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Volume 30, Issue 3
Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres

Shu Wang & Niu Haiping

DOI: 10.4208/jpde.v30.n3.4

J. Part. Diff. Eq., 30 (2017), pp. 232-253.

Published online: 2017-08

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  • Abstract
We study the singular structure of a kind of three dimensional non-selfsimilar global solutions and their interaction for quasilinear hyperbolic conservation laws. The initial discontinuity is two disjoint unit spheres and initial data just contain two different constant states, the global solutions and some new phenomena are discovered. We give the solutions in 0 ‹ t ‹ T^* and T^* ‹ t, and at t=T^*, the two basic shock waves and the constant state u_ are disappeared.Then, we find a new shock wave between two rarefaction by R-H condition. Finally, we show the limit of the solution when t → ∞. A technique is proposed to construct the three dimensional shock wave without dimensional reduction or coordinate transformation.
  • Keywords

Initial discontinuity Burgers equation elementary wave global solution

  • AMS Subject Headings

35L65, 35L67

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wangshu@bjut.edu.cn (Shu Wang)

niuhaiping@email.bjut.edu.cn (Niu Haiping)

  • BibTex
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  • TXT
@Article{JPDE-30-232, author = {Wang , Shu and Haiping , Niu}, title = {Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres}, journal = {Journal of Partial Differential Equations}, year = {2017}, volume = {30}, number = {3}, pages = {232--253}, abstract = { We study the singular structure of a kind of three dimensional non-selfsimilar global solutions and their interaction for quasilinear hyperbolic conservation laws. The initial discontinuity is two disjoint unit spheres and initial data just contain two different constant states, the global solutions and some new phenomena are discovered. We give the solutions in 0 ‹ t ‹ T^* and T^* ‹ t, and at t=T^*, the two basic shock waves and the constant state u_ are disappeared.Then, we find a new shock wave between two rarefaction by R-H condition. Finally, we show the limit of the solution when t → ∞. A technique is proposed to construct the three dimensional shock wave without dimensional reduction or coordinate transformation.}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v30.n3.4}, url = {http://global-sci.org/intro/article_detail/jpde/10467.html} }
TY - JOUR T1 - Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres AU - Wang , Shu AU - Haiping , Niu JO - Journal of Partial Differential Equations VL - 3 SP - 232 EP - 253 PY - 2017 DA - 2017/08 SN - 30 DO - http://doi.org/10.4208/jpde.v30.n3.4 UR - https://global-sci.org/intro/article_detail/jpde/10467.html KW - Initial discontinuity KW - Burgers equation KW - elementary wave KW - global solution AB - We study the singular structure of a kind of three dimensional non-selfsimilar global solutions and their interaction for quasilinear hyperbolic conservation laws. The initial discontinuity is two disjoint unit spheres and initial data just contain two different constant states, the global solutions and some new phenomena are discovered. We give the solutions in 0 ‹ t ‹ T^* and T^* ‹ t, and at t=T^*, the two basic shock waves and the constant state u_ are disappeared.Then, we find a new shock wave between two rarefaction by R-H condition. Finally, we show the limit of the solution when t → ∞. A technique is proposed to construct the three dimensional shock wave without dimensional reduction or coordinate transformation.
Wang , Shu and Haiping , Niu. (2017). Solutions to a 3D Burgers Equation with Initial Discontinuity That Are Two Disjoint Spheres. Journal of Partial Differential Equations. 30 (3). 232-253. doi:10.4208/jpde.v30.n3.4
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