{"id":{"repo_id":"maryland","oai_identifier":"oai:drum.lib.umd.edu:1903/7229"},"canonical_url":"https://search.dev.ndltd.org/etd/maryland/oai:drum.lib.umd.edu:1903/7229","repository":{"repo_id":"maryland","name":"University of Maryland","base_url":"https://api.drum.lib.umd.edu/server/oai/request"},"display":{"title":"Critical thresholds in Eulerian dynamics","abstract":"In this thesis, we study the critical regularity phenomena in Eulerian dynamics, $u_t+u\\cdot \\nabla u=F(u,D u,\\cdots), $ here $F$ represents a general force acting on the flow and by regularity we seek to obtain a large set of sub-critical initial data. We analyze three prototype models, ranging from the one-dimensional Euler-Poisson equations to two-dimensional system of Burgers equations to three-dimensional, four-dimensional and even higher-dimensional restricted Euler systems. We begin with the one-dimensional Euler-Poisson equations, where $F$ is the Poisson forcing term together with the usual $\\gamma$-law pressure. We prove that global regularity of the Euler-Poisson equations with $\\gamma\\geq 1$ depends on whether or not the initial configuration crosses an intrinsic critical threshold. Next, we discuss multi-dimensional examples. The first multi-dimensional example that we focus our attention on is the two dimensional pressureless flow, where $F=\\epsilon \\Delta u$. Our analysis shows that there is a uniform \\textsl{BV} bound of the solutions $u^{\\epsilon}$. Moreover, if the initial velocity gradient $\\nabla u_0$ does not have negative eigenvalues, then its vanishing viscosity limit is the smooth solution of the corresponding equations of the inviscid fluid flow. The second multi-dimensional example we discuss here is the restricted Euler dynamics, where $\\nabla F= \\displaystyle\\frac{1}{n} \\mathrm {tr} (\\nabla u)^2I_{n\\times n}$\\,. Our analysis shows that for the three-dimensional case, the finite-time breakdown of the restricted Euler system is generic, and for the four-dimensional case, there is a surprising global existence for sub-critical initial data. Further analysis extends the above result to the general $n$-dimensional ($n>4$) restricted Euler system.","abstract_html":"In this thesis, we study the critical regularity phenomena in Eulerian dynamics, <span class=\"etd-inline-math\">u<sub>t</sub>+u\\cdot \\nabla u=F(u,D u,\\cdots), </span> here $F$ represents a general force acting on the flow and by regularity we seek to obtain a large set of sub-critical initial data. We analyze three prototype models, ranging from the one-dimensional Euler-Poisson equations to two-dimensional system of Burgers equations to three-dimensional, four-dimensional and even higher-dimensional restricted Euler systems. We begin with the one-dimensional Euler-Poisson equations, where $F$ is the Poisson forcing term together with the usual <span class=\"etd-inline-math\">&gamma;</span>-law pressure. We prove that global regularity of the Euler-Poisson equations with <span class=\"etd-inline-math\">&gamma;\\geq 1</span> depends on whether or not the initial configuration crosses an intrinsic critical threshold. Next, we discuss multi-dimensional examples. The first multi-dimensional example that we focus our attention on is the two dimensional pressureless flow, where <span class=\"etd-inline-math\">F=&epsilon; \\Delta u</span>. Our analysis shows that there is a uniform \\textsl{BV} bound of the solutions <span class=\"etd-inline-math\">u<sup>&epsilon;</sup></span>. Moreover, if the initial velocity gradient <span class=\"etd-inline-math\">\\nabla u<sub>0</sub></span> does not have negative eigenvalues, then its vanishing viscosity limit is the smooth solution of the corresponding equations of the inviscid fluid flow. The second multi-dimensional example we discuss here is the restricted Euler dynamics, where <span class=\"etd-inline-math\">\\nabla F= \\displaystyle\\frac{1}{n} \\mathrm {tr} (\\nabla u)<sup>2</sup>I<sub>n\\times n</sub></span>\\,. Our analysis shows that for the three-dimensional case, the finite-time breakdown of the restricted Euler system is generic, and for the four-dimensional case, there is a surprising global existence for sub-critical initial data. Further analysis extends the above result to the general $n$-dimensional ($n&gt;4$) restricted Euler