\u5bf9\u8fb9\u754c\u6761\u4ef6, \u6211\u4eec\u7684\u4e92\u8865\u6761\u4ef6\u5982\u4e0b: \u5047\u8bbe$\\mathrm{Re}p\\geq -\\delta_1\\lvert \\zeta \\rvert^2$, \u5176\u4e2d$0<\\delta_1<\\delta$\u4e14$\\lvert p \\rvert^2+\\lvert \\zeta \\rvert^4>0$, \u6ce8\u610f\u8fd9\u91cc$p$\u4e3a\u590d\u6570. \u5219\u53ef\u4ee5\u8bc1\u660e<\/li>\n<\/ul>\n$$
\nL(x,t,i(\\zeta+\\tau\\nu),p)
\n$$
\n\u4f5c\u4e3a$\\tau$\u7684\u51fd\u6570\u6709$n$\u4e2a\u865a\u90e8\u4e3a\u6b63\u7684\u6839$\\tau_s^+$, \u4ee5\u53ca\u865a\u90e8\u4e3a\u8d1f\u7684\u6839$\\tau_s^-$, \u5176\u4e2d$\\zeta=\\zeta(x)$\u662f$x\\in\\partial\\Omega$\u5904\u7684\u5207\u5411\u91cf, \u800c$\\nu=\\nu(x)$\u4e3a$x\\in\\partial\\Omega$\u5904\u7684\u5355\u4f4d\u5185\u6cd5\u5411\u91cf.<\/p>\n
\u4e8b\u5b9e\u4e0a, \u5728\u6211\u4eec\u7684\u60c5\u5f62, \u76f4\u63a5\u9a8c\u8bc1\u77e5
\n$$
\nL(x,t,i(\\zeta+\\tau\\nu),p)=0
\n$$
\n\u7684\u6839\u5206\u522b\u4e3a
\n$$\\begin{aligned}
\n \\tau^\\pm_2&=\\cdots=\\tau^\\pm_n=\\pm i\\sqrt{p+\\lvert \\zeta \\rvert^2},\\\\
\n \\tau^\\pm_1&=\\frac{\\frac{2(\\alpha-1)\\nabla_\\nu w\\cdot\\nabla_\\zeta w}{1+\\lvert \\nabla w \\rvert^2}\\pm\\sqrt{\\left( \\frac{2(\\alpha-1)(\\nabla_\\nu w\\cdot\\nabla_\\zeta w)}{1+\\lvert \\nabla w \\rvert^2} \\right)^2-4\\left( 1+\\frac{(\\alpha-1)\\lvert \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2} \\right)\\left( p+\\lvert \\zeta \\rvert^2+\\frac{(\\alpha-1)\\lvert \\nabla_\\zeta w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2} \\right)}}{2\\left( 1+\\frac{(\\alpha-1)\\lvert \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2} \\right)}\\\\
\n &=\\pm i\\sqrt{p+\\lvert \\zeta \\rvert^2}+o(\\alpha-1).
\n\\end{aligned}$$
\n\u5bf92-\u503c\u51fd\u6570$\\sqrt{p+\\lvert \\zeta \\rvert^2}$\u6211\u4eec\u53ef\u4ee5\u9009\u53d6\u5176\u4e2d\u4e00\u4e2a\u5206\u652f, \u4f7f\u5f97\u5176\u5b9e\u90e8\u662f\u5927\u4e8e\u96f6\u7684. \u4e3a\u65b9\u4fbf\u8d77\u89c1, \u6211\u4eec\u4ee4$\\tau_0=i\\sqrt{p+\\lvert \\zeta \\rvert^2}$.<\/p>\n
\u73b0\u5728, \u6211\u4eec\u4ee4
\n$$
\nL_0^+(\\zeta,\\tau,p)=(\\tau-\\tau_1^+)\\cdots(\\tau-\\tau_n^+)=(\\tau-\\tau_1^+)(\\tau-\\tau_0)^{n-1},
\n$$
\n\u4ee5\u53ca
\n$$
\nR(\\zeta,\\tau,p)=B_0(i\\zeta,i\\tau,p)\\hat L_0(i\\zeta,i\\tau,p),
\n$$
\n\u5176\u4e2d
\n$$
\nB_0(i\\zeta,i\\tau,p)=B_0(x_0,t_0,i(\\zeta+\\tau\\nu),p)=\\mathrm{diag}(\\tau,\\cdots,\\tau,1)_{n\\times n},\\quad (x_0,t_0)\\in\\partial\\Omega\\times(0,T),
\n$$
\n\u800c
\n$$
\nL_0(i\\zeta,i\\tau,p)=L_0(x_0,t_0,i(\\zeta+\\tau\\nu),p)=(p+\\lvert \\zeta \\rvert^2+\\tau^2)E+X_\\tau^TX_\\tau,
\n$$
\n\u5176\u4e2d
\n$$
\nX_\\tau=\\sqrt{\\frac{\\alpha-1}{1+\\lvert \\nabla w \\rvert^2}} (\\nabla_\\zeta w+\\tau\\nabla_\\nu w).
