{"id":318,"date":"2017-05-01T10:22:55","date_gmt":"2017-05-01T01:22:55","guid":{"rendered":"http:\/\/192.168.99.111\/kyeonghun\/?page_id=318"},"modified":"2017-09-22T09:01:50","modified_gmt":"2017-09-22T00:01:50","slug":"%ec%a0%84%ea%b3%b5%ec%86%8c%ea%b0%9c","status":"publish","type":"page","link":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/%ec%a0%84%ea%b3%b5%ec%86%8c%ea%b0%9c\/","title":{"rendered":"\uc804\uacf5\uc18c\uac1c"},"content":{"rendered":"

 <\/p>\n

Stochastic Partial Differential Equations<\/strong> (SPDE in short):<\/p>\n

\u00a0\ud3b8\ubbf8\ubd84\ubc29\uc815\uc2dd (PDE)\uc740\u00a0\uac70\uc2dc\uc801\uc778 \uad00\uc810\uc5d0\uc11c \ud604\uc0c1\uc758 \ud3c9\uade0\uc801\uc778 \uc6c0\uc9c1\uc784\uc744 \u00a0\ubb18\uc0ac\ud55c\ub2e4. \ubc18\uba74 SPDE\ub294 \ubd88\uaddc\uce59\ud55c\u00a0\ub3d9\ub825\uc774 \uad6c\ud604\ub41c \ub79c\ub364\ub178\uc774\uc988\uac00 \uc801\uc6a9\ub41c \ubc29\uc815\uc2dd\uc73c\ub85c \u00a0PDE\ub97c \ud3ec\ud568\ud558\ub294 \ubcf4\ub2e4 \uc77c\ubc18\uc801\uc778 \ud615\ud0dc\uc758 \ubc29\uc815\uc2dd\uc774\ub2e4. \uc27d\uac8c \ub9d0\ud558\uc790\uba74 <\/p>\n

  <\/span>   <\/span>\"\[\text{SPDE theory=PDE theory+Probability}\]\"<\/p>\n

\uc774\ub77c\uace0 \ud560\uc218 \uc788\ub2e4.<\/p>\n

Deterministic and Stochastic \u00a0Science \ube44\uad50:<\/p>\n

\"\"<\/p>\n

PDE \uc640 SPDE \ube44\uad50:<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0\u00a0\"\"<\/p>\n

\uc8fc\uc694 \uc5f0\uad6c\ub418\ub294 SPDE\uc758 \uc77c\ubc18\uc801\uc778 \ud615\ud0dc\ub294 Parabolic PDE\uc778<\/p>\n

\n

  <\/span>   <\/span>\"\[du=(Lu+f(u))dt, \,\,t>0\quad;\,\, u(0,\cdot)=u_0\]\"<\/p>\n

\ub97c \uc77c\ubc18\ud654\ud55c<\/p>\n

\n

  <\/span>   <\/span>\"\[du=(Lu+f(u))dt+\sum_k (\Lambda^ku+g^k(u))dW^k_t, \,\,t>0 \quad\, ;\,u(0,\cdot)=u_0\]\"<\/p>\n

\uc758 \ud615\ud0dc\uc774\ub2e4. \u00a0\uc704\uc5d0\uc11c \"L, \Lambda^k\"\ub294 (\ub79c\ub364\ud55c) \ubbf8\ubd84\uc5f0\uc0b0\uc790\ub4e4\uc774\uace0 \uc2dc\ub9ac\uc988\uc5d0 \ub098\ud0c0\ub098\ub294 \uac83\ub4e4\uc740 \ud655\ub960\uacfc\uc815(\ud639\uc740 \ub178\uc774\uc988)\ub4e4\uc778 \"W^k_t\"\uc5d0 \ub300\ud55c \ud655\ub960\uc801\ubd84(stochastic integral)\ub4e4\uc774\ub2e4.<\/p>\n

\uc608\ub97c \ub4e4\uc5b4 parabolic PDE\uc758 \ub300\ud45c\uac00 \ub418\ub294 \uc5f4\ubc29\uc815\uc2dd(heat equation)<\/p>\n

\n

  <\/span>   <\/span>\"\[<\/p>\n<\/p>\n

\uc744 \uc77c\ubc18\ud654\ud558\uc5ec, \ub79c\ub364 \ub178\uc774\uc988\uc758 \uc601\ud5a5\ud558\uc5d0\uc11c\uc758 \uc5f4\uc804\ub3c4\ub97c \ub098\ud0c0\ub0b4\ub294 \ud655\ub960\uc5f4\ubc29\uc815\uc2dd(stochastic heat equation)<\/p>\n

