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 Mat. Sb. (N.S.), 1970, Volume 81(123), Number 2, Pages 228–255 (Mi msb3372)

First order quasilinear equations in several independent variables

S. N. Kruzhkov

Abstract: In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations
$$u_t+\sum_{i=1}^n\frac d{dx_i}\varphi_i(t,x,u)+\psi(t,x,u)=0$$
in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the “vanishing viscosity method”; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in $L_1$ of the solution of this problem which do not depend on small viscosity.
Bibliography: 22 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 10:2, 217–243

Bibliographic databases:

UDC: 517.944
MSC: 35K45, 35A05, 26A42

Citation: S. N. Kruzhkov, “First order quasilinear equations in several independent variables”, Mat. Sb. (N.S.), 81(123):2 (1970), 228–255; Math. USSR-Sb., 10:2 (1970), 217–243

Citation in format AMSBIB
\Bibitem{Kru70} \by S.~N.~Kruzhkov \paper First order quasilinear equations in several independent variables \jour Mat. Sb. (N.S.) \yr 1970 \vol 81(123) \issue 2 \pages 228--255 \mathnet{http://mi.mathnet.ru/msb3372} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=267257} \zmath{https://zbmath.org/?q=an:0202.11203|0215.16203} \transl \jour Math. USSR-Sb. \yr 1970 \vol 10 \issue 2 \pages 217--243 \crossref{https://doi.org/10.1070/SM1970v010n02ABEH002156} 

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161. Ancona F., Marson A., “On the attainable set for scalar nonlinear conservation laws with boundary control”, SIAM J. Control Optim., 36:1 (1998), 290–312
162. Tang Tao, “Convergence analysis for operator-splitting methods applied to conservation laws with stiff source terms”, SIAM J. Numer. Anal., 35:5 (1998), 1939–1968
163. Harten A., Lax P.D., Levermore C.D., Morokoff W.J., “Convex entropies and hyperbolicity for general Euler equations”, SIAM J. Numer. Anal., 35:6 (1998), 2117–2127
164. Yang Huanan, “On wavewise entropy inequality for high resolution schemes. II. Fully discrete MUSCL schemes with exact evolution in small time”, SIAM J. Numer. Anal., 36:1 (1998), 1–31
165. N. S. Bakhvalov, M. I. Zelikin, A. S. Kalashnikov, V. L. Kamynin, O. A. Oleinik, E. Yu. Panov, N. S. Petrosyan, V. M. Tikhomirov, A. V. Faminskii, V. N. Chubarikov, “Stanislav Nikolaevich Kruzhkov (obituary)”, Russian Math. Surveys, 53:5 (1998), 1071–1078
166. K. Karlsen, “The corrected operator splitting approach applied to a nonlinear advection-diffusion problem”, Computer Methods in Applied Mechanics and Engineering, 167:3-4 (1998), 239
167. Laurent Gosse, “A priori error estimate for a well-balanced scheme designed for inhomogeneous scalar conservation laws”, C.R. Acad. Sci. Ser. I Math., 327:5 (1998), 467
168. B. Perthame, “Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure”, Journal de Mathématiques Pures et Appliquées, 77:10 (1998), 1055
169. Yuan Hongjun, “Existence and Uniqueness of BV Solutions for a Conservation Law withσ-Finite Borel Measures as Initial Conditions”, Journal of Differential Equations, 146:1 (1998), 90
170. Yuan Hongjun, “Source-type solutions of a singular conservation law with absorption”, Nonlinear Analysis: Theory, Methods & Applications, 32:4 (1998), 467
171. Changjiang Zhu, “Existence of the Entropy Solution for a Viscoelastic Model”, Journal of Differential Equations, 146:1 (1998), 22
172. Alberto Bressan, Wen Shen, “Uniqueness for discontinuous ODE and conservation laws”, Nonlinear Analysis: Theory, Methods & Applications, 34:5 (1998), 637
173. Jörg Härterich, “Attractors of Viscous Balance Laws: Uniform Estimates for the Dimension”, Journal of Differential Equations, 142:1 (1998), 188
174. K Ito, Y Yan, “Viscous Scalar Conservation Law with Nonlinear Flux Feedback and Global Attractors”, Journal of Mathematical Analysis and Applications, 227:1 (1998), 271
175. R Bürger, W.L Wendland, “Existence, Uniqueness, and Stability of Generalized Solutions of an Initial-Boundary Value Problem for a Degenerating Quasilinear Parabolic Equation”, Journal of Mathematical Analysis and Applications, 218:1 (1998), 207
176. Yuan Hongjun, “The Cauchy Problem for a Singular Conservation Law with Measures as Initial Conditions”, Journal of Mathematical Analysis and Applications, 225:2 (1998), 427
177. Antonin Chambolle, Bradley J. Lucier, “Un principe du maximum pour des opérateurs monotones”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 326:7 (1998), 823
178. Shen Wen, Tveito A., Winther R., “On the zero relaxation limit for a system modeling the motions of a viscoelastic solid”, SIAM J. Math. Anal., 30:5 (1999), 1115–1135
179. Sun Wen Tao, Zhang Huai Yu, “Finite element method for two-phase immiscible flow”, Numer. Methods Partial Differential Equations, 15:4 (1999), 407–416
180. Liu Tai-Ping, Yang Tong, “Well-posedness theory for hyperbolic conservation laws”, Comm. Pure Appl. Math., 52:12 (1999), 1553–1586
181. Terracina A., “A free boundary problem for scalar conservation laws”, SIAM J. Math. Anal., 30:5 (1999), 985–1009
182. Tadmor E., Tang Tao, “Pointwise error estimates for scalar conservation laws with piecewise smooth solutions”, SIAM J. Numer. Anal., 36:6 (1999), 1739–1758
183. E. Yu. Panov, “Property of strong precompactness for bounded sets of measure-valued solutions of a first-order quasilinear equation”, Sb. Math., 190:3 (1999), 427–446
184. E. Yu. Panov, “A non-local theory of generalized entropy solutions of the Cauchy problem for a class of hyperbolic systems of conservation laws”, Izv. Math., 63:1 (1999), 129–179
185. J. Carrillo, P. Wittbold, “Unicité des solutions renormalisées de problèmes elliptiques-paraboliques”, C.R. Acad. Sci. Ser. I Math., 328:1 (1999), 23–28
186. L GOSSE, “Sur la stabilité des approximations implicites des lois de conservation scalaires non homogènes”, C.R. Acad. Sci. Ser. I Math., 329:1 (1999), 79
187. M. A. Katsoulakis, G. Kossioris, Ch. Makridakis, “Convergence and error estimates of relaxation schemes for multidimensional conservation laws”, Communications in Partial Differential Equations, 24:3-4 (1999), 395
188. D. Amadori, P. Baiti, P.G. LeFloch, B. Piccoli, “Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws”, Journal of Differential Equations, 151:2 (1999), 345
189. Peng Zhang, Tong Zhang, “Generalized Characteristic Analysis and Guckenheimer Structure”, Journal of Differential Equations, 152:2 (1999), 409
190. Fabio Ancona, Andrea Marson, “Scalar non-linear conservation laws with integrable boundary data”, Nonlinear Analysis: Theory, Methods & Applications, 35:6 (1999), 687
191. B Cockburn, G Gripenberg, “Continuous Dependence on the Nonlinearities of Solutions of Degenerate Parabolic Equations”, Journal of Differential Equations, 151:2 (1999), 231