\n$$
\n\u6ce8\u610f\u5230$E+X_\\tau^TX_\\tau$\u7684\u4f34\u968f\u77e9\u9635
\n$$
\n(E+X_\\tau^TX_\\tau)^*=(1+X_\\tau X_\\tau^T)E-X_\\tau^TX_\\tau.
\n$$
\n\u6216\u8005
\n$$
\n(\\lambda E+X_\\tau^TX_\\tau)^*=(\\lambda^{n-1}+\\lambda^{n-2} X_\\tau X_\\tau^T)E-\\lambda^{n-2} X_\\tau^TX_\\tau.
\n$$
\n\u56e0\u6b64
\n$$
\n\\hat L_0(i\\zeta,i\\tau,p)=\\left( (p+\\lvert \\zeta \\rvert^2+\\tau^2)^{n-1}+(p+\\lvert \\zeta \\rvert^2+\\tau^2)^{n-2}X_\\tau X_\\tau^T \\right)E
\n-(p+\\lvert \\zeta \\rvert^2+\\tau^2)^{n-2}X_\\tau^T X_\\tau.
\n$$
\n\u6ce8\u610f\u5230
\n$$
\n\\begin{aligned}
\n X_\\tau X_\\tau^T&=\\lvert X_\\tau \\rvert^2=(\\alpha-1)\\frac{\\lvert \\nabla_\\zeta w+\\tau \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2}\\\\
\n &=(\\alpha-1)\\frac{1}{1+\\lvert \\nabla w \\rvert^2}\\left( \\tau^2\\lvert \\nabla_\\nu w \\rvert^2+2\\tau\\nabla_\\zeta w\\nabla_\\nu w+\\lvert \\nabla_\\nu w \\rvert^2 \\right).
\n\\end{aligned}
\n$$<\/p>\n
\u6211\u4eec\u9996\u5148\u5bf9$n=2$\u6765\u8fdb\u884c\u9a8c\u8bc1. \u6b64\u65f6
\n$$
\nL_0^+(i\\zeta,i\\tau,p)=(\\tau-\\tau_0)(\\tau-\\tau_1^+),
\n$$
\n\u800c
\n$$
\nB_0(i\\zeta,i\\tau,p)=\\mathrm{diag}(\\tau,1),
\n$$
\n\u4ee5\u53ca
\n$$
\n\\begin{aligned}
\n \\hat L_0(i\\zeta,i\\tau,p)&=\\left( p+\\lvert \\zeta \\rvert^2+\\tau^2+(\\alpha-1)\\frac{\\tau^2\\lvert \\nabla_\\nu w \\rvert^2+2\\tau\\nabla_\\zeta w\\nabla_\\nu w^T+\\lvert \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2} \\right)E\\\\
\n &\\qquad-(\\alpha-1)\\frac{\\tau^2\\nabla_\\nu w^T\\nabla_\\nu w+2\\tau\\nabla_\\nu w^T\\nabla_\\zeta w+\\nabla_\\zeta w^T\\nabla_\\zeta w}{1+\\lvert \\nabla w \\rvert^2}.
