{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/34289"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/34289","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Discrete Fourier restriction phenomenon associated with some periodic dispersive equations","abstract":"\"The thesis consists of six chapters. In Chapter 1, we will briefly introduce the background of the topic, as well as some results we already know. The next five chapters can be divided into two parts. The first part is about the discrete Fourier restriction phenomenon. In Chapter 2, we consider the discrete Fourier restriction phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain's level set result on Strichartz estimates associated with periodic Schrodinger equations. Some sharp estimates on $L^\\frac{2(d+2)}{d}$ norms of certain exponential sums in higher dimensional cases are established. In Chapter 3 we further discuss the discrete Fourier restriction problem associated with higher order dispersive equations, with the method developed in Chapter 2. We obtain some sharp bound on the size of the Fourier transform of a function for large indices. Some new Strichartz estimates of this type are obtained. Also, we can use the method to prove some exponential sum estimates, which are classic in number theory. The second part of the thesis is about the local well-posedness of some dispersive equations. In Chapter 4, we prove the local well-posedness of the periodic gKdV equations. The method we apply here is a generalization of Bourgain's \"\"denominator manipulation\"\". With this idea, we further discuss a more general type of KdV equation in Chapter 5. We establish the local well-posedness of the periodic KdV equations with nonlinear terms $F(u)u_x$, provided $F\\in C^5$ and the initial data $u_0\\in H^s$ with $s>1/2$ (1/2 is sharp). In Chapter 6 we focus on the local well-posedness of the periodic fifth order KdV type dispersive equations with nonlinear terms $P_1(u)u_x + P_2(u)u_x^2$, provided the initial data $u_0\\in H^s$ with $s>1$. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. Some Strichartz estimates derived in Chapter 3 are used in the proof. Also, a couple of counterexamples are given to exhibit the sharpness of the indices.\"","abstract_html":"&quot;The thesis consists of six chapters. In Chapter 1, we will briefly introduce the background of the topic, as well as some results we already know. The next five chapters can be divided into two parts. The first part is about the discrete Fourier restriction phenomenon. In Chapter 2, we consider the discrete Fourier restriction phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain&#x27;s level set result on Strichartz estimates associated with periodic Schrodinger equations. Some sharp estimates on <span class=\"etd-inline-math\">L<sup>\\</sup>frac{2(d+2)}{d}</span> norms of certain exponential sums in higher dimensional cases are established. In Chapter 3 we further discuss the discrete Fourier restriction problem associated with higher order dispersive equations, with the method developed in Chapter 2. We obtain some sharp bound on the size of the Fourier transform of a function for large indices. Some new Strichartz estimates of this type are obtained. Also, we can use the method to prove some exponential sum estimates, which are classic in number theory. The second part of the thesis is about the local well-posedness of some dispersive equations. In Chapter 4, we prove the local well-posedness of the periodic gKdV equations. The method we apply here is a generalization of Bourgain&#x27;s &quot;&quot;denominator manipulation&quot;&quot;. With this idea, we further discuss a more general type of KdV equation in Chapter 5. We establish the local well-posedness of the periodic KdV equations with nonlinear terms <span class=\"etd-inline-math\">F(u)u<sub>x</sub></span>, provided <span class=\"etd-inline-math\">F\\in C<sup>5</sup></span> and the initial data <span class=\"etd-inline-math\">u<sub>0</sub>\\in H<sup>s</sup></span> with $s&gt;1/2$ (1/2 is sharp). In Chapter 6 we focus on the local well-posedness of the periodic fifth order KdV type dispersive equations with nonlinear terms <span class=\"etd-inline-math\">P<sub>1</sub>(u)u<sub>x</sub> + P<sub>2</sub>(u)u<sub>x</sub><sup>2</sup></span>, provided the initial data <span class=\"etd-inline-math\">u<sub>0</sub>\\in H<sup>s</sup></span> with $s&gt;1$. Here <span class=\"etd-inline-math\">P<sub>1</sub>(u)</span> and <span class=\"etd-inline-math\">P<sub>2</sub>(u)</span> are polynomials of $u$. Some Strichartz estimates derived in Chapter 3 are used in the proof. Also, a couple of counterexamples are given to exhibit the sharpness of the indices.&quot;","abstract_has_math":true,"creators":["Hu, Yi"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Li, Xiaochun","Erdogan, M. Burak","Tzirakis, Nikolaos","Berndt, Bruce C."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-09-18T21:09:44Z","date_published":"2012-09-18T21:09:44Z","updated_at":"2026-07-22T22:25:31Z","subjects":["discrete Fourier restriction","periodic dispersive equations","Schrodinger equations","Korteweg-de Vries (KdV) equations","fifth order Korteweg-de Vries (KdV) equations"],"languages":["en"],"rights":["Copyright 2012 Yi Hu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/34289","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Li, Xiaochun","Erdogan, M. Burak","Tzirakis, Nikolaos","Berndt, Bruce C."]