Global solutions of the two-dimensional Riemann problem with four-shock interactions for the Euler equations of potential flow Article Swipe
Gui‐Qiang Chen
,
Alexander J. Cliffe
,
Feimin Huang
,
Song Liu
,
Qin Wang
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.4171/jems/1622
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.4171/jems/1622
We present a rigorous approach and related techniques to construct global solutions of the two-dimensional (2-D) Riemann problem with four-shock interactions for the Euler equations of potential flow. With the introduction of three critical angles: the vacuum critical angle from the compatibility conditions, and the sonic and detachment angles—whose existence and uniqueness follow from our rigorous proof of the strict monotonicity of the steady detachment and sonic angles for 2-D steady potential flow with respect to the upstream Mach number, we classify all configurations of the Riemann solutions for the interaction of two forward and two backward shocks, including the subsonic-subsonic reflection configuration that has not emerged in the previous results. To achieve this, we recast the 2-D Riemann problem as a shock reflection-diffraction problem with respect to a symmetry line, which is further reformulated as a free boundary problem for a second-order quasilinear equation of mixed elliptic-hyperbolic type. The difficulties arise from the degenerate ellipticity of the nonlinear equation near the sonic boundaries, the nonlinearity of the free boundary condition, the singularity of the solution near the corners of the domain, and the geometric properties of the free boundary. To solve the problem, we analyze the solutions of a quasilinear degenerate elliptic equation by using the maximum principle for the mixed boundary value problem, the theory of oblique derivative boundary value problems, uniform a priori estimates, and a sophisticated iteration method. To the best of our knowledge, this is the first rigorous result for the 2-D Riemann problem with four-shock interactions for the Euler equations. The approach and techniques developed for the Riemann problem with four-shock interactions should be useful in solving other 2-D Riemann problems for more general Euler equations and related nonlinear hyperbolic systems of conservation laws.
Related Topics
Concepts
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.4171/jems/1622
- OA Status
- diamond
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4409431640
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4409431640Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.4171/jems/1622Digital Object Identifier
- Title
-
Global solutions of the two-dimensional Riemann problem with four-shock interactions for the Euler equations of potential flowWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2025Year of publication
- Publication date
-
2025-04-14Full publication date if available
- Authors
-
Gui‐Qiang Chen, Alexander J. Cliffe, Feimin Huang, Song Liu, Qin WangList of authors in order
- Landing page
-
https://doi.org/10.4171/jems/1622Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
diamondOpen access status per OpenAlex
- OA URL
-
https://doi.org/10.4171/jems/1622Direct OA link when available
- Concepts
-
Mathematics, Riemann problem, Euler equations, Riemann hypothesis, Flow (mathematics), Euler's formula, Shock (circulatory), Mathematical analysis, Riemann solver, Geometry, Mechanics, Internal medicine, Physics, Finite volume method, MedicineTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4409431640 |
|---|---|
| doi | https://doi.org/10.4171/jems/1622 |
| ids.doi | https://doi.org/10.4171/jems/1622 |
| ids.openalex | https://openalex.org/W4409431640 |
| fwci | 0.0 |
| type | article |
| title | Global solutions of the two-dimensional Riemann problem with four-shock interactions for the Euler equations of potential flow |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T10940 |
| topics[0].field.id | https://openalex.org/fields/26 |
| topics[0].field.display_name | Mathematics |
| topics[0].score | 0.9954000115394592 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2604 |
| topics[0].subfield.display_name | Applied Mathematics |
| topics[0].display_name | Navier-Stokes equation solutions |
| topics[1].id | https://openalex.org/T10095 |