\n\\end{aligned}$$
\n\u56e0\u6b64\u82e5\u8bb0
\n$$
\nR(i\\zeta,i\\tau,p) :=B_0(i\\zeta,i\\tau,p)\\hat L_0(i\\zeta,i\\tau,p),
\n$$
\n\u5219
\n$$
\n\\begin{aligned}
\n R_{11}&=\\tau\\left( p+\\lvert \\zeta \\rvert^2+\\tau^2+(\\alpha-1)\\frac{\\tau^2\\lvert \\nabla_\\nu w \\rvert^2+2\\tau\\nabla_\\zeta w\\nabla_\\nu w^T+\\lvert \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2} \\right)\\\\
\n &\\qquad-(\\alpha-1)\\tau\\frac{\\tau^2\\nabla_\\nu w^1\\nabla_\\nu w^1+2\\tau\\nabla_\\nu w^1\\nabla_\\zeta w^1+\\nabla_\\zeta w^1\\nabla_\\zeta w^1}{1+\\lvert \\nabla w \\rvert^2}\\\\
\n R_{12}&=-(\\alpha-1)\\tau \\frac{\\tau^2\\nabla_\\nu w^1\\nabla_\\nu w^2+2\\tau\\nabla_\\nu w^1\\nabla_\\zeta w^2+\\nabla_\\zeta w^1\\nabla_\\zeta w^2}{1+\\lvert \\nabla w \\rvert^2}\\\\
\n R_{21}&=-(\\alpha-1)\\frac{\\tau^2\\nabla_\\nu w^2\\nabla_\\nu w^1+2\\tau\\nabla_\\nu w^2\\nabla_\\zeta w^1+\\nabla_\\zeta w^2\\nabla_\\zeta w^1}{1+\\lvert \\nabla w \\rvert^2}\\\\
\n R_{22}&=\\left( p+\\lvert \\zeta \\rvert^2+\\tau^2+(\\alpha-1)\\frac{\\tau^2\\lvert \\nabla_\\nu w \\rvert^2+2\\tau\\nabla_\\zeta w\\nabla_\\nu w^T+\\lvert \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2} \\right)\\\\
\n &\\qquad-(\\alpha-1)\\frac{\\tau^2\\nabla_\\nu w^2\\nabla_\\nu w^2+2\\tau\\nabla_\\nu w^2\\nabla_\\zeta w^2+\\nabla_\\zeta w^2\\nabla_\\zeta w^2}{1+\\lvert \\nabla w \\rvert^2}.
\n\\end{aligned}
\n$$
\n\u6ce8\u610f\u5230
\n$$
\np+\\lvert \\zeta \\rvert^2+\\tau_1^{+2}+(\\alpha-1)\\frac{\\lvert \\nabla_{\\zeta+{\\tau_1^+}\\nu}w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2}=0,
\n$$
\n\u56e0\u6b64
\n$$
\n\\begin{aligned}
\n &p+\\lvert \\zeta \\rvert^2+\\tau^2+(\\alpha-1)\\frac{\\tau^2\\lvert \\nabla_\\nu w \\rvert^2+2\\tau\\nabla_\\zeta w\\nabla_\\nu w^T+\\lvert \\nabla_\\nu w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2}\\\\
\n &\\qquad=\\tau^2-\\tau_1^{+2}+(\\alpha-1)\\frac{(\\tau^2-\\tau_1^{+2})\\lvert \\nabla_\\nu w \\rvert^2+2(\\tau-\\tau_1^+)\\nabla_\\zeta w\\nabla_\\nu w^T}{1+\\lvert \\nabla w \\rvert^2}.
