Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator
In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><m...
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2025-02-01
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| author | Lyailya Zhapsarbayeva Dongming Wei Bagyzhan Bagymkyzy |
| author_facet | Lyailya Zhapsarbayeva Dongming Wei Bagyzhan Bagymkyzy |
| author_sort | Lyailya Zhapsarbayeva |
| collection | DOAJ |
| description | In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mrow><mo>(</mo><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mo>+</mo><mi>u</mi><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><msub><mo>∂</mo><mi>x</mi></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="|" close="|"><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> augmented with the initial condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mi>L</mi></mrow></semantics></math></inline-formula>, and the boundary condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is the density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> the viscosity, <i>u</i> the velocity of the fluid, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We show that this initial boundary problem has an unique solution in the Buchner space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><msub><mi>W</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula> for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions. |
| format | Article |
| id | doaj-art-89083ee19fdb481099cc2cb4759801f1 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-89083ee19fdb481099cc2cb4759801f12025-08-20T02:04:48ZengMDPI AGMathematics2227-73902025-02-0113570810.3390/math13050708Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace OperatorLyailya Zhapsarbayeva0Dongming Wei1Bagyzhan Bagymkyzy2Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, 2, Satbaev St., Astana 010000, KazakhstanDepartment of Mathematics, Nazarbayev University, 53, Kabanbay batyr Ave., Astana 010000, KazakhstanDepartment of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, 2, Satbaev St., Astana 010000, KazakhstanIn this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mrow><mo>(</mo><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mo>+</mo><mi>u</mi><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><msub><mo>∂</mo><mi>x</mi></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="|" close="|"><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> augmented with the initial condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mi>L</mi></mrow></semantics></math></inline-formula>, and the boundary condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is the density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> the viscosity, <i>u</i> the velocity of the fluid, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We show that this initial boundary problem has an unique solution in the Buchner space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><msub><mi>W</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula> for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions.https://www.mdpi.com/2227-7390/13/5/708<i>p</i>-Laplacianpower-law non-Newtonian fluid modelexistence and uniquenessBurgers’ equationBochner spaceSobolev space |
| spellingShingle | Lyailya Zhapsarbayeva Dongming Wei Bagyzhan Bagymkyzy Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator Mathematics <i>p</i>-Laplacian power-law non-Newtonian fluid model existence and uniqueness Burgers’ equation Bochner space Sobolev space |
| title | Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator |
| title_full | Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator |
| title_fullStr | Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator |
| title_full_unstemmed | Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator |
| title_short | Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator |
| title_sort | existence and uniqueness of the viscous burgers equation with the p laplace operator |
| topic | <i>p</i>-Laplacian power-law non-Newtonian fluid model existence and uniqueness Burgers’ equation Bochner space Sobolev space |
| url | https://www.mdpi.com/2227-7390/13/5/708 |
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