192. José Carrillo, Petra Wittbold, “Uniqueness of Renormalized Solutions of Degenerate Elliptic–Parabolic Problems”, Journal of Differential Equations, 156:1 (1999), 93
193. Runhild Aae Klausen, Nils Henrik Risebro, “Stability of Conservation Laws with Discontinuous Coefficients”, Journal of Differential Equations, 157:1 (1999), 41
194. Gui-Qiang Chen, Hermano Frid, “Large-Time Behavior of Entropy Solutions of Conservation Laws”, Journal of Differential Equations, 152:2 (1999), 308
195. M. Bertsch, J. Goncerzewicz, D. Hilhorst, “Large time behaviour of solutions of scalar viscous and nonviscous conservation laws”, Applied Mathematics Letters, 12:3 (1999), 83
196. Sh. Kawashima, Sh. Nishibata, “Cauchy problem for a model system of the radiating gas: weak solutions with a jump and classical solutions”, Math. Models Methods Appl. Sci., 09:01 (1999), 69
197. Otto F., “Evolution of Microstructure in Unstable Porous Media Flow: a Relaxational Approach”, Commun. Pure Appl. Math., 52:7 (1999), 873–915
198. Ph.G.. LeFloch, Roberto Natalini, “Conservation laws with vanishing nonlinear diffusion and dispersion11This work was partially carried out during a visit of the first author to the Istituto per le Applicazioni del Calcolo”, Nonlinear Analysis: Theory, Methods & Applications, 36:2 (1999), 213
199. J Santos, P de Oliveira, “A converging finite volume scheme for hyperbolic conservation laws with source terms”, Journal of Computational and Applied Mathematics, 111:1-2 (1999), 239
200. Towers J.D., “Convergence of a difference scheme for conservation laws with a discontinuous flux”, SIAM J. Numer. Anal., 38:2 (2000), 681–698
201. Gosse L., Makridakis Ch., “Two a posteriori error estimates for one-dimensional scalar conservation laws”, SIAM J. Numer. Anal., 38:3 (2000), 964–988
202. Chainais-Hillairet C., “Second-order finite-volume schemes for a non-linear hyperbolic equation: error estimate”, Math. Methods Appl. Sci., 23:5 (2000), 467–490
203. Galaktionov V.A., Peletier L.A., Vazquez J.L., “Asymptotics of the fast-diffusion equation with critical exponent”, SIAM J. Math. Anal., 31:5 (2000), 1157–1174
204. Ben Moussa B., Vila J.P., “Convergence of SPH method for scalar nonlinear conservation laws”, SIAM J. Numer. Anal., 37:3 (2000), 863–887
205. Karlsen K.H., Risebro N.H., “Corrected operator splitting for nonlinear parabolic equations”, SIAM J Numer Anal, 37:3 (2000), 980–1003
206. Evje S., Karlsen K.H., “Monotone difference approximations of BV solutions to degenerate convection-diffusion equations”, SIAM J. Numer. Anal., 37:6 (2000), 1838–1860
207. E. Yu. Panov, “On the theory of generalized entropy solutions of the Cauchy problem for a class of non-strictly hyperbolic systems of conservation laws”, Sb. Math., 191:1 (2000), 121–150
208. L. Gosse, “A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms”, Computers & Mathematics with Applications, 39:9-10 (2000), 135–159
209. B. Nkonga, “On the conservative and accurate CFD approximations for moving meshes and moving boundaries”, Computer Methods in Applied Mechanics and Engineering, 190:13-14 (2000), 1801
210. Miguel Escobedo, Eduard Feireisl, Philippe Laurengot, “Large time behaviour for degenerate parabolic equations with dominating convective term”, Communications in Partial Differential Equations, 25:1-2 (2000), 73
211. Huijiang Zhao, R. A. Admas, “Existence and convergence of solutions for the generalized BBM-burgers equations with dissipative term 2: the multidimensional case”, Applicable Analysis, 75:1-2 (2000), 107
212. Ahmed Noussair, “Analysis of nonlinear resonance in conservation laws with point sources and well-balanced scheme”, Nonlinear Analysis: Theory, Methods & Applications, 42:8 (2000), 1431
213. Lung-an Ying, Pingwen Zhang, “Vanishing Curvature Viscosity for Front Propagation”, Journal of Differential Equations, 161:2 (2000), 289
214. Haitao Fan, Shi Jin, Zhen-huan Teng, “Zero reaction limit for hyperbolic conservation laws with source terms”, Journal of Differential Equations, 168:2 (2000), 270–294
215. D. Hilhorst, M.A. Peletier, “Convergence to Travelling Waves in a Reaction-Diffusion System Arising in Contaminant Transport”, Journal of Differential Equations, 163:1 (2000), 89
216. R. Bürger, S. Evje, K.Hvistendahl Karlsen, “On Strongly Degenerate Convection–Diffusion Problems Modeling Sedimentation–Consolidation Processes”, Journal of Mathematical Analysis and Applications, 247:2 (2000), 517
217. Sh. Nishibata, “Asymptotic behavior of solutions to a model system of radiating gas with discontinuous initial data”, Math. Models Methods Appl. Sci., 10:08 (2000), 1209–1231
218. Küther M., “Error estimates for the staggered Lax–Friedrichs scheme on unstructured grids”, SIAM J. Numer. Anal., 39:4 (2001), 1269–1301
219. Frid H., “Periodic and almost periodic solutions of conservation laws: Global existence and decay”, Bol. Soc. Brasil. Mat. (N.S.), 32:1 (2001), 1–35
220. Kim Yong Jung, Tzavaras A.E., “Diffusive $N$-waves and metastability in the Burgers equation”, SIAM J. Math. Anal., 33:3 (2001), 607–633
221. Bianchini S., “Stability of $L^\infty$ solutions for hyperbolic systems with coinciding shocks and rarefactions”, SIAM J. Math. Anal., 33:4 (2001), 959–981
222. Jakobsen E.R., Karlsen K.H., Risebro N.H., “On the convergence rate of operator splitting for Hamilton–Jacobi equations with source terms”, SIAM J. Numer. Anal., 39:2 (2001), 499–518
223. A. E. Biryuk, “Spectral Properties of Solutions of the Burgers Equation with Small Dissipation”, Funct. Anal. Appl., 35:1 (2001), 1–12
224. J. Ildefonso Díaz, Gonzalo Galiano, Ansgar Jüngel, “On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions”, Nonlinear Analysis: Real World Applications, 2:3 (2001), 305–336
225. Ingo Thomas, Thomas Sonar, “On a Second Order Residual Estimator for Numerical Schemes for Nonlinear Hyperbolic Conservation Laws”, Journal of Computational Physics, 171:1 (2001), 227
226. Christian Klingenberg, Nils Henrik Risebro, “Stability of a Resonant System of Conservation Laws Modeling Polymer Flow with Gravitation”, Journal of Differential Equations, 170:2 (2001), 344
227. R. Bürger, C. Liu, W.L. Wendland, “Existence and Stability for Mathematical Models of Sedimentation–Consolidation Processes in Several Space Dimensions”, Journal of Mathematical Analysis and Applications, 264:2 (2001), 288
228. L.A. Monthé, “A study of splitting scheme for hyperbolic conservation laws with source terms”, Journal of Computational and Applied Mathematics, 137:1 (2001), 1
229. Thierry Cazenave, Flávio Dickstein, “On the Influence of Boundary Conditions on Flows in Porous Media”, Journal of Mathematical Analysis and Applications, 253:1 (2001), 79
