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Partial differential inclusions of transport type with state constraints

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  • The focus is on the existence of weak solutions to the quasilinear first-order partial differential inclusion

    $\partial_t \; f \:\: ∈ \:\: - {\rm{div}}_{\mathbf{x}} \big( \mathcal{G}(t, f) \; f \big) + \mathcal{U}(t, f) · f + \mathcal{W}(t, f)$

    with values in $L^p({{\mathbb{R}}^{N}})$ for $p ∈ (1, ∞)$. The solution is to satisfy state constraints in addition, i.e., all its values belong to a given set $\mathcal{V} \subset L^p({{\mathbb{R}}^{N}})$ of constraints. We specify sufficient conditions such that every function in $\mathcal{V}$ initializes at least one weak solution with all its values in $\mathcal{V}$(so-called weak invariance a.k.a. viability of $\mathcal{V}$). Due to the regularity assumptions about the set-valued coefficient mappings, these solutions prove to be renormalized (in the sense of Di Perna and Lions).

    Mathematics Subject Classification: Primary: 35F25, 35R70; Secondary: 35B30, 35D30, 47J35.


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  •   A. Aggarwal , R. M. Colombo  and  P. Goatin , Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015) , 963-983.  doi: 10.1137/140975255.
      H. W. Alt, Linear Functional Analysis. An Application-Oriented Introduction, London: Springer, 2016, Translated from the 6th German edition by Robert Nürnberg. doi: 10.1007/978-1-4471-7280-2.
      L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of Variations and Nonlinear Partial Differential Equations (eds. B. Dacorogna and P. Marcellini), vol. 1927 of Lecture Notes in Math., Springer, Berlin, 2008, 1–41. doi: 10.1007/978-3-540-75914-0_1.
      L. Ambrosio , Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications, Rev. Mat. Complut., 30 (2017) , 427-450.  doi: 10.1007/s13163-017-0244-3.
      L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in Transport Equations and Multi-D Hyperbolic Conservation Laws (eds. F. Ancona, S. Bianchini, R. M. Colombo, C. De Lellis, A. Marson and A. Montanari), vol. 5 of Lect. Notes Unione Mat. Ital., Springer, Berlin, 2008, 3–57. doi: 10.1007/978-3-540-76781-7_1.
      L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, vol. 25 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.
      J.-P. Aubin and H. Frankowska, Feedback controls for uncertain systems, in Modeling, Estimation and Control of Systems with Uncertainty (Sopron, 1990) (eds. G. B. Di Masi, A. Gombani and A. B. Kurzhansky), vol. 10 of Progr. Systems Control Theory, Birkh¨auser Boston, Boston, MA, 1991, 1–21.
      J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.
      J.-P. Aubin , Mutational equations in metric spaces, Set-Valued Anal., 1 (1993) , 3-46.  doi: 10.1007/BF01039289.
      J.-P. Aubin, Mutational and Morphological Analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999, Tools for shape evolution and morphogenesis. doi: 10.1007/978-1-4612-1576-9.
      J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, Viability Theory, 2nd edition, Springer, Heidelberg, 2011, New directions. doi: 10.1007/978-3-642-16684-6.
      J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1984, Set-valued maps and viability theory. doi: 10.1007/978-3-642-69512-4.
      J.-P. Aubin and A. Désilles, Traffic Networks as Information Systems, Mathematical Engineering, Springer-Verlag, Berlin Heidelberg, 2017, A viability approach. doi: 10.1007/978-3-642-54771-3.
      J.-P. Aubin and H. Frankowska, Set-valued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1990.
      D. Averna , Lusin type theorems for multifunctions, Scorza Dragoni's property and Carathéodory selections, Boll. Unione Mat. Ital., VII. Ser., A, 8 (1994) , 193-202. 
      V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
      V. I. Bogachev, Measure Theory. Vol. I, II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.
      R. Borsche , R. M. Colombo , M. Garavello  and  A. Meurer , Differential equations modeling crowd interactions, J. Nonlinear Sci., 25 (2015) , 827-859.  doi: 10.1007/s00332-015-9242-0.
      D. Bothe , Multivalued differential equations on graphs, Nonlinear Anal., 18 (1992) , 245-252.  doi: 10.1016/0362-546X(92)90062-J.
      D. Bothe , Flow invariance for perturbed nonlinear evolution equations, Abstr. Appl. Anal., 1 (1996) , 417-433.  doi: 10.1155/S1085337596000231.
      H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
      O. Cârjǎ  and  M. D. P. Monteiro Marques , Weak tangency, weak invariance, and Carathéodory mappings, J. Dynam. Control Systems, 8 (2002) , 445-461.  doi: 10.1023/A:1020765401015.
      O. Cârjǎ, M. Necula and I. I. Vrabie, Viability, Invariance and Applications, vol. 207 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2007.
      C. Castaing , M. Moussaoui  and  A. Syam , Multivalued differential equations on closed convex sets in Banach spaces, Set-Valued Anal., 1 (1993) , 329-353.  doi: 10.1007/BF01027824.
      C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, vol. 580 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1977.
      F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.
      R. M. Colombo and G. Guerra, Nonlocal sources in hyperbolic balance laws with applications, in Hyperbolic Problems: Theory, Numerics, Applications (eds. S. Benzoni-Gavage and D. Serre), Springer, Berlin, 2008,577–584. doi: 10.1007/978-3-540-75712-2_56.
      R. M. Colombo , A. Corli  and  M. D. Rosini , Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007) , 449-461.  doi: 10.1002/zamm.200710327.
      R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023, 34pp. doi: 10.1142/S0218202511500230.
      R. M. Colombo  and  G. Guerra , Hyperbolic balance laws with a dissipative non local source, Commun. Pure Appl. Anal., 7 (2008) , 1077-1090.  doi: 10.3934/cpaa.2008.7.1077.
      R. M. Colombo  and  M. Lécureux-Mercier , Nonlocal crowd dynamics models for several populations, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012) , 177-196.  doi: 10.1016/S0252-9602(12)60011-3.
      G. Corbin, A. Hunt, F. Schneider, A. Klar and C. Surulescu, Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum, Math. Models Methods Appl. Sci., 28 (2018), 1771–1880, URL https://arXiv.org/abs/1801.02842. doi: 10.1142/S0218202518400055.
      K. Deimling, Multivalued Differential Equations, vol. 1 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228.
      J. Diestel, Remarks on weak compactness in $L_1(μ, X)$, Glasg. Math. J., 18 (1977), 87–91, URL http://dx.doi.org/10.1017/S0017089500003074.
      R. J. DiPerna  and  P.-L. Lions , Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989) , 511-547.  doi: 10.1007/BF01393835.
      J. Dyson , S. A. Gourley  and  G. F. Webb , A non-local evolution equation model of cell-cell adhesion in higher dimensional space, J. Biol. Dyn., 7 (2013) , 68-87.  doi: 10.1080/17513758.2012.755572.
      C. Engwer , T. Hillen , M. Knappitsch  and  C. Surulescu , Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015) , 551-582.  doi: 10.1007/s00285-014-0822-7.
      C. Engwer , A. Hunt  and  C. Surulescu , Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, Math. Med. Biol., 33 (2016) , 435-459.  doi: 10.1093/imammb/dqv030.
      C. Engwer , M. Knappitsch  and  C. Surulescu , A multiscale model for glioma spread including cell-tissue interactions and proliferation, Math. Biosci. Eng., 13 (2016) , 443-460.  doi: 10.3934/mbe.2015011.
      C. Engwer , C. Stinner  and  C. Surulescu , On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, Math. Models Methods Appl. Sci., 27 (2017) , 1355-1390.  doi: 10.1142/S0218202517400188.
      H. Frankowska , S. Plaskacz  and  T. Rze & uchowski , Measurable viability theorems and the Hamilton-Jacobi-Bellman equation, J. Differential Equations, 116 (1995) , 265-305.  doi: 10.1006/jdeq.1995.1036.
      H. Frankowska  and  S. Plaskacz , A measurable upper semicontinuous viability theorem for tubes, Nonlinear Anal., 26 (1996) , 565-582.  doi: 10.1016/0362-546X(94)00299-W.
      H. G. Garnir, M. De Wilde and J. Schmets, Analyse Fonctionnelle. Tome II. Measure et intégration dans l'espace Euclidien $E_{n}$, Birkhäuser Verlag, Basel-Stuttgart, 1972, Lehrb¨ucher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Band 37.
      A. Gerisch  and  M. A. J. Chaplain , Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion, J. Theoret. Biol., 250 (2008) , 684-704.  doi: 10.1016/j.jtbi.2007.10.026.
      P. Goatin  and  S. Scialanga , Well-posedness and finite volume approximations of the lwr traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016) , 107-121.  doi: 10.3934/nhm.2016.11.107.
      J. W. Green  and  F. A. Valentine , On the Arzelà-Ascoli theorem, Math. Mag., 34 (1960/1961) , 199-202.  doi: 10.2307/2687984.
      G. Haddad , Monotone trajectories of differential inclusions and functional-differential inclusