Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three. (12th February 2015)

• Record Type: Journal Article
• Authors: Wu, Zhigang; Qin, Yuming
• Is Part Of: Mathematical methods in the applied sciences. Volume 38:Number 13(2015:Sep. 15)
• Page Start: 2864
• Page End: 2875
• Publication Date: 2015-02-12
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3269
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma3269-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="mma3269-para-0001">By rewriting a bipolar Euler–Poisson equations with damping into a Euler equation with damping coupled with a Euler–Poisson equation with damping and using a new spectral analysis, we obtain the optimal decay results of the solutions in <italic>L</italic><sup>2</sup> norm. More precisely, the velocities <italic>u</italic><sub>1</sub> and <italic>u</italic><sub>2</sub> decay at the <italic>L</italic><sup>2</sup>−rate <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23tff9r8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:mma:media:mma3269:mma3269-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>, which is faster than the normal <italic>L</italic><sup>2</sup>‐rate <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23tff9vx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:mma:media:mma3269:mma3269-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> for the heat equation and the Navier–Stokes equations. In addition, the decay rates of the disparities of two densities <italic>ρ</italic><sub>1</sub>−<italic>ρ</italic><sub>2</sub> and the disparity of two velocities <italic>u</italic><sub>1</sub>−<italic>u</italic><sub>2</sub> could reach to <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23tff9wg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:mma:media:mma3269:mma3269-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>19</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23tff9st" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:mma:media:mma3269:mma3269-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>17</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> in <italic>L</italic><sup>2</sup> norm, respectively. Copyright © 2015 John Wiley & Sons, Ltd.</p> </abstract>