\n\\end{aligned}
\n$$
\n\u4ee4
\n$$\\begin{aligned}
\n a&=(\\alpha-1)\\frac{|\\nabla_\\nu w|^2}{1+\\lvert \\nabla w \\rvert^2},\\\\
\n a_{11}&=(\\alpha-1)\\frac{|\\nabla_\\nu w^1|^2}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n a_{12}=a_{21}=(\\alpha-1)\\frac{\\nabla_\\nu w^1\\nabla_\\nu w^2}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n a_{22}=(\\alpha-1)\\frac{|\\nabla_\\nu w^2|^2}{1+\\lvert \\nabla w \\rvert^2},\\\\
\n b&=(\\alpha-1)\\frac{\\nabla_\\nu w\\nabla_\\zeta w^T}{1+\\lvert \\nabla w \\rvert^2},\\\\
\n b_{11}&=(\\alpha-1)\\frac{\\nabla_\\nu w^1\\nabla_\\zeta w^1}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n b_{12}=(\\alpha-1)\\frac{\\nabla_\\nu w^1\\nabla_\\zeta w^2}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n b_{21}=(\\alpha-1)\\frac{\\nabla_\\nu w^2\\nabla_\\zeta w^1}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n b_{22}=(\\alpha-1)\\frac{\\nabla_\\nu w^2\\nabla_\\zeta w^2}{1+\\lvert \\nabla w \\rvert^2},\\\\
\n c&=(\\alpha-1)\\frac{\\lvert \\nabla_\\zeta w \\rvert^2}{1+\\lvert \\nabla w \\rvert^2},\\\\
\n c_{11}&=(\\alpha-1)\\frac{\\lvert \\nabla_\\zeta w^1 \\rvert^2}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n c_{12}=c_{21}=(\\alpha-1)\\frac{ \\nabla_\\zeta w^1\\nabla_\\zeta w^2}{1+\\lvert \\nabla w \\rvert^2},\\quad
\n c_{22}=(\\alpha-1)\\frac{\\lvert \\nabla_\\zeta w^2 \\rvert^2}{1+\\lvert \\nabla w \\rvert^2},
\n \\end{aligned}$$
\n \u5219
\n $$\\begin{aligned}
\n R_{11}&=\\tau \\left( (\\tau^2-\\tau_1^{+2})(1+a)+2(\\tau-\\tau_1^+)b \\right)-a_{11}\\tau^3-2b_{11}\\tau^2-c_{11}\\tau,\\\\
\n R_{12}&=-a_{12}\\tau^3-2b_{12}\\tau^2-c_{12}\\tau,\\\\
\n R_{21}&=-a_{21}\\tau^2-2b_{21}\\tau-c_{21},\\\\
\n R_{22}&=\\left( (\\tau^2-\\tau_1^{+2})(1+a)+2(\\tau-\\tau_1^+)b \\right)-a_{22}\\tau^2-2b_{22}\\tau-c_{22}.
\n \\end{aligned}$$
\n \u56e0\u6b64, \u6a21\u6389$L_0^+=(\\tau-\\tau_1^+)(\\tau-\\tau_0)$, \u6211\u4eec\u5f97\u5230
\n $$
\n \\begin{aligned}
\n R_{11}’&=\\tau\\left(\\tau_0^2(1+a-a_{11})+\\tau_0\\tau_1^+(1+a-a_{11})-a_{11}\\tau_1^{+2}-2b_{11}(\\tau_0+\\tau_1^+)+2b\\tau_0-c_{11}\\right)\\\\
\n &\\qquad-\\tau_0\\tau_1^+\\left( (1+a-a_{11})(\\tau_0+\\tau_1^+)-2b_{11}+2b \\right),\\\\
\n R_{12}’&=-\\tau\\left( a_{12}(\\tau_0^2+\\tau_0\\tau_1^++\\tau_1^+)+2b_{12}(\\tau_0+\\tau_1^+)+c_{12} \\right)\\\\
\n &\\qquad-\\tau_0\\tau_1^+\\left( a_{12}(\\tau_0+\\tau_1^+)+2b_{12} \\right),\\\\
\n R_{21}’&=-\\tau\\left( a_{12}(\\tau_0+\\tau_1^+)+2b_{21} \\right)\\\\
\n &\\qquad+a_{12}\\tau_0\\tau_1^+-c_{12},\\\\
\n R_{22}’&=\\tau\\left( (1+a-a_{22})(\\tau_0+\\tau_1^+)-2b_{22}+2b \\right)\\\\
\n &\\qquad-\\tau_1^+\\left( (1+a)\\tau_1^++(1+a-a_{22})\\tau_0^++2b \\right)-c_{22}.
\n \\end{aligned}
\n $$
\n \u6211\u4eec\u7684\u8fb9\u754c\u4e92\u8865\u6761\u4ef6\u8981\u6c42
\n $$
\n \\mathrm{rank}R’=2.
\n $$
\n \u8fd9\u5728$\\alpha=1$\u7684\u60c5\u5f62, \u53ea\u9700\u8981$\\tau_0\\neq0$\u5373\u53ef.<\/p>\n
\n- \u5bf9\u521d\u503c\u6761\u4ef6, \u6211\u4eec\u7684\u4e92\u8865\u6761\u4ef6\u5982\u4e0b: \u6ce8\u610f\u5230, \u6211\u4eec\u7684\u521d\u503c\u6761\u4ef6\u6539\u5199\u6210\u4e00\u822c\u5f62\u5f0f\u4e3a<\/li>\n<\/ul>\n
$$
\n \\mathcal{C}u|_{t=0}=\\Psi,
\n$$
\n\u5373
\n$$
\n\\sum_{\\beta=1}^n C_{q\\beta}(x,\\partial_x,\\partial_t)u^\\beta|_{t=0}=\\psi_q(x), \\quad q=1,2,\\ldots,n.
\n$$
\n\u5728\u6211\u4eec\u7684\u60c5\u5f62, $C_{q\\beta}=\\delta_{q\\beta}$, \u56e0\u6b64\u521d\u503c\u7b97\u5b50$\\mathcal{C}$\u7684\u4e3b\u90e8$\\mathcal{C}^0=\\mathcal{C}$. \u5982\u679c\u6211\u4eec\u8bb0\u4ee5$C^0_{q\\beta}=\\delta_{q\\beta}$\u4e3a\u5143\u7d20\u7684\u77e9\u9635\u4e3a$C^0$, \u5219
\n$$
\nC^0(x,i\\xi,p)=E_n.
\n$$<\/p>\n
\u56de\u5fc6, $L_0$\u4e3a$l_{\\beta\\gamma}^0(x,t,i\\xi,p)$\u4f5c\u4e3a$(\\beta,\\gamma)$\u5143\u7d20\u7684\u77e9\u9635. \u5728\u6211\u4eec\u7684\u60c5\u5f62,
\n$$
\nL_0(x,0,0,p)=(\\delta_{\\beta\\gamma}p)_{n\\times n}=p E_n.
\n$$
\n\u82e5\u8bb0$\\hat L_0(x,t,i\\xi,p)$\u4e3a$L_0(x,t,i\\xi,p)$\u7684\u4f34\u968f\u77e9\u9635\u4e3a$C^0$,
\n\u5219\u5728\u6211\u4eec\u7684\u60c5\u5f62
\n$$
\n\\hat L_0(x,0,0,p)=p^{n-1}E_n.
\n$$
\n\u800c\u6240\u8c13\u7684\u521d\u503c\u6761\u4ef6\u7684\u4e92\u8865\u6027\u6761\u4ef6\u5373\u662f\u8bf4\u77e9\u9635
\n$$
\nC_0(x,0,p)\\hat L_0(x,0,0,p)=p^{n-1}E_n
\n$$
\n\u7684\u884c\u5411\u91cf\u6a21\u6389\u9996\u9879\u7cfb\u6570\u4e3a1\u7684\u591a\u5f62\u5f0f(\u5373$\\det L_0(x,0,0,0)$\u7684\u6700\u9ad8\u6b21\u9879)$p^n$\u662f\u7ebf\u6027\u65e0\u5173\u7684.<\/p>\n
\n\t
\u53c2\u8003\u6587\u732e<\/div>\n\t
<\/ol><\/div>","protected":false},"excerpt":{"rendered":"\u6211\u4eec\u9996\u5148\u6765\u770b$\\alpha$\u8c03\u548c\u6620\u7167\u6d41, \u5176\u65b9\u7a0b\u53ef\u4ee5\u5199\u4f5c $$\\left\\{\\begin{aligned} \\… \u7ee7\u7eed\u9605\u8bfb$\\alpha$-\u8c03\u548c\u6620\u7167\u6d41\u7684\u521d\u8fb9\u503c\u95ee\u9898\u4e4b\u5b58\u5728\u6027<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[139,140,31],"class_list":["post-703","post","type-post","status-publish","format-standard","hentry","category-math","tag-alpha-diaoheyingzhao","tag-hubutiaojian","tag-31","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/703","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=703"}],"version-history":[{"count":2,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/703\/revisions"}],"predecessor-version":[{"id":705,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/703\/revisions\/705"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}