z
" /> functions. In this paper, the viscosity term is changed to <path id="x64" d="M517 51v-25q-101 -18 -155 -38v65l-61 -36q-50 -29 -78 -29q-68 0 -126.5 58t-58.5 155q0 103 77 175.5t180 72.5q36 0 67 -12v143q0 48 -11.5 58.5t-65.5 14.5v25q79 9 156 34v-597q0 -38 7.5 -48t36.5 -13zM362 85v275q-35 51 -103 51q-20 0 -40.5 -8.5t-42 -28
t-35 -57.5t-13.5 -89q0 -83 42 -130t96 -47q51 0 96 34z" /> , and it is proved that if initial data is slightly smooth (but belonging to is sufficient), then a weak solution exists uniquely in the space of BV functions.