230. F. Andreu, C. Ballester, V. Caselles, J.M. Mazón, “The Dirichlet Problem for the Total Variation Flow”, Journal of Functional Analysis, 180:2 (2001), 347
231. Thomas Hillen, Christian Rohde, Frithjof Lutscher, “Existence of Weak Solutions for a Hyperbolic Model of Chemosensitive Movement”, Journal of Mathematical Analysis and Applications, 260:1 (2001), 173
232. Laurent Lévi, Fabrice Peyroutet, “A Time-Fractional Step Method for Conservation Law Related Obstacle Problems”, Advances in Applied Mathematics, 27:4 (2001), 768
233. F. Peyroutet, “Splitting method applied to hyperbolic problem with source term”, Applied Mathematics Letters, 14:1 (2001), 99
234. Alain Yves Le Roux, Marie Noëlle Le Roux, “Convergence d'un schéma à profils stationnaires pour les équations quasi linéaires du premier ordre avec termes sources”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 333:7 (2001), 703
235. F. Peyroutet, M. Madaune-Tort, “Error estimate for a splitting method applied to convection-reaction equations”, Math. Models Methods Appl. Sci., 11:06 (2001), 1081–1100
236. L. Gosse, “A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms”, Math. Models Methods Appl. Sci., 11:02 (2001), 339–365
237. Kim Yong-Jung, “Piecewise self-similar solutions and a numerical scheme for scalar conservation laws”, SIAM J. Numer. Anal., 40:6 (2002), 2105–2132
238. Dias J.-P., LeFloch Ph.G., “Some existence results for conservation laws with source-term”, Math. Methods Appl. Sci., 25:13 (2002), 1149–1160
239. Giga Y., “Viscosity solutions with shocks”, Comm. Pure Appl. Math., 55:4 (2002), 431–480
240. A. Yu. Goritskii, E. Yu. Panov, “Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a First-Order Quasilinear Equation”, Proc. Steklov Inst. Math., 236 (2002), 110–123
241. E. Yu. Panov, “Maximum and minimum generalized entropy solutions to the Cauchy problem for a first-order quasilinear equation”, Sb. Math., 193:5 (2002), 727–743
242. E. Yu. Panov, “On generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation in the class of locally summable functions”, Izv. Math., 66:6 (2002), 1171–1218
243. E. Yu. Panov, “O statisticheskikh resheniyakh zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Matem. modelirovanie, 14:3 (2002), 17–26
244. Laurent Levi, “Obstacle problems for scalar conservation laws”, ESAIM: M2AN, 35:3 (2002), 575
245. Mario Ohlberger, “A posteriorierror estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations”, ESAIM: M2AN, 35:2 (2002), 355
246. Kenneth Hvistendahl Karlsen, Nils Henrik Risebro, “Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients”, ESAIM: M2AN, 35:2 (2002), 239
247. Seok Hwang, Athanasios E. Tzavaras, “Kinetic decomposition of approximate solutions to conservation laws: application to relaxation and diffusion-dispersion approximations”, Comm. Partial Differential Equations, 27:5-6 (2002), 1229–1254
248. Kenneth Hvistendahl Karlsen, Nils Henrik Risebro, “Unconditionally Stable Methods for Hamilton–Jacobi Equations”, Journal of Computational Physics, 180:2 (2002), 710
249. Kenneth Hvistendahl Karlsen, Nils Henrik Risebro, “A note on front tracking and the equivalence between viscosity solutions of Hamilton–Jacobi equations and entropy solutions of scalar conservation laws”, Nonlinear Analysis: Theory, Methods & Applications, 50:4 (2002), 455
250. G. Bellettini, V. Caselles, M. Novaga, “The Total Variation Flow in N”, Journal of Differential Equations, 184:2 (2002), 475
251. Wancheng Sheng, “Two-Dimensional Riemann Problem for Scalar Conservation Laws”, Journal of Differential Equations, 183:1 (2002), 239
252. Jose Carrillo, Petra Wittbold, “Renormalized Entropy Solutions of Scalar Conservation Laws with Boundary Condition”, Journal of Differential Equations, 185:1 (2002), 137
253. Kenneth H. Karlsen, Mario Ohlberger, “A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations”, Journal of Mathematical Analysis and Applications, 275:1 (2002), 439
254. J. Nieto, J. Soler, F. Poupaud, “About uniqueness of weak solutions to first order quasi-linear equations”, Math. Models Methods Appl. Sci., 12:11 (2002), 1599–1615
255. Steinar Evje, Kenneth H Karlsen, “An Error Estimate for Viscous Approximate Solutions of Degenerate Parabolic Equations”, Journal of Nonlinear Mathematical Physics, 9:3 (2002), 262
256. Coronel A., James F., Sepúlveda M., “Numerical identification of parameters for a model of sedimentation processes”, Inverse Problems, 19:4 (2003), 951–972
257. Liu Hongxia, Pan Tao, “Interaction of elementary waves for scalar conservation laws on a bounded domain”, Math. Methods Appl. Sci., 26:7 (2003), 619–632
258. Zhu Changjiang, Duan Renjun, “Existence and uniqueness of entropy solution to initial boundary value problem for the inviscid Burgers equation”, J. Phys. A, 36:8 (2003), 2099–2107
259. S. I. Pokhozhaev, “Multidimensional scalar conservation laws”, Sb. Math., 194:1 (2003), 151–164
260. S. I. Pokhozhaev, “On a priori Estimates and Gradient Catastrophes of Smooth Solutions to Hyperbolic Systems of Conservation Laws”, Proc. Steklov Inst. Math., 243 (2003), 247–277
261. Bressan A., “An ill posed Cauchy problem for a hyperbolic system in two space dimensions”, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103–117
262. Galaktionov V.A., Shishkov A.E., “Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations”, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1075–1119
263. Stefan Ulbrich, “Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws”, Systems & Control Letters, 48:3-4 (2003), 313
264. Wenxian Shen, “Traveling waves in time periodic lattice differential equations”, Nonlinear Analysis: Theory, Methods & Applications, 54:2 (2003), 319
265. Kazuo Kobayasi, “The equivalence of weak solutions and entropy solutions of nonlinear degenerate second-order equations”, Journal of Differential Equations, 189:2 (2003), 383
266. Haitao Fan, Shi Jin, Judith R. Miller, “Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms”, Journal of Differential Equations, 189:1 (2003), 267–291
267. Manuel Portilheiro, “Weak solutions for equations defined by accretive operators II: relaxation limits”, Journal of Differential Equations, 195:1 (2003), 66
268. Wladimir Neves, “Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains”, Journal of Differential Equations, 192:2 (2003), 360
269. Tong Li, “Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow”, Journal of Differential Equations, 190:1 (2003), 131
270. Seok Hwang, “Kinetic decomposition for kinetic models of BGK type”, Journal of Differential Equations, 190:2 (2003), 353
271. Yuan Hongjun, Zheng Xiaoyu, “Existence and uniqueness for a quasilinear hyperbolic equation with σ-finite Borel measures as initial conditions”, Journal of Mathematical Analysis and Applications, 277:1 (2003), 27
