Document Zbl 1538.35218

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Appl. Math. Comput.</a> <a href="/?q=in%3A501254" title="Articles in this Issue">5, No. 3, 1148-1179 (2023)</a>. </div> <div class="abstract">Summary: The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved. By studying the interactions among of the delta-shock, vacuum, and contact discontinuity, fourteen kinds of structures of Riemann solutions are obtained. The compound wave solutions consisting of delta-shocks, vacuums, and contact discontinuities are found. The single and double closed vacuum cavitations develop in solutions. Furthermore, it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data. Finally, the numerical results completely coinciding with theoretical analysis are presented.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7729064">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L67" title="MSC2020">35L67</a> </td> <td class="space"> Shocks and singularities for hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L65" title="MSC2020">35L65</a> </td> <td class="space"> Hyperbolic conservation laws </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L45" title="MSC2020">35L45</a> </td> <td class="space"> Initial value problems for first-order hyperbolic systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L80" title="MSC2020">35L80</a> </td> <td class="space"> Degenerate hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A05" title="MSC2020">78A05</a> </td> <td class="space"> Geometric optics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aperturbed+Riemann+problem">perturbed Riemann problem</a>; <a href="/?q=ut%3Ageometrical+optics">geometrical optics</a>; <a href="/?q=ut%3Adelta-shock+wave">delta-shock wave</a>; <a href="/?q=ut%3Avacuum">vacuum</a>; <a href="/?q=ut%3Acompound+wave">compound wave</a>; <a href="/?q=ut%3Anumerical+results">numerical results</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1538.35218" data-ciurl="/ci/07729064" data-biburl="/bibtex/07729064.bib" data-amsurl="/amsrefs/07729064.bib" data-xmlurl="/xml/07729064.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07729064.pdf" title="Zbl 1538.35218 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s42967-022-00192-3" aria-label="DOI for “The perturbed Riemann problem for a geometrical optics system”" title="10.1007/s42967-022-00192-3">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Chang, T.; Hsiao, L., The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics (1989), Essex: Longman Scientific & Technical, Essex · <a href="/0698.76078" class="nowrap">Zbl 0698.76078</a></td> </tr><tr> <td>[2]</td> <td class="space">Engquist, B.; Runborg, O., Multi-phase computations in geometrical optics, J. Comput. Appl. Math., 74, 175-192 (1996) · <a href="/0947.78001" class="nowrap">Zbl 0947.78001</a> · <a href="https://doi.org/10.1016/0377-0427(96)00023-4" class="nowrap">doi:10.1016/0377-0427(96)00023-4</a></td> </tr><tr> <td>[3]</td> <td class="space">Guo, L.; Pan, L.; Yin, G., The perturbed Riemann problem and delta contact discontinuity in chromatography equations, Nonlinear Anal. TMA., 106, 110-123 (2014) · <a href="/1295.35321" class="nowrap">Zbl 1295.35321</a> · <a href="https://doi.org/10.1016/j.na.2014.04.016" class="nowrap">doi:10.1016/j.na.2014.04.016</a></td> </tr><tr> <td>[4]</td> <td class="space">Keyfit, B.; Kranzer, H.; Keyfitz, BL; Shearer, M., A strictly hyperbolic system of conservation laws admitting singular shocks, Nonlinear Evolution Equations That Change Type, 107-125 (1990), Berlin: Springer, Berlin · <a href="/0718.76071" class="nowrap">Zbl 0718.76071</a> · <a href="https://doi.org/10.1007/978-1-4613-9049-7_9" class="nowrap">doi:10.1007/978-1-4613-9049-7_9</a></td> </tr><tr> <td>[5]</td> <td class="space">Korchinski, DJ, Solution of a Riemann problem for a $2\times 2$ system of conservation laws possessing no classical weak solution (1977), Garden City, NY: Ph.D.Thesis. Adelphi University, Garden City, NY</td> </tr><tr> <td>[6]</td> <td class="space">Le Floch, P.; Keyfitz, BL; Shearer, M., An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations That Change Type, 126-138 (1990), Berlin: Springer, Berlin · <a href="/0727.35083" class="nowrap">Zbl 0727.35083</a> · <a href="https://doi.org/10.1007/978-1-4613-9049-7_10" class="nowrap">doi:10.1007/978-1-4613-9049-7_10</a></td> </tr><tr> <td>[7]</td> <td class="space">Li, J.; Zhang, T.; Yang, S., The two-dimensional riemann problem in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics (1998), London: Longman, London · <a href="/0935.76002" class="nowrap">Zbl 0935.76002</a></td> </tr><tr> <td>[8]</td> <td class="space">Nedeljkov, M.; Oberguggenberger, M., Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl., 344, 1143-1157 (2008) · <a href="/1155.35059" class="nowrap">Zbl 1155.35059</a> · <a href="https://doi.org/10.1016/j.jmaa.2008.03.040" class="nowrap">doi:10.1016/j.jmaa.2008.03.040</a></td> </tr><tr> <td>[9]</td> <td class="space">Qu, A.; Wang, Z., Stability of the Riemann solutions for a Chaplygin gas, J. Math. Anal. Appl., 409, 347-361 (2014) · <a href="/1306.35073" class="nowrap">Zbl 1306.35073</a> · <a href="https://doi.org/10.1016/j.jmaa.2013.07.018" class="nowrap">doi:10.1016/j.jmaa.2013.07.018</a></td> </tr><tr> <td>[10]</td> <td class="space">Shelkovich, V., The Riemann problem admitting $\delta , \delta^{\prime }$-shocks, and vacuum states (the vanishing viscosity approach), J. Differ. Equ., 231, 459-500 (2006) · <a href="/1108.35117" class="nowrap">Zbl 1108.35117</a> · <a href="https://doi.org/10.1016/j.jde.2006.08.003" class="nowrap">doi:10.1016/j.jde.2006.08.003</a></td> </tr><tr> <td>[11]</td> <td class="space">Shen, C.; Sun, M., Interactions of delta shock waves for the transport equations with split delta functions, J. Math. Anal. Appl., 351, 747-755 (2009) · <a href="/1159.35042" class="nowrap">Zbl 1159.35042</a> · <a href="https://doi.org/10.1016/j.jmaa.2008.11.005" class="nowrap">doi:10.1016/j.jmaa.2008.11.005</a></td> </tr><tr> <td>[12]</td> <td class="space">Shen, C.; Sun, M., Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws, Nonlinear Anal. TMA., 73, 3284-3294 (2010) · <a href="/1207.35057" class="nowrap">Zbl 1207.35057</a> · <a href="https://doi.org/10.1016/j.na.2010.07.008" class="nowrap">doi:10.1016/j.na.2010.07.008</a></td> </tr><tr> <td>[13]</td> <td class="space">Sheng, W.; Zhang, T., The Riemann problem for transportation equation in gas dynamics, Mem. Amer. Math. Soc., 137, 654 (1999) · <a href="/0913.35082" class="nowrap">Zbl 0913.35082</a></td> </tr><tr> <td>[14]</td> <td class="space">Sun, M., Interactions of delta shock waves for the chromatography equations, Appl. Math. Lett., 26, 631-637 (2013) · <a href="/1264.35142" class="nowrap">Zbl 1264.35142</a> · <a href="https://doi.org/10.1016/j.aml.2013.01.002" class="nowrap">doi:10.1016/j.aml.2013.01.002</a></td> </tr><tr> <td>[15]</td> <td class="space">Shu, C-W; Quarteroni, A., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, 325-432 (1998), Berlin: Springer, Berlin · <a href="/0927.65111" class="nowrap">Zbl 0927.65111</a> · <a href="https://doi.org/10.1007/BFb0096355" class="nowrap">doi:10.1007/BFb0096355</a></td> </tr><tr> <td>[16]</td> <td class="space">Tan, D.; Zhang, T., Two-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws: I. Four-J cases, II. Initial data involving some rarefaction waves, J. Differ. Equ., 111, 203-254, 255-283 (1994) · <a href="/0803.35086" class="nowrap">Zbl 0803.35086</a> · <a href="https://doi.org/10.1006/jdeq.1994.1082" class="nowrap">doi:10.1006/jdeq.1994.1082</a></td> </tr><tr> <td>[17]</td> <td class="space">Tan, D.; Zhang, T.; Zheng, Y., Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conversation laws, J. Differ. Equ., 112, 1-32 (1994) · <a href="/0804.35077" class="nowrap">Zbl 0804.35077</a> · <a href="https://doi.org/10.1006/jdeq.1994.1093" class="nowrap">doi:10.1006/jdeq.1994.1093</a></td> </tr><tr> <td>[18]</td> <td class="space">Yang, H., Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differ. Equ., 159, 447-484 (1999) · <a href="/0948.35079" class="nowrap">Zbl 0948.35079</a> · <a href="https://doi.org/10.1006/jdeq.1999.3629" class="nowrap">doi:10.1006/jdeq.1999.3629</a></td> </tr><tr> <td>[19]</td> <td class="space">Yang, H., Generalized plane delta shock waves for $n$-dimensional zero-pressure gas dynamics, J. Math. Anal. Appl., 260, 18-35 (2001) · <a href="/0985.35044" class="nowrap">Zbl 0985.35044</a> · <a href="https://doi.org/10.1006/jmaa.2000.7426" class="nowrap">doi:10.1006/jmaa.2000.7426</a></td> </tr><tr> <td>[20]</td> <td class="space">Yang, H.; Cheng, H., Riemann problem for a geometrical optics system, Acta Math. Sin., 30, 1846-1860 (2014) · <a href="/1307.35167" class="nowrap">Zbl 1307.35167</a> · <a href="https://doi.org/10.1007/s10114-014-3561-1" class="nowrap">doi:10.1007/s10114-014-3561-1</a></td> </tr><tr> <td>[21]</td> <td class="space">Yang, H.; Hu, R.; Sun, Y., The Riemann problem with three constant initial states for one-dimensional zero-pressure gas dynamics, Southeast Asian Bull. Math., 32, 1-9 (2008)</td> </tr><tr> <td>[22]</td> <td class="space">Yang, H., Li, S.: Riemann solutions containing compound waves for a geometrical optics system by the viscosity method, to appear</td> </tr><tr> <td>[23]</td> <td class="space">Yang, H.; Sun, W., The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws, Nonlinear Anal. TMA., 67, 3041-3049 (2007) · <a href="/1120.35067" class="nowrap">Zbl 1120.35067</a> · <a href="https://doi.org/10.1016/j.na.2006.09.057" class="nowrap">doi:10.1016/j.na.2006.09.057</a></td> </tr><tr> <td>[24]</td> <td class="space">Yang, H.; Zhang, Y., New developments of delta shock waves and its applications in systems of conservation laws, J. Differ. Equ., 252, 5951-5993 (2012) · <a href="/1248.35127" class="nowrap">Zbl 1248.35127</a> · <a href="https://doi.org/10.1016/j.jde.2012.02.015" class="nowrap">doi:10.1016/j.jde.2012.02.015</a></td> </tr><tr> <td>[25]</td> <td class="space">Yang, H.; Zhang, Y., Delta shock waves with Dirac delta function in both components for systems of conservation laws, J. Differ. Equ., 257, 4369-4402 (2014) · <a href="/1304.35422" class="nowrap">Zbl 1304.35422</a> · <a href="https://doi.org/10.1016/j.jde.2014.08.009" class="nowrap">doi:10.1016/j.jde.2014.08.009</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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