system.","abstract_has_math":true,"creators":["Wei, Dongming"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Tadmor, Eitan"],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007-07-06","date_published":"2007-07-06","updated_at":"2026-07-24T03:02:13Z","subjects":[],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1903/7229","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Tadmor, Eitan"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Wei, Dongming"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2007-09-28T14:58:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2007-09-28T14:58:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2007-07-06"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1903/7229"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we study the critical regularity phenomena in Eulerian dynamics, $u_t+u\\cdot \\nabla u=F(u,D u,\\cdots), $ here $F$ represents a general force acting on the flow and by regularity we seek to obtain a large set of sub-critical initial data. We analyze three prototype models, ranging from the one-dimensional Euler-Poisson equations to two-dimensional system of Burgers equations to three-dimensional, four-dimensional and even higher-dimensional restricted Euler systems. We begin with the one-dimensional Euler-Poisson equations, where $F$ is the Poisson forcing term together with the usual $\\gamma$-law pressure. We prove that global regularity of the Euler-Poisson equations with $\\gamma\\geq 1$ depends on whether or not the initial configuration crosses an intrinsic critical threshold. Next, we discuss multi-dimensional examples. The first multi-dimensional example that we focus our attention on is the two dimensional pressureless flow, where $F=\\epsilon \\Delta u$. Our analysis shows that there is a uniform \\textsl{BV} bound of the solutions $u^{\\epsilon}$. Moreover, if the initial velocity gradient $\\nabla u_0$ does not have negative eigenvalues, then its vanishing viscosity limit is the smooth solution of the corresponding equations of the inviscid fluid flow. The second multi-dimensional example we discuss here is the restricted Euler dynamics, where $\\nabla F= \\displaystyle\\frac{1}{n} \\mathrm {tr} (\\nabla u)^2I_{n\\times n}$\\,. Our analysis shows that for the three-dimensional case, the finite-time breakdown of the restricted Euler system is generic, and for the four-dimensional case, there is a surprising global existence for sub-critical initial data. Further analysis extends the above result to the general $n$-dimensional ($n>4$) restricted Euler system."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Critical thresholds in Eulerian dynamics"]}]}],"canonical_facts":{"dc:contributor.advisor":["Tadmor, Eitan"],"dc:contributor.department":["Mathematics"],"dc:creator":["Wei, Dongming"],"dc:date.accessioned":["2007-09-28T14:58:28Z"],"dc:date.available":["2007-09-28T14:58:28Z"],"dc:date.issued":["2007-07-06"],"dc:description.abstract":["In this thesis, we study the critical regularity phenomena in Eulerian dynamics, $u_t+u\\cdot \\nabla u=F(u,D u,\\cdots), $ here $F$ represents a general force acting on the flow and by regularity we seek to obtain a large set of sub-critical initial data. We analyze three prototype models, ranging from the one-dimensional Euler-Poisson equations to two-dimensional system of Burgers equations to three-dimensional, four-dimensional and even higher-dimensional restricted Euler systems. We begin with the one-dimensional Euler-Poisson equations, where $F$ is the Poisson forcing term together with the usual $\\gamma$-law pressure. We prove that global regularity of the Euler-Poisson equations with $\\gamma\\geq 1$ depends on whether or not the initial configuration crosses an intrinsic critical threshold. Next, we discuss multi-dimensional examples. The first multi-dimensional example that we focus our attention on is the two dimensional pressureless flow, where $F=\\epsilon \\Delta u$. Our analysis shows that there is a uniform \\textsl{BV} bound of the solutions $u^{\\epsilon}$. Moreover, if the initial velocity gradient $\\nabla u_0$ does not have negative eigenvalues, then its vanishing viscosity limit is the smooth solution of the corresponding equations of the inviscid fluid flow. The second multi-dimensional example we discuss here is the restricted Euler dynamics, where $\\nabla F= \\displaystyle\\frac{1}{n} \\mathrm {tr} (\\nabla u)^2I_{n\\times n}$\\,. Our analysis shows that for the three-dimensional case, the finite-time breakdown of the restricted Euler system is generic, and for the four-dimensional case, there is a surprising global existence for sub-critical initial data. Further analysis extends the above result to the general $n$-dimensional ($n>4$) restricted Euler system."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1903/7229"],"dc:language.iso":["en_US"],"dc:title":["Critical thresholds in Eulerian dynamics"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T03:02:13Z"}