272. Corrado Lattanzio, Pierangelo Marcati, “Global well-posedness and relaxation limits of a model for radiating gas”, Journal of Differential Equations, 190:2 (2003), 439
273. A.L. Amadori, R. Natalini, “Entropy solutions to a strongly degenerate anisotropic convection–diffusion equation with application to utility theory”, Journal of Mathematical Analysis and Applications, 284:2 (2003), 511
274. Vicente Martínez, Antonio Marquina, “Computation of travelling wave solutions of scalar conservation laws with a stiff source term”, Computers & Fluids, 32:8 (2003), 1161
275. Panov E.Y., “To the theory of generalized entropy solutions of the Cauchy problem for a first order quasilinear equation in the class of locally integrable functions”, Hyperbolic Problems: Theory, Numerics, Applications, 2003, 789–796
276. Laurent Gosse, “Time-Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System”, Math. Models Methods Appl. Sci., 13:08 (2003), 1081
277. Aregba-Driollet D., Milišić V., “Kinetic approximation of a boundary value problem for conservation laws”, Numer. Math., 97:4 (2004), 595–633
278. Bürger R., Karlsen K.H., Risebro N.H., Towers J.D., “Monotone difference approximations for the simulation of clarifier-thickener units”, Comput. Vis. Sci., 6:2-3 (2004), 83–91
279. Seok Hwang, “Kinetic decomposition for singularly perturbed higher order partial differential equations”, Journal of Differential Equations, 200:2 (2004), 191
280. F.R Guarguaglini, V Milišić, A Terracina, “A discrete BGK approximation for strongly degenerate parabolic problems with boundary conditions”, Journal of Differential Equations, 202:2 (2004), 183
281. Volker G. Jakubowski, Petra Wittbold, “On a nonlinear elliptic–parabolic integro-differential equation with L1-data”, Journal of Differential Equations, 197:2 (2004), 427
282. Peidong Lei, Zhuoqun Wu, Jingxue Yin, “Boundary value problem for a class of degenerate quasilinear parabolic equations with singularity”, Journal of Mathematical Analysis and Applications, 296:1 (2004), 209
283. M Remešı́ková, “Solution of convection–diffusion problems with nonequilibrium adsorption”, Journal of Computational and Applied Mathematics, 169:1 (2004), 101
284. Changjiang Zhu, Huijiang Zhao, “Well-posedness of the global entropy solution to the Cauchy problem of a hyperbolic conservation laws with relaxation”, Journal of Mathematical Analysis and Applications, 291:2 (2004), 438
285. Juan Nieto, “Hydrodynamical limit for a drift-diffusion system modeling large-population dynamics”, Journal of Mathematical Analysis and Applications, 291:2 (2004), 716
286. R. Bürger, K.H. Karlsen, N.H. Risebro, J.D. Towers, “Numerical methods for the simulation of continuous sedimentation in ideal clarifier-thickener units”, International Journal of Mineral Processing, 73:2-4 (2004), 209
287. R. Bürger, J.J.R. Damasceno, K.H. Karlsen, “A mathematical model for batch and continuous thickening of flocculated suspensions in vessels with varying cross-section”, International Journal of Mineral Processing, 73:2-4 (2004), 183
288. Th Katsaounis, B Perthame, C Simeoni, “Upwinding sources at interfaces in conservation laws”, Applied Mathematics Letters, 17:3 (2004), 309
289. K. H. Karlsen, N. H. Risebro, J. D. Towers, “Front tracking for scalar balance equations”, J. Hyperbolic Differ. Equ., 01:01 (2004), 115–148
290. H. Frid, “Asymptotic stability of non-planar Riemann solutions for multi-D systems of conservation laws with symmetric nonlinearities”, J. Hyperbolic Differ. Equ., 01:03 (2004), 567–579
291. Mishra S., “Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function”, SIAM J. Numer. Anal., 43:2 (2005), 559–577
292. Breuß M., “The implicit upwind method for 1-D scalar conservation laws with continuous fluxes”, SIAM J. Numer. Anal., 43:3 (2005), 970–986
293. Andreu F., Caselles V., Mazón J.M., “A strongly degenerate quasilinear equation: the parabolic case”, Arch. Ration. Mech. Anal., 176:3 (2005), 415–453
294. Jovanović V., Rohde Ch., “Finite-volume schemes for Friedrichs systems in multiple space dimensions: a priori anda posteriori error estimates”, Numer. Methods Partial Differential Equations, 21:1 (2005), 104–131
295. Dai Wen-Rong, Kong De-Xing, “Analysis of singularities and development of shocks for a class of multidimensional hyperbolic systems of conservation laws”, Math. Methods Appl. Sci., 28:13 (2005), 1585–1611
296. Lin Fang-Hua, Liu Chun, Zhang Ping, “On hydrodynamics of viscoelastic fluids”, Comm. Pure Appl. Math., 58:11 (2005), 1437–1471
297. Coclite G.M., Risebro N.H., “Conservation laws with time dependent discontinuous coefficients”, SIAM J. Math. Anal., 36:4 (2005), 1293–1309
298. I. V. Kuznetsov, “Entropy solutions to a second order forward-backward parabolic differential equation”, Siberian Math. J., 46:3 (2005), 467–488
299. Panov E.Yu., “Existence of strong traces for generalized solutions of multidimensional scalar conservation laws”, J. Hyperbolic Differ. Equ., 2:4 (2005), 885–908
300. Danilov V., Mitrovic D., “Weak asymptotics of shock wave formation process”, Nonlinear Anal., 61:4 (2005), 613–635
301. Luigi Ambrosio, François Bouchut, Camillo De Lellis, “Well-Posedness for a Class of Hyperbolic Systems of Conservation Laws in Several Space Dimensions”, Communications in Partial Differential Equations, 29:9-10 (2005), 1635–1651
302. Rachel Levy, Michael Shearer, “Kinetics and nucleation for driven thin film flow”, Physica D: Nonlinear Phenomena, 209:1-4 (2005), 145
303. Azmy S. Ackleh, Kazufumi Ito, “Measure-valued solutions for a hierarchically size-structured population”, Journal of Differential Equations, 217:2 (2005), 431
304. Satoru Takagi, “On renormalized dissipative solutions for conservation laws”, Nonlinear Analysis: Theory, Methods & Applications, 63:5-7 (2005), e2483
305. Haitao Fan, Jörg M. Härterich, “Conservation laws with a degenerate source: Traveling waves, large-time behavior and zero relaxation limit”, Nonlinear Analysis: Theory, Methods & Applications, 63:8 (2005), 1042
306. S. Berres, R. Bürger, A. Coronel, M. Sepúlveda, “Numerical identification of parameters for a strongly degenerate convection–diffusion problem modelling centrifugation of flocculated suspensions”, Applied Numerical Mathematics, 52:4 (2005), 311
307. Alain-Yves Le Roux, Marie-Noelle Le Roux, “Numerical solution of a nonlinear reaction diffusion equation”, Journal of Computational and Applied Mathematics, 173:2 (2005), 211
308. R. Bürger, K.H. Karlsen, N.H. Risebro, “A relaxation scheme forcontinuous sedimentation in ideal clarifier-thickener units”, Computers & Mathematics with Applications, 50:7 (2005), 993
309. A. Visintin, “Quasilinear first-order PDEs with hysteresis”, Journal of Mathematical Analysis and Applications, 312:2 (2005), 401