| topics[1].field.id | https://openalex.org/fields/31 |
| topics[1].field.display_name | Physics and Astronomy |
| topics[1].score | 0.9872000217437744 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/3103 |
| topics[1].subfield.display_name | Astronomy and Astrophysics |
| topics[1].display_name | Cosmology and Gravitation Theories |
| topics[2].id | https://openalex.org/T10173 |
| topics[2].field.id | https://openalex.org/fields/22 |
| topics[2].field.display_name | Engineering |
| topics[2].score | 0.9829000234603882 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2206 |
| topics[2].subfield.display_name | Computational Mechanics |
| topics[2].display_name | Computational Fluid Dynamics and Aerodynamics |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C33923547 |
| concepts[0].level | 0 |
| concepts[0].score | 0.8967714309692383 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[0].display_name | Mathematics |
| concepts[1].id | https://openalex.org/C40709475 |
| concepts[1].level | 3 |
| concepts[1].score | 0.7454166412353516 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q2152221 |
| concepts[1].display_name | Riemann problem |
| concepts[2].id | https://openalex.org/C38409319 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6072974801063538 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q375175 |
| concepts[2].display_name | Euler equations |
| concepts[3].id | https://openalex.org/C199479865 |
| concepts[3].level | 2 |
| concepts[3].score | 0.6055726408958435 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q205966 |
| concepts[3].display_name | Riemann hypothesis |
| concepts[4].id | https://openalex.org/C38349280 |
| concepts[4].level | 2 |
| concepts[4].score | 0.5982404351234436 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q1434290 |
| concepts[4].display_name | Flow (mathematics) |
| concepts[5].id | https://openalex.org/C62884695 |
| concepts[5].level | 2 |
| concepts[5].score | 0.5823604464530945 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q184871 |
| concepts[5].display_name | Euler's formula |
| concepts[6].id | https://openalex.org/C2781300812 |
| concepts[6].level | 2 |
| concepts[6].score | 0.4915705621242523 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q178061 |
| concepts[6].display_name | Shock (circulatory) |
| concepts[7].id | https://openalex.org/C134306372 |
| concepts[7].level | 1 |
| concepts[7].score | 0.4258379340171814 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[7].display_name | Mathematical analysis |
| concepts[8].id | https://openalex.org/C120796332 |
| concepts[8].level | 3 |
| concepts[8].score | 0.41624271869659424 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q6153788 |
| concepts[8].display_name | Riemann solver |
| concepts[9].id | https://openalex.org/C2524010 |
| concepts[9].level | 1 |
| concepts[9].score | 0.11310353875160217 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q8087 |
| concepts[9].display_name | Geometry |
| concepts[10].id | https://openalex.org/C57879066 |
| concepts[10].level | 1 |
| concepts[10].score | 0.07783070206642151 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q41217 |
| concepts[10].display_name | Mechanics |
| concepts[11].id | https://openalex.org/C126322002 |
| concepts[11].level | 1 |
| concepts[11].score | 0.0 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q11180 |
| concepts[11].display_name | Internal medicine |
| concepts[12].id | https://openalex.org/C121332964 |
| concepts[12].level | 0 |
| concepts[12].score | 0.0 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[12].display_name | Physics |
| concepts[13].id | https://openalex.org/C50478463 |
| concepts[13].level | 2 |
| concepts[13].score | 0.0 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q1401936 |
| concepts[13].display_name | Finite volume method |
| concepts[14].id | https://openalex.org/C71924100 |
| concepts[14].level | 0 |
| concepts[14].score | 0.0 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q11190 |
| concepts[14].display_name | Medicine |
| keywords[0].id | https://openalex.org/keywords/mathematics |
| keywords[0].score | 0.8967714309692383 |
| keywords[0].display_name | Mathematics |
| keywords[1].id | https://openalex.org/keywords/riemann-problem |
| keywords[1].score | 0.7454166412353516 |
| keywords[1].display_name | Riemann problem |