\n

  <\/span>   <\/span>\"\[<\/p>\n<\/p>\n

\u00a0\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \u00a0 \ub2e4\ub978 \uc7ac\ubbf8\uc788\ub294 \uc608\ub85c \ud655\ub960\ub098\ube44\uc544\uc2a4\ud1a1\uc2a4\ubc29\uc815\uc2dd\uc744 \uc0dd\uac01\ud574\ubcfc \uc218 \uc788\ub2e4. \u00a0\ud655\ub960\ub098\ube44\uc544\uc2a4\ud1a1\uc2a4\ubc29\uc815\uc2dd(stochastic Navier-Stokes equation)\uc740 \uc815\ud655\ud788 \uce21\uc815 \ud560 \uc218 \uc5c6\ub294 \ubd88\ud655\uc2e4(random)\ud55c \uc678\ubd80\uc758 \ud798\uc774 \ud750\ub974\ub294 \uc720\uccb4\uc5d0 \uac00\ud574\uc9c8 \ub54c \uc720\uccb4\uc758 \ud750\ub984\uc744 \uc124\uba85\ud558\uae30 \uc704\ud55c \ubaa8\ub378\ub85c\uc368 \ub2e4\uc74c\uacfc \uac19\uc740 \ud615\ud0dc\uc774\ub2e4:<\/p>\n

\n

  <\/span>   <\/span>\"\[<\/p>\n

\uc5ec\uae30\uc11c \"u\"\ub294 \uc720\uccb4\uc758 velocity, \"p\"\ub294 pressure\ub97c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\ud655\ub960\ud3b8\ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\ub294 \u00a0\ud3b8\ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\uac00 \uac00\uc9c0\ub294 \ud574\uc11d\ud559\uc801 \ud2b9\uc131\uacfc \ubd88\ud655\uc2e4\uc131\uc774 \ud3ec\ud568\ub41c \ud655\ub960\ub860\uc801 \uc131\uc9c8\uc744 \ub3d9\uc2dc\uc5d0 \uac00\uc9c4\ub2e4. \uc774\ub85c \uc778\ud574 PDE \uc774\ub860\ub4e4\uc5d0\uc11c \uc8fc\ub85c \ub2e4\ub8e8\uc5b4\uc9c0\ub294 \uc874\uc7ac\uc131, \uc720\uc77c\uc131, \uc815\uce59\uc131(Regularity)\uc758 \ubb38\uc81c\ub4e4\uc740 SPDE\uc758 \uacbd\uc6b0\uc5d0 \ud574\uac00 \uac00\uc9c0\ub294 \ubd88\ud655\uc2e4\uc131 \ub54c\ubb38\uc5d0 \u00a0\uc131\uc9c8 \uaddc\uba85\uc5d0 \u00a0\ub9ce\uc740 \uc5b4\ub824\uc6c0\uc774 \uc218\ubc18\ub41c\ub2e4. \uac00\uc7a5 \uac04\ub2e8\ud55c \uc608\ub97c \ub4e0\ub2e4\uba74 \uad6c\uac04 \"I=(0,10)\"\uc5d0\uc11c zero boundary condition\uacfc initial data \"u(0,\cdot)=\sin \frac{2\pi x}{10}\"\uc774 \uc8fc\uc5b4\uc9c4 heat equation\uacfc \u00a0stochastic heat equation \uc758 \ud574\uc758 simulation\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub9e4\uc6b0 \ub2e4\ub978 \ud328\ud134\uc744 \ubcf4\uc778\ub2e4:<\/p>\n

\"\"<\/p>\n

\n","protected":false},"excerpt":{"rendered":"

  Stochastic Partial Differential Equations (SPDE in short): \u00a0\ud3b8\ubbf8\ubd84\ubc29\uc815\uc2dd (PDE)\uc740\u00a0\uac70\uc2dc\uc801\uc778 \uad00\uc810\uc5d0\uc11c \ud604\uc0c1\uc758 \ud3c9\uade0\uc801\uc778 \uc6c0\uc9c1\uc784\uc744 \u00a0\ubb18\uc0ac\ud55c\ub2e4. \ubc18\uba74 SPDE\ub294 \ubd88\uaddc\uce59\ud55c\u00a0\ub3d9\ub825\uc774 \uad6c\ud604\ub41c \ub79c\ub364\ub178\uc774\uc988\uac00 \uc801\uc6a9\ub41c \ubc29\uc815\uc2dd\uc73c\ub85c \u00a0PDE\ub97c \ud3ec\ud568\ud558\ub294 \ubcf4\ub2e4 \uc77c\ubc18\uc801\uc778 \ud615\ud0dc\uc758 \ubc29\uc815\uc2dd\uc774\ub2e4. \uc27d\uac8c \ub9d0\ud558\uc790\uba74     \uc774\ub77c\uace0 \ud560\uc218 \uc788\ub2e4. Deterministic and Stochastic \u00a0Science \ube44\uad50: … \uacc4\uc18d \uc77d\uae30 →<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ngg_post_thumbnail":0,"footnotes":""},"class_list":["post-318","page","type-page","status-publish","hentry"],"distributor_meta":false,"distributor_terms":false,"distributor_media":false,"distributor_original_site_name":"Kyeonghun Kim\u2019s homepage","distributor_original_site_url":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun","push-errors":false,"_links":{"self":[{"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/pages\/318","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/comments?post=318"}],"version-history":[{"count":51,"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/pages\/318\/revisions"}],"predecessor-version":[{"id":579,"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/pages\/318\/revisions\/579"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/kyeonghun\/wp-json\/wp\/v2\/media?parent=318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}