},{"key":"dc:creator","label":"Author","values":["Hu, Yi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-09-18T21:09:44Z","2012-08"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["discrete Fourier restriction","periodic dispersive equations","Schrodinger equations","Korteweg-de Vries (KdV) equations","fifth order Korteweg-de Vries (KdV) equations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 Yi Hu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/34289"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"The thesis consists of six chapters. In Chapter 1, we will briefly introduce the background of the topic, as well as some results we already know. The next five chapters can be divided into two parts. The first part is about the discrete Fourier restriction phenomenon. In Chapter 2, we consider the discrete Fourier restriction phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain's level set result on Strichartz estimates associated with periodic Schrodinger equations. Some sharp estimates on $L^\\frac{2(d+2)}{d}$ norms of certain exponential sums in higher dimensional cases are established. In Chapter 3 we further discuss the discrete Fourier restriction problem associated with higher order dispersive equations, with the method developed in Chapter 2. We obtain some sharp bound on the size of the Fourier transform of a function for large indices. Some new Strichartz estimates of this type are obtained. Also, we can use the method to prove some exponential sum estimates, which are classic in number theory. The second part of the thesis is about the local well-posedness of some dispersive equations. In Chapter 4, we prove the local well-posedness of the periodic gKdV equations. The method we apply here is a generalization of Bourgain's \"\"denominator manipulation\"\". With this idea, we further discuss a more general type of KdV equation in Chapter 5. We establish the local well-posedness of the periodic KdV equations with nonlinear terms $F(u)u_x$, provided $F\\in C^5$ and the initial data $u_0\\in H^s$ with $s>1/2$ (1/2 is sharp). In Chapter 6 we focus on the local well-posedness of the periodic fifth order KdV type dispersive equations with nonlinear terms $P_1(u)u_x + P_2(u)u_x^2$, provided the initial data $u_0\\in H^s$ with $s>1$. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. Some Strichartz estimates derived in Chapter 3 are used in the proof. Also, a couple of counterexamples are given to exhibit the sharpness of the indices.\"","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-06-28T18:00:26Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Hu_Yi.pdf: 405643 bytes, checksum: 6f1fccdd877069601b17bb429a341f56 (MD5)","Made available in DSpace on 2012-09-18T21:09:44Z (GMT). No. of bitstreams: 2 Hu_Yi.pdf: 405655 bytes, checksum: 354feb6968a09db915198898d70f9c15 (MD5) license.txt: 4052 bytes, checksum: 10c5289869bdb4dd025f10d56f1eda14 (MD5)"]},{"key":"dc:title","label":"Title","values":["Discrete Fourier restriction phenomenon associated with some periodic dispersive equations"]}]}],"canonical_facts":{"dc:contributor":["Li, Xiaochun","Erdogan, M. Burak","Tzirakis, Nikolaos","Berndt, Bruce C."],"dc:creator":["Hu, Yi"],"dc:date":["2012-09-18T21:09:44Z","2012-08"],"dc:description":["\"The thesis consists of six chapters. In Chapter 1, we will briefly introduce the background of the topic, as well as some results we already know. The next five chapters can be divided into two parts. The first part is about the discrete Fourier restriction phenomenon. In Chapter 2, we consider the discrete Fourier restriction phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain's level set result on Strichartz estimates associated with periodic Schrodinger equations. Some sharp estimates on $L^\\frac{2(d+2)}{d}$ norms of certain exponential sums in higher dimensional cases are established. In Chapter 3 we further discuss the discrete Fourier restriction problem associated with higher order dispersive equations, with the method developed in Chapter 2. We obtain some sharp bound on the size of the Fourier transform of a function for large indices. Some new Strichartz estimates of this type are obtained. Also, we can use the method to prove some exponential sum estimates, which are classic in number theory. The second part of the thesis is about the local well-posedness of some dispersive equations. In Chapter 4, we prove the local well-posedness of the periodic gKdV equations. The method we apply here is a generalization of Bourgain's \"\"denominator manipulation\"\". With this idea, we further discuss a more general type of KdV equation in Chapter 5. We establish the local well-posedness of the periodic KdV equations with nonlinear terms $F(u)u_x$, provided $F\\in C^5$ and the initial data $u_0\\in H^s$ with $s>1/2$ (1/2 is sharp). In Chapter 6 we focus on the local well-posedness of the periodic fifth order KdV type dispersive equations with nonlinear terms $P_1(u)u_x + P_2(u)u_x^2$, provided the initial data $u_0\\in H^s$ with $s>1$. Here $P_1(u)$ and $P_2(u)$ are polynomials of $u$. Some Strichartz estimates derived in Chapter 3 are used in the proof. Also, a couple of counterexamples are given to exhibit the sharpness of the indices.\"","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-06-28T18:00:26Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Hu_Yi.pdf: 405643 bytes, checksum: 6f1fccdd877069601b17bb429a341f56 (MD5)","Made available in DSpace on 2012-09-18T21:09:44Z (GMT). No. of bitstreams: 2 Hu_Yi.pdf: 405655 bytes, checksum: 354feb6968a09db915198898d70f9c15 (MD5) license.txt: 4052 bytes, checksum: 10c5289869bdb4dd025f10d56f1eda14 (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/34289"],"dc:language":["en"],"dc:rights":["Copyright 2012 Yi Hu"],"dc:subject":["discrete Fourier restriction","periodic dispersive equations","Schrodinger equations","Korteweg-de Vries (KdV) equations","fifth order Korteweg-de Vries (KdV) equations"],"dc:title":["Discrete Fourier restriction phenomenon associated with some periodic dispersive equations"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:31Z"}