| keywords[2].id | https://openalex.org/keywords/euler-equations |
| keywords[2].score | 0.6072974801063538 |
| keywords[2].display_name | Euler equations |
| keywords[3].id | https://openalex.org/keywords/riemann-hypothesis |
| keywords[3].score | 0.6055726408958435 |
| keywords[3].display_name | Riemann hypothesis |
| keywords[4].id | https://openalex.org/keywords/flow |
| keywords[4].score | 0.5982404351234436 |
| keywords[4].display_name | Flow (mathematics) |
| keywords[5].id | https://openalex.org/keywords/eulers-formula |
| keywords[5].score | 0.5823604464530945 |
| keywords[5].display_name | Euler's formula |
| keywords[6].id | https://openalex.org/keywords/shock |
| keywords[6].score | 0.4915705621242523 |
| keywords[6].display_name | Shock (circulatory) |
| keywords[7].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[7].score | 0.4258379340171814 |
| keywords[7].display_name | Mathematical analysis |
| keywords[8].id | https://openalex.org/keywords/riemann-solver |
| keywords[8].score | 0.41624271869659424 |
| keywords[8].display_name | Riemann solver |
| keywords[9].id | https://openalex.org/keywords/geometry |
| keywords[9].score | 0.11310353875160217 |
| keywords[9].display_name | Geometry |
| keywords[10].id | https://openalex.org/keywords/mechanics |
| keywords[10].score | 0.07783070206642151 |
| keywords[10].display_name | Mechanics |
| language | en |
| locations[0].id | doi:10.4171/jems/1622 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S88235189 |
| locations[0].source.issn | 1435-9855, 1435-9863 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | 1435-9855 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | True |
| locations[0].source.display_name | Journal of the European Mathematical Society |
| locations[0].source.host_organization | https://openalex.org/P4310316982 |
| locations[0].source.host_organization_name | European Mathematical Society |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310316982 |
| locations[0].source.host_organization_lineage_names | European Mathematical Society |
| locations[0].license | cc-by |
| locations[0].pdf_url | |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Journal of the European Mathematical Society |
| locations[0].landing_page_url | https://doi.org/10.4171/jems/1622 |
| indexed_in | crossref, doaj |
| authorships[0].author.id | https://openalex.org/A5018452636 |
| authorships[0].author.orcid | https://orcid.org/0000-0001-5146-3839 |
| authorships[0].author.display_name | Gui‐Qiang Chen |
| authorships[0].countries | GB |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I40120149 |
| authorships[0].affiliations[0].raw_affiliation_string | University of Oxford, Oxford, UK |
| authorships[0].institutions[0].id | https://openalex.org/I40120149 |
| authorships[0].institutions[0].ror | https://ror.org/052gg0110 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I40120149 |
| authorships[0].institutions[0].country_code | GB |
| authorships[0].institutions[0].display_name | University of Oxford |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Gui-Qiang G. Chen |
| authorships[0].is_corresponding | False |
| authorships[0].raw_affiliation_strings | University of Oxford, Oxford, UK |
| authorships[1].author.id | https://openalex.org/A5112924207 |
| authorships[1].author.orcid | |
| authorships[1].author.display_name | Alexander J. Cliffe |
| authorships[1].countries | GB |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I40120149 |
| authorships[1].affiliations[0].raw_affiliation_string | University of Oxford, Oxford, UK |
| authorships[1].institutions[0].id | https://openalex.org/I40120149 |
| authorships[1].institutions[0].ror | https://ror.org/052gg0110 |
| authorships[1].institutions[0].type | education |
| authorships[1].institutions[0].lineage | https://openalex.org/I40120149 |
| authorships[1].institutions[0].country_code | GB |
| authorships[1].institutions[0].display_name | University of Oxford |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Alexander J. Cliffe |
| authorships[1].is_corresponding | False |
| authorships[1].raw_affiliation_strings | University of Oxford, Oxford, UK |
| authorships[2].author.id | https://openalex.org/A5101996813 |
| authorships[2].author.orcid | https://orcid.org/0000-0002-9490-7118 |
| authorships[2].author.display_name | Feimin Huang |
| authorships[2].countries | CN |
| authorships[2].affiliations[0].institution_ids | https://openalex.org/I19820366 |
| authorships[2].affiliations[0].raw_affiliation_string | Chinese Academy of Sciences, Beijing, P. R. China |
| authorships[2].institutions[0].id | https://openalex.org/I19820366 |
| authorships[2].institutions[0].ror | https://ror.org/034t30j35 |
| authorships[2].institutions[0].type | government |
| authorships[2].institutions[0].lineage | https://openalex.org/I19820366 |
| authorships[2].institutions[0].country_code | CN |
| authorships[2].institutions[0].display_name | Chinese Academy of Sciences |
| authorships[2].author_position | middle |
| authorships[2].raw_author_name | Feimin Huang |
| authorships[2].is_corresponding | False |
| authorships[2].raw_affiliation_strings | Chinese Academy of Sciences, Beijing, P. R. China |
| authorships[3].author.id | https://openalex.org/A5100339600 |
| authorships[3].author.orcid | https://orcid.org/0000-0002-8719-0833 |
| authorships[3].author.display_name | Song Liu |
| authorships[3].countries | HK |
| authorships[3].affiliations[0].institution_ids | https://openalex.org/I14243506 |
| authorships[3].affiliations[0].raw_affiliation_string | The Hong Kong Polytechnic University, Hong Kong, P. R. China |
| authorships[3].institutions[0].id | https://openalex.org/I14243506 |
| authorships[3].institutions[0].ror | https://ror.org/0030zas98 |
| authorships[3].institutions[0].type | education |
| authorships[3].institutions[0].lineage | https://openalex.org/I14243506 |
| authorships[3].institutions[0].country_code | HK |
| authorships[3].institutions[0].display_name | Hong Kong Polytechnic University |
| authorships[3].author_position | middle |
| authorships[3].raw_author_name | Song Liu |
| authorships[3].is_corresponding | False |
| authorships[3].raw_affiliation_strings | The Hong Kong Polytechnic University, Hong Kong, P. R. China |
| authorships[4].author.id | https://openalex.org/A5100422808 |
| authorships[4].author.orcid | https://orcid.org/0000-0002-8391-4996 |
| authorships[4].author.display_name | Qin Wang |
| authorships[4].countries | CN |
| authorships[4].affiliations[0].institution_ids | https://openalex.org/I189210763 |
| authorships[4].affiliations[0].raw_affiliation_string | Yunnan University, Kunming, P. R. China |
| authorships[4].institutions[0].id | https://openalex.org/I189210763 |
| authorships[4].institutions[0].ror | https://ror.org/0040axw97 |
| authorships[4].institutions[0].type | education |
| authorships[4].institutions[0].lineage | https://openalex.org/I189210763 |
| authorships[4].institutions[0].country_code | CN |
| authorships[4].institutions[0].display_name | Yunnan University |
| authorships[4].author_position | last |
| authorships[4].raw_author_name | Qin Wang |
| authorships[4].is_corresponding | False |
| authorships[4].raw_affiliation_strings | Yunnan University, Kunming, P. R. China |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://doi.org/10.4171/jems/1622 |
| open_access.oa_status | diamond |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Global solutions of the two-dimensional Riemann problem with four-shock interactions for the Euler equations of potential flow |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T10940 |
| primary_topic.field.id | https://openalex.org/fields/26 |
| primary_topic.field.display_name | Mathematics |
| primary_topic.score | 0.9954000115394592 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2604 |
| primary_topic.subfield.display_name | Applied Mathematics |
| primary_topic.display_name | Navier-Stokes equation solutions |
| related_works | https://openalex.org/W2085037704, https://openalex.org/W3174853118, https://openalex.org/W2379245068, https://openalex.org/W2010276774, https://openalex.org/W1663225052, https://openalex.org/W2161856882, https://openalex.org/W1990653832, https://openalex.org/W2133986346, https://openalex.org/W3164279712, https://openalex.org/W4385807557 |
| cited_by_count | 0 |
| locations_count | 1 |
| best_oa_location.id | doi:10.4171/jems/1622 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S88235189 |
| best_oa_location.source.issn | 1435-9855, 1435-9863 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | 1435-9855 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | True |