310. Nathalie Lanson, Jean-Paul Vila, “Convergence des méthodes particulaires renormalisées pour les systèmes de Friedrichs”, Comptes Rendus Mathematique, 340:6 (2005), 465
311. M. Campos Pinto, A. Cohen, W. Dahmen, R .Devore, “On the stability of nonlinear conservation laws in the Hausdorff metric”, J. Hyperbolic Differ. Equ., 02:01 (2005), 25–38
312. M. Campos Pinto, A. Cohen, P. Petrushev, “High order geometric smoothness for conservation laws”, J. Hyperbolic Differ. Equ., 02:01 (2005), 39–59
313. E. Tadmor, M. Rascle, P. Bagnerini, “Compensated compactness for 2D conservation laws”, J. Hyperbolic Differ. Equ., 02:03 (2005), 697–712
314. Adimurthi, S. Mishra, G. D. V. Gowda, “Optimal entropy solutions for conservation laws with discontinuous flux-functions”, J. Hyperbolic Differ. Equ., 02:04 (2005), 783–837
315. J. Ehrt, J. Härterich, “Asymptotic behavior of spatially inhomogeneous balance laws”, J. Hyperbolic Differ. Equ., 02:03 (2005), 645–672
316. Eymard R., Galloueët Th., “Analytical and numerical study of a model of erosion and sedimentation”, SIAM J. Numer. Anal., 43:6 (2006), 2344–2370
317. Jovanović V., Rohde Ch., “Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws”, SIAM J. Numer. Anal., 43:6 (2006), 2423–2449
318. Frolkovič P., Kačur J., “Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D”, Comput. Geosci., 10:3 (2006), 279–290
319. Visintin A., “Quasilinear parabolic P.D.E.s with discontinuous hysteresis”, Ann. Mat. Pura Appl. (4), 185:4 (2006), 487–519
320. Du Tao, Wu Zi-Niu, Wang Bing, “On steady state computation of turbulent flows using $k-\varepsilon$ models approximated by the time splitting method”, Int. J. Numer. Methods Fluids, 51:1 (2006), 77–115
321. Ye Xiao Ping, Lin Long Wei, “Error bounds for Glimm difference approximations for scalar conservation laws without convexity”, Acta Math. Sin. (Engl. Ser.), 22:4 (2006), 1271–1282
322. Droniou J., Imbert C., “Fractal first-order partial differential equations”, Arch. Ration. Mech. Anal., 182:2 (2006), 299–331
323. Popov B., Trifonov O., “One-sided stability and convergence of the Nessyahu–Tadmor scheme”, Numer. Math., 104:4 (2006), 539–559
324. Strub I.S., Bayen A.M., “Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling”, Internat. J. Robust Nonlinear Control, 16:16 (2006), 733–748
325. Omrane A., “Error estimates for scalar conservation laws by a kinetic approach”, Int. J. Math. Math. Sci., 2006, 41510, 15 pp.
326. E. Yu. Panov, “On well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations”, J. Math. Sci., 150:6 (2008), 2578–2587
327. M. V. Korobkov, E. Yu. Panov, “Isentropic solutions of quasilinear equations of the first order”, Sb. Math., 197:5 (2006), 727–752
328. S. A. Sazhenkov, “The genuinely nonlinear Graetz–Nusselt ultraparabolic equation”, Siberian Math. J., 47:2 (2006), 355–375
329. Giuseppe Maria Coclite, Kenneth Hvistendahl Karlsen, “A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation”, Communications in Partial Differential Equations, 31:8 (2006), 1253
330. Kaouther Ammar, Petra Wittbold, Jose Carrillo, “Scalar conservation laws with general boundary condition and continuous flux function”, Journal of Differential Equations, 228:1 (2006), 111
331. Arturo de Pablo, Guillermo Reyes, “Long time behaviour for a nonlinear first-order equation”, Nonlinear Analysis: Theory, Methods & Applications, 65:2 (2006), 284
332. John M. Hong, “An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem”, Journal of Differential Equations, 222:2 (2006), 515
333. Gui-Qiang Chen, Stéphane Junca, Michel Rascle, “Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws”, Journal of Differential Equations, 222:2 (2006), 439
334. B ANDREIANOV, N IGBIDA, “Revising uniqueness for a nonlinear diffusion–convection equation”, Journal of Differential Equations, 227:1 (2006), 69
335. Seok Hwang, “Nonlinear diffusive–dispersive limits for scalar multidimensional conservation laws”, Journal of Differential Equations, 225:1 (2006), 90
336. Raimund Bürger, Aníbal Coronel, Mauricio Sepúlveda, “On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels”, Applied Numerical Mathematics, 56:10-11 (2006), 1397
337. Anne-Laure Dalibard, “Kinetic formulation for heterogeneous scalar conservation laws”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 23:4 (2006), 475
338. Giuseppe M. Coclite, Kenneth H. Karlsen, “On the well-posedness of the Degasperis–Procesi equation”, Journal of Functional Analysis, 233:1 (2006), 60
339. Seok Hwang, “Fluid-dynamic limit for the Ruijgrok–Wu model of the Boltzmann equation”, Journal of Mathematical Analysis and Applications, 315:1 (2006), 327
340. R. Bürger, F. Concha, K.H. Karlsen, A. Narváez, “Numerical simulation of clarifier-thickener units treating ideal suspensions with a flux density function having two inflection points”, Mathematical and Computer Modelling, 44:3-4 (2006), 255
341. F. Bouchut, H. Frid, “Finite difference schemes with cross derivatives correctors for multidimensional parabolic systems”, J. Hyperbolic Differ. Equ., 03:01 (2006), 27–52
342. F. Lagoutière, “Large time behavior of numerical solutions of scalar conservation laws”, J. Hyperbolic Differ. Equ., 03:04 (2006), 631–648
343. L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, J. Soler, “Exact solutions and dynamics of globally coupled oscillators”, Math. Models Methods Appl. Sci., 16:12 (2006), 1919–1959
344. Hua Zhang, Wan-cheng Sheng, “Guckenheimer structure of solution of Riemann problem with four pieces of constants in two space dimensions for scalar conservation laws”, J. of Shanghai Univ., 10:4 (2006), 305
345. Nishikawa M., Nishibata Sh., “Convergence rates toward the travelling waves for a model system of the radiating gas”, Math. Methods Appl. Sci., 30:6 (2007), 649–663
346. Liu Hong Xia, Pan Tao, “Construction of solutions and $L^1$-error estimates of viscous methods for scalar conservation laws with boundary”, Acta Math Sinica, 23:3 (2007), 393–410
347. Herty M., Klar A., Piccoli B., “Existence of solutions for supply chain models based on partial differential equations”, SIAM J. Math. Anal., 39:1 (2007), 160–173
348. Tang Huazhong, Warnecke G., “On convergence of a domain decomposition method for a scalar conservation law”, SIAM J. Numer. Anal., 45:4 (2007), 1453–1471
349. Dalibard A.-L., “Kinetic formulation for a parabolic conservation law. Application to homogenization”, SIAM J. Math. Anal., 39:3 (2007), 891–915
350. Eymard R., Gallouët Th., “A partial differential inequality in geological models”, Chin. Ann. Math. Ser. B, 28:6 (2007), 709–736
351. Dalibard A.-L., “Initial layer for the Homogenization of a Conservation Law with Vanishing Viscosity”, Arch. Ration. Mech. Anal., 185:3 (2007), 515–543
352. Moutari S., Rascle M., “A hybrid Lagrangian model based on the Aw–Rascle traffic flow model”, SIAM J. Appl. Math., 68:2 (2007), 413–436
353. Echenim N., Clement F., Sorine M., “Multiscale modeling of follicular ovulation as a reachability problem”, Multiscale Model. Simul., 6:3 (2007), 895–912
354. Sheng W., Zhang Tong, “A cartoon for the climbing ramp problem of a shock and von Neumann paradox”, Arch. Ration. Mech. Anal., 184:2 (2007), 243–255
355. Sjoberg D., “On uniqueness and continuity for the quasi-linear, bianisotropic maxwell equations, using an entropy condition”, Progress In Electromagnetics Research, 71 (2007), 317–339
356. Jun Young-Bae, Kim Young-Hee, Oh Kyong-Ah, “Subtraction algebras with additional conditions”, Commun. Korean Math. Soc., 22:1 (2007), 1–7
357. Hwang Seok, “Convergence of approximate solutions to scalar conservation laws by degenerate diffusion”, Commun. Korean Math. Soc., 22:1 (2007), 145–155
358. Panov E.Yu., “Existence of strong traces for quasi-solutions of multidimensional conservation laws”, J. Hyperbolic Differ. Equ., 4:4 (2007), 729–770
359. Karlsen K.H., Rascle M., Tadmor E., “On the existence and compactness of a two-dimensional resonant system of conservation laws”, Commun. Math. Sci., 5:2 (2007), 253–265
360. Jianwen Zhang, “Well-posedness for a class of nonlinear parabolic equations with strong degeneracy”, Nonlinear Analysis: Theory, Methods & Applications, 67:1 (2007), 270
361. Giuseppe M. Coclite, Kenneth H. Karlsen, “On the uniqueness of discontinuous solutions to the Degasperis–Procesi equation”, Journal of Differential Equations, 234:1 (2007), 142
362. Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda, “Conservation law with the flux function discontinuous in the space variable—II”, Journal of Computational and Applied Mathematics, 203:2 (2007), 310
363. Raimund Bürger, Ariel Narváez, “Steady-state, control, and capacity calculations for flocculated suspensions in clarifier–thickeners”, International Journal of Mineral Processing, 84:1-4 (2007), 274
364. Matania Ben-Artzi, Philippe G. LeFloch, “Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 24:6 (2007), 989
365. Raimund Bürger, Hermano Frid, Kenneth H. Karlsen, “On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition”, Journal of Mathematical Analysis and Applications, 326:1 (2007), 108
366. M. Remešíková, “Numerical solution of two-dimensional convection–diffusion–adsorption problems using an operator splitting scheme”, Applied Mathematics and Computation, 184:1 (2007), 116
367. M. Bendahmane, K. H. Karlsen, J. M. Urbano, “On a two-sidedly degenerate chemotaxis model with volume-filling effect”, Math. Models Methods Appl. Sci., 17:05 (2007), 783–804
368. Bürger R., García A., Karlsen K.H., Towers J.D., “A family of numerical schemes for kinematic flows with discontinuous flux”, J. Engrg. Math., 60:3-4 (2008), 387–425
369. Lanson N., Vila J.-P., “Renormalized meshfree schemes. II. Convergence for scalar conservation laws”, SIAM J. Numer. Anal., 46:4 (2008), 1935–1964
370. Bürger R., Ruiz R., Schneider K., Sepúlveda M., “Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension”, M2AN Math. Model. Numer. Anal., 42:4 (2008), 535–563
371. Bürger R., Ruiz R., Schneider K., Sepúlveda M.A., “Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux”, J. Engrg. Math., 60:3-4 (2008), 365–385
372. V. A. Galaktionov, S. I. Pokhozhaev, “Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves”, Comput. Math. Math. Phys., 48:10 (2008), 1784–1810
373. Galaktionov V.A., “Nonlinear dispersion equations: smooth deformations, compactions, and extensions to higher orders”, Comput. Math. Math. Phys., 48:10 (2008), 1823–1856
374. Lin Gui-cheng, Sheng Wan-cheng, “Godunov's method for initial-boundary value problem of scalar conservation laws”, J. Shanghai Univ., 12:4 (2008), 298–301
375. Yao Ai-di, Sheng Wan-cheng, “Initial-boundary value problem of nonlinear hyperbolic system for conservation laws with delta-shock waves”, J. Shanghai Univ., 12:4 (2008), 306–310
376. Galaktionov V.A., “On higher-order viscosity approximations of odd-order nonlinear PDEs”, J. Engrg. Math., 60:2 (2008), 173–208
377. Shearer M., Gray J.M.N.T., Thornton A.R., “Stable solutions of a scalar conservation law for particle-size segregation in dense granular avalanches”, European J. Appl. Math., 19:1 (2008), 61–86
378. A. V. Gasnikov, “Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves”, Comput. Math. Math. Phys., 48:8 (2008), 1376–1405
379. Pohozaev S.I., “On the nonexistence of global solutions of the Hamilton–Jacobi equation”, Differ. Equ., 44:10 (2008), 1467–1477
380. Greg Norgard, Kamran Mohseni, “A regularization of the Burgers equation using a filtered convective velocity”, J. Phys. A: Math. Theor, 41:34 (2008), 344016
381. Romain Nguyen van yen, Marie Farge, Dmitry Kolomenskiy, Kai Schneider, Nick Kingsbury, “Wavelets meet Burgulence: CVS-filtered Burgers equation”, Physica D: Nonlinear Phenomena, 237:14-17 (2008), 2151
382. Kaouther Ammar, “On nonlinear diffusion problems with strong degeneracy”, Journal of Differential Equations, 244:8 (2008), 1841
383. Gui-Qiang Chen, Nadine Even, Christian Klingenberg, “Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems”, Journal of Differential Equations, 245:11 (2008), 3095
384. Jin Feng, David Nualart, “Stochastic scalar conservation laws”, Journal of Functional Analysis, 255:2 (2008), 313
385. Wen Shen, Zhengfu Xu, “Vanishing viscosity approximation to hyperbolic conservation laws”, Journal of Differential Equations, 244:7 (2008), 1692
386. J. Kačur, B. Malengier, M. Remešíkov, “Convergence of an operator splitting method on a bounded domain for a convection–diffusion–reaction system”, Journal of Mathematical Analysis and Applications, 348:2 (2008), 894
387. Philippe G. LeFloch, Baver Okutmustur, “Hyperbolic conservation laws on manifolds with limited regularity”, Comptes Rendus Mathematique, 346:9-10 (2008), 539
388. L. Caravenna, “An entropy based Glimm-type functional”, J. Hyperbolic Differ. Equ., 05:03 (2008), 643–662
389. Ambrosio L., Frid H., “Multiscale Young measures in almost periodic homogenization and applications”, Arch. Ration. Mech. Anal., 192:1 (2009), 37–85
390. Dalibard A.-L., “Homogenization of non-linear scalar conservation laws”, Arch. Ration. Mech. Anal., 192:1 (2009), 117–164
391. Ohlberger M., “A review of a posteriori error control and adaptivity for approximations of non-linear conservation laws”, Internat. J. Numer. Methods Fluids, 59:3 (2009), 333–354
392. Danilov V.G., Mitrovic D., “Smooth approximations of global in time solutions to scalar conservation laws”, Abstr. Appl. Anal., 2009, 350762, 26 pp.
393. Bürger R., Karlsen K.H., Towers J.D., “An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections”, SIAM J. Numer. Anal., 47:3 (2009), 1684–1712
394. LeFloch Ph.G., Okutmustur B., Neves W., “Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes”, Acta Math. Sin. (Engl. Ser.), 25:7 (2009), 1041–1066
395. Mitrovic D., Bojkovic V., Danilov V.G., “Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process”, Math. Meth. Appl. Sci., 33:7 (2009), 904–921
396. N. N. Subbotina, E. A. Kolpakova, “On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics”, Proc. Steklov Inst. Math. (Suppl.), 268, suppl. 1 (2010), S222–S239
397. Frid H., Silva J., “Homogenization of nonlinear PDEs in the Fourier–Stieltjes algebras”, SIAM J. Math. Anal., 41:4 (2009), 1589–1620
398. A. V. Gasnikov, “Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity”, Izv. Math., 73:6 (2009), 1111–1148
399. Holden H., Karlsen K.H., Mitrovic D., “Zero diffusion-dispersion-smoothing limits for a scalar conservation law with discontinuous flux function”, Int. J. Differ. Equ., 2009, 279818, 33 pp.
400. Jia Zhi, Yao Ai-di, “Solutions to a hyperbolic system of conservation laws on two boundaries”, Appl. Math. J. Chinese Univ. Ser. B, 24:4 (2009), 495–502
401. Aleksić J., Mitrovic D., Pilipović S., “Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media”, J. Evol. Equ., 9:4 (2009), 809–828
402. Holden H., Karlsen K.H., Mitrovic D., Panov E.Yu., “Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function”, Acta Math. Sci. Ser. B Engl. Ed., 29:6 (2009), 1573–1612
403. Panov E.Yu., “On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux”, J. Differential Equations, 247:10 (2009), 2821–2870
404. Antontsev S., Díaz J.I., “On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 103:1 (2009), 201–214
405. Panov E.Yu., “On infinite-dimensional Keyfitz-Kranzer systems of conservation laws”, Differ. Equ., 45:2 (2009), 274–278
406. Lyashko A.D., Fedotov E.M., “Galerkin-Petrov limit schemes for the convection-diffusion equation”, Differ. Equ., 45:7 (2009), 1063–1073
407. Ambrosio L., Crippa G., Figalli A., Spinolo L.V., “Some New Well-Posedness Results for Continuity and Transport Equations, and Applications to the Chromatography System”, SIAM Journal on Mathematical Analysis, 41:5 (2009), 1890–1920
408. B. Andreianov, M. Bendahmane, K.H. Karlsen, S. Ouaro, “Well-posedness results for triply nonlinear degenerate parabolic equations”, Journal of Differential Equations, 247:1 (2009), 277
409. Frederike Kissling, Philippe G. LeFloch, Christian Rohde, “A kinetic decomposition for singular limits of non-local conservation laws”, Journal of Differential Equations, 247:12 (2009), 3338
410. Hassan Ibrahim, “Existence and uniqueness for a nonlinear parabolic/Hamilton–Jacobi coupled system describing the dynamics of dislocation densities”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26:2 (2009), 415
411. Mostafa Bendahmane, Raimund Bürger, Ricardo Ruiz-Baier, Kai Schneider, “Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems”, Applied Numerical Mathematics, 59:7 (2009), 1668
412. Luigi Ambrosio, Hermano Frid, Jean Silva, “Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications”, Journal of Functional Analysis, 256:6 (2009), 1962
413. Adimurthi, G.D. Veerappa Gowda, Jérôme Jaffré, “Monotonization of flux, entropy and numerical schemes for conservation laws”, Journal of Mathematical Analysis and Applications, 352:1 (2009), 427
414. Xiaozhou Yang, Tao Wei, “New structures for non-selfsimilar solutions of multi-dimensional conservation laws”, Acta Mathematica Scientia, 29:5 (2009), 1182
415. S. Diehl, “A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients”, J. Hyperbolic Differ. Equ., 06:01 (2009), 127–159
416. E. Comparini, R. Dal Passo, C. Pescatore, M. Ughi, “On a model for the propagation of isotopic disequilibrium by diffusion”, Math. Models Methods Appl. Sci., 19:08 (2009), 1277–1294
417. Kolb O., Lang J., Bales P., “An implicit box scheme for subsonic compressible flow with dissipative source term”, Numer. Algorithms, 53:2-3 (2010), 293–307
418. Kondo C.I., Webler C.M., “Higher-Order for the Multidimensional Generalized BBM-Burgers Equation: Existence and Convergence Results”, Acta Appl. Math., 111:1 (2010), 45–64
419. Galaktionov V.A., “Shock waves and compactons for fifth-order non-linear dispersion equations”, European J. Appl. Math., 21:1 (2010), 1–50
420. Panov E.Yu., “Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux”, Arch. Ration. Mech. Anal., 195:2 (2010), 643–673
421. May L.B.H., Shearer M., Daniels K.E., “Scalar Conservation Laws with Nonconstant Coefficients with Application to Particle Size Segregation in Granular Flow”, J. Nonlinear Sci., 2010
422. Mamaghani M., Enchéry G., Chainais-Hillairet C., “Development of a refinement criterion for adaptive mesh refinement in steam-assisted gravity drainage simulation”, Comput. Geosci., 2010
423. D'Apice C., Kogut P.I., Manzo R., “Efficient controls for traffic flow on networks”, J. Dyn. Control Syst., 16:3 (2010), 407–437
424. Ch. Srinivasa Rao, Manoj K. Yadav, “Large-time behaviour of solutions of the inviscid non-planar Burgers equation”, J. Eng. Math., 2010
425. Feireisl E., “Mathematical Models of Incompressible Fluids as Singular Limits of Complete Fluid Systems”, Milan J. Math., 2010
426. Mi-Ho Giga, Yoshikazu Giga, “Very singular diffusion equations: second and fourth order problems”, Japan J. Indust. Appl. Math, 27:3 (2010), 323–345
427. E. A. Kolpakova, “Obobschennyi metod kharakteristik v teorii uravnenii Gamiltona–Yakobi i zakonov sokhraneniya”, Tr. IMM UrO RAN, 16, no. 5, 2010, 95–102
428. Crippa G., Spinolo L.V., “An Overview on Some Results Concerning the Transport Equation and its Applications to Conservation Laws”, Communications on Pure and Applied Analysis, 9:5 (2010), 1283–1293
429. Francesco Bigolin, Francesco Serra Cassano, “Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs”, Advances in Calculus of Variations, 3:1 (2010), 69
430. V. A. Garanzha, “Polyconvex potentials, invertible deformations, and a thermodynamically consistent formulation of the equations of the nonlinear theory of elasticity”, Comput. Math. Math. Phys., 50:9 (2010), 1561–1587
431. B. Andreianov, M. Bendahmane, K. H. Karlsen, “Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations”, J. Hyperbolic Differ. Equ., 07:01 (2010), 1–67
432. H. Frankowska, “On LeFloch's solutions to the initial-boundary value problem for scalar conservation laws”, J. Hyperbolic Differ. Equ., 07:03 (2010), 503–543
433. Mitrovic D., “Existence and Stability of a Multidimensional Scalar Conservation Law with Discontinuous Flux”, Netw. Heterog. Media, 5:1 (2010), 163–188
434. Brenier Ya., “Hidden Convexity in Some Nonlinear PDEs From Geomety and Physics”, J. Convex Anal., 17:3-4, SI (2010), 945–959
435. Victor A. Galaktionov, “Single Point Gradient Blow-Up and Nonuniqueness for a Third-Order Nonlinear Dispersion Equation: Third-Order Nonlinear Dispersion Equation”, Stud. Appl. Math., 126:2 (2011), 103–143
436. Corrado Lattanzio, Amelio Maurizi, Benedetto Piccoli, “Moving Bottlenecks in Car Traffic Flow: A PDE-ODE Coupled Model”, SIAM J. Math. Anal, 43:1 (2011), 50
437. Boris Andreianov, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro, “A Theory of $L^1$-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux”, Arch. Rational Mech. Anal., 201:1 (2011), 27–86
438. S. Blandin, D. Work, P. Goatin, B. Piccoli, A. Bayen, “A General Phase Transition Model for Vehicular Traffic”, SIAM J. Appl. Math, 71:1 (2011), 107
439. F. Betancourt, R. Bürger, K. H. Karlsen, E. M. Tory, “On nonlocal conservation laws modelling sedimentation”, Nonlinearity, 24:3 (2011), 855
440. Simone Cifani, Espen R. Jakobsen, “Entropy solution theory for fractional degenerate convection–diffusion equations”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 28:3 (2011), 413–441
441. Darko Mitrovic, “New entropy conditions for scalar conservation laws with discontinuous flux”, DCDS-A, 30:4 (2011), 1191
442. Guodong Wang, “An Engquist–Osher type finite difference scheme with a discontinuous flux function in space”, Journal of Computational and Applied Mathematics, 235:17 (2011), 4966–4977
443. João Paulo Dias, Mário Figueira, Hermano Frid, “Vanishing viscosity with short wave–long wave interactions for multi-D scalar conservation laws”, Journal of Differential Equations, 251:3 (2011), 492–503
444. Paulo Amorim, Philippe G. LeFloch, Wladimir Neves, “A geometric approach to error estimates for conservation laws posed on a spacetime”, Nonlinear Analysis: Theory, Methods & Applications, 74:15 (2011), 4898–4917
445. P. V. Lysuho, E. Yu. Panov, “Renormalized entropy solutions to the Cauchy problem for first order quasilinear conservation laws in the class of periodic functions”, J. Math. Sci., 177:1 (2011), 27–49
446. V. Caselles, “On the entropy conditions for some flux limited diffusion equations”, Journal of Differential Equations, 250:8 (2011), 3311
447. N. V. Chemetov, L. K. Arruda, “$L_p$-solvability of a full superconductive model”, Nonlinear Analysis: Real World Applications, 12:4 (2011), 2118–2129
448. Vicent Caselles, “An existence and uniqueness result for flux limited diffusion equations”, DCDS-A, 31:4 (2011), 1151
449. Huijiang Zhao, “Existence and uniqueness of the global admissible solution for a viscoelastic model with relaxation”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 126:05 (2011), 1113
450. Jörg Härterich, “Heteroclinic orbits between rotating waves in hyperbolic balance laws”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 129:03 (2011), 519
451. Adrian T. Hill, Endre Süli, “Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 125:02 (2011), 439
452. Pierangelo Marcati, Roberto Natalini, “Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 125:01 (2011), 115
453. G.M. Coclite, M.M. Coclite, “Conservation laws with singular nonlocal sources”, Journal of Differential Equations, 250:10 (2011), 3831
454. Marco Di Francesco, Peter A. Markowich, Jan-Frederik Pietschmann, Marie-Therese Wolfram, “On the Hughes' model for pedestrian flow: The one-dimensional case”, Journal of Differential Equations, 250:3 (2011), 1334
455. Danilov V.G., Mitrovic D., “Shock Wave Formation Process for a Multidimensional Scalar Conservation Law”, Quart Appl. Math., 69:4 (2011), 613–634
456. Jimenez J., “Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux”, J. Evol. Equ., 11:3 (2011), 553–576
457. Panov E.Yu., “On the Dirichlet Problem for First Order Quasilinear Equations on a Manifold”, Trans. Amer. Math. Soc., 363:5 (2011), 2393–2446
458. Barbara Lee Keyfitz, “Singular shocks: retrospective and prospective”, Confluentes Math., 03:03 (2011), 445–470
459. Lysukho P.V., Panov E.Yu., “O suschestvovanii i edinstvennosti neogranichennykh entropiinykh reshenii zadachi koshi dlya kvazilineinykh zakonov sokhraneniya pervogo poryadka”, Differentsialnye uravneniya, 47:1 (2011), 103–111
460. Mitrovic D., Ivec I., “A Generalization of H-Measures and Application on Purely Fractional Scalar Conservation Laws”, Commun. Pure Appl. Anal., 10:6 (2011), 1617–1627
461. Theodoros Katsaounis, Chiara Simeoni, “Three-points interfacial quadrature for geometrical source terms on nonuniform grids”, Calcolo, 49:3 (2012), 149–176
462. C. Bahadoran, “Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries”, Commun. Math. Phys, 310:1 (2012), 1–24
463. Debora Amadori, M. Di Francesco, “The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions”, Acta Mathematica Scientia, 32:1 (2012), 259
464. Yachun Li, Qin Wang, “Homogeneous Dirichlet problems for quasilinear anisotropic degenerate parabolic–hyperbolic equations”, Journal of Differential Equations, 252:9 (2012), 4719–4741
465. Gui-Qiang Chen, Qian Ding, Kenneth H. Karlsen, “On Nonlinear Stochastic Balance Laws”, Arch Rational Mech Anal, 204:3 (2012), 707–743
466. Chun Shen, Meina Sun, “The Bifurcation Phenomenon for Scalar Conservation Laws with Discontinuous Flux Functions”, Acta Appl. Math., 121:1 (2012), 69–80
467. Nathaël Alibaud, Simone Cifani, Espen R. Jakobsen, “Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations”, SIAM J. Math. Anal, 44:2 (2012), 603
468. Ionel Sorin Ciuperca, Arnaud Heibig, Liviu Iulian Palade, “Existence and uniqueness results for the Doi–Edwards polymer melt model: the case of the (full) nonlinear configurational probability density equation”, Nonlinearity, 25:4 (2012), 991
469. Svenn Tveit, Ivar Aavatsmark, “Errors in the upstream mobility scheme for countercurrent two-phase flow in heterogeneous porous media”, Comput. Geosci., 16:3 (2012), 809–825
470. Beixiang Fang, Pingfan Tang, Ya-Guang Wang, “The Riemann problem of the Burgers equation with a discontinuous source term”, Journal of Mathematical Analysis and Applications, 395:1 (2012), 307–335
471. N. Besse, “Global weak solutions for the relativistic waterbag continuum”, Math. Models Methods Appl. Sci., 22:01 (2012), 1150001, 43 pp.
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