| best_oa_location.source.display_name | Journal of the European Mathematical Society |
| best_oa_location.source.host_organization | https://openalex.org/P4310316982 |
| best_oa_location.source.host_organization_name | European Mathematical Society |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310316982 |
| best_oa_location.source.host_organization_lineage_names | European Mathematical Society |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Journal of the European Mathematical Society |
| best_oa_location.landing_page_url | https://doi.org/10.4171/jems/1622 |
| primary_location.id | doi:10.4171/jems/1622 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S88235189 |
| primary_location.source.issn | 1435-9855, 1435-9863 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | 1435-9855 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | True |
| primary_location.source.display_name | Journal of the European Mathematical Society |
| primary_location.source.host_organization | https://openalex.org/P4310316982 |
| primary_location.source.host_organization_name | European Mathematical Society |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310316982 |
| primary_location.source.host_organization_lineage_names | European Mathematical Society |
| primary_location.license | cc-by |
| primary_location.pdf_url | |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Journal of the European Mathematical Society |
| primary_location.landing_page_url | https://doi.org/10.4171/jems/1622 |
| publication_date | 2025-04-14 |
| publication_year | 2025 |
| referenced_works_count | 0 |
| abstract_inverted_index.a | 2, 121, 128, 136, 141, 199, 224, 228 |
| abstract_inverted_index.To | 111, 190, 232 |
| abstract_inverted_index.We | 0 |
| abstract_inverted_index.as | 120, 135 |
| abstract_inverted_index.be | 269 |
| abstract_inverted_index.by | 204 |
| abstract_inverted_index.in | 107, 271 |
| abstract_inverted_index.is | 132, 239 |
| abstract_inverted_index.of | 12, 25, 31, 57, 61, 84, 91, 145, 156, 166, 173, 179, 186, 198, 217, 235, 287 |
| abstract_inverted_index.to | 8, 75, 127 |
| abstract_inverted_index.we | 80, 114, 194 |
| abstract_inverted_index.2-D | 69, 117, 246, 274 |
| abstract_inverted_index.The | 149, 256 |
| abstract_inverted_index.all | 82 |
| abstract_inverted_index.and | 5, 43, 46, 50, 65, 94, 182, 227, 258, 282 |
| abstract_inverted_index.for | 21, 68, 88, 140, 209, 244, 252, 261, 277 |
| abstract_inverted_index.has | 104 |
| abstract_inverted_index.not | 105 |
| abstract_inverted_index.our | 54, 236 |
| abstract_inverted_index.the | 13, 22, 29, 35, 40, 44, 58, 62, 76, 85, 89, 99, 108, 116, 153, 157, 161, 164, 167, 171, 174, 177, 180, 183, 187, 192, 196, 206, 210, 215, 233, 240, 245, 253, 262 |
| abstract_inverted_index.two | 92, 95 |
| abstract_inverted_index.Mach | 78 |
| abstract_inverted_index.With | 28 |
| abstract_inverted_index.best | 234 |
| abstract_inverted_index.flow | 72 |
| abstract_inverted_index.free | 137, 168, 188 |
| abstract_inverted_index.from | 39, 53, 152 |
| abstract_inverted_index.more | 278 |
| abstract_inverted_index.near | 160, 176 |
| abstract_inverted_index.that | 103 |
| abstract_inverted_index.this | 238 |
| abstract_inverted_index.with | 18, 73, 125, 249, 265 |
| abstract_inverted_index.(2-D) | 15 |
| abstract_inverted_index.Euler | 23, 254, 280 |
| abstract_inverted_index.angle | 38 |
| abstract_inverted_index.arise | 151 |
| abstract_inverted_index.first | 241 |
| abstract_inverted_index.flow. | 27 |
| abstract_inverted_index.laws. | 289 |
| abstract_inverted_index.line, | 130 |
| abstract_inverted_index.mixed | 146, 211 |
| abstract_inverted_index.other | 273 |
| abstract_inverted_index.proof | 56 |
| abstract_inverted_index.shock | 122 |
| abstract_inverted_index.solve | 191 |
| abstract_inverted_index.sonic | 45, 66, 162 |
| abstract_inverted_index.this, | 113 |
| abstract_inverted_index.three | 32 |
| abstract_inverted_index.type. | 148 |
| abstract_inverted_index.using | 205 |
| abstract_inverted_index.value | 213, 221 |
| abstract_inverted_index.which | 131 |
| abstract_inverted_index.angles | 67 |
| abstract_inverted_index.follow | 52 |
| abstract_inverted_index.global | 10 |
| abstract_inverted_index.priori | 225 |
| abstract_inverted_index.recast | 115 |
| abstract_inverted_index.result | 243 |
| abstract_inverted_index.should | 268 |
| abstract_inverted_index.steady | 63, 70 |
| abstract_inverted_index.strict | 59 |
| abstract_inverted_index.theory | 216 |
| abstract_inverted_index.useful | 270 |
| abstract_inverted_index.vacuum | 36 |
| abstract_inverted_index.Riemann | 16, 86, 118, 247, 263, 275 |
| abstract_inverted_index.achieve | 112 |
| abstract_inverted_index.analyze | 195 |
| abstract_inverted_index.angles: | 34 |
| abstract_inverted_index.corners | 178 |
| abstract_inverted_index.domain, | 181 |
| abstract_inverted_index.emerged | 106 |
| abstract_inverted_index.forward | 93 |
| abstract_inverted_index.further | 133 |
| abstract_inverted_index.general | 279 |
| abstract_inverted_index.maximum | 207 |
| abstract_inverted_index.method. | 231 |
| abstract_inverted_index.number, | 79 |
| abstract_inverted_index.oblique | 218 |
| abstract_inverted_index.present | 1 |
| abstract_inverted_index.problem | 17, 119, 124, 139, 248, 264 |
| abstract_inverted_index.related | 6, 283 |
| abstract_inverted_index.respect | 74, 126 |
| abstract_inverted_index.shocks, | 97 |
| abstract_inverted_index.solving | 272 |
| abstract_inverted_index.systems | 286 |
| abstract_inverted_index.uniform | 223 |
| abstract_inverted_index.approach | 4, 257 |
| abstract_inverted_index.backward | 96 |
| abstract_inverted_index.boundary | 138, 169, 212, 220 |
| abstract_inverted_index.classify | 81 |
| abstract_inverted_index.critical | 33, 37 |
| abstract_inverted_index.elliptic | 202 |
| abstract_inverted_index.equation | 144, 159, 203 |
| abstract_inverted_index.previous | 109 |
| abstract_inverted_index.problem, | 193, 214 |
| abstract_inverted_index.problems | 276 |
| abstract_inverted_index.results. | 110 |
| abstract_inverted_index.rigorous | 3, 55, 242 |
| abstract_inverted_index.solution | 175 |
| abstract_inverted_index.symmetry | 129 |
| abstract_inverted_index.upstream | 77 |
| abstract_inverted_index.boundary. | 189 |
| abstract_inverted_index.construct | 9 |
| abstract_inverted_index.developed | 260 |
| abstract_inverted_index.equations | 24, 281 |
| abstract_inverted_index.existence | 49 |
| abstract_inverted_index.geometric | 184 |
| abstract_inverted_index.including | 98 |
| abstract_inverted_index.iteration | 230 |
| abstract_inverted_index.nonlinear | 158, 284 |
| abstract_inverted_index.potential | 26, 71 |
| abstract_inverted_index.principle | 208 |
| abstract_inverted_index.problems, | 222 |
| abstract_inverted_index.solutions | 11, 87, 197 |
| abstract_inverted_index.condition, | 170 |
| abstract_inverted_index.degenerate | 154, 201 |
| abstract_inverted_index.derivative | 219 |
| abstract_inverted_index.detachment | 47, 64 |
| abstract_inverted_index.equations. | 255 |
| abstract_inverted_index.estimates, | 226 |
| abstract_inverted_index.four-shock | 19, 250, 266 |
| abstract_inverted_index.hyperbolic | 285 |
| abstract_inverted_index.knowledge, | 237 |
| abstract_inverted_index.properties | 185 |
| abstract_inverted_index.reflection | 101 |
| abstract_inverted_index.techniques | 7, 259 |
| abstract_inverted_index.uniqueness | 51 |
| abstract_inverted_index.boundaries, | 163 |
| abstract_inverted_index.conditions, | 42 |
| abstract_inverted_index.ellipticity | 155 |
| abstract_inverted_index.interaction | 90 |
| abstract_inverted_index.quasilinear | 143, 200 |
| abstract_inverted_index.singularity | 172 |
| abstract_inverted_index.conservation | 288 |
| abstract_inverted_index.difficulties | 150 |
| abstract_inverted_index.interactions | 20, 251, 267 |
| abstract_inverted_index.introduction | 30 |
| abstract_inverted_index.monotonicity | 60 |
| abstract_inverted_index.nonlinearity | 165 |
| abstract_inverted_index.reformulated | 134 |
| abstract_inverted_index.second-order | 142 |
| abstract_inverted_index.compatibility | 41 |
| abstract_inverted_index.configuration | 102 |
| abstract_inverted_index.sophisticated | 229 |
| abstract_inverted_index.angles—whose | 48 |
| abstract_inverted_index.configurations | 83 |
| abstract_inverted_index.two-dimensional | 14 |
| abstract_inverted_index.subsonic-subsonic | 100 |
| abstract_inverted_index.elliptic-hyperbolic | 147 |
| abstract_inverted_index.reflection-diffraction | 123 |
| cited_by_percentile_year | |
| countries_distinct_count | 3 |
| institutions_distinct_count | 5 |
| citation_normalized_percentile.value | 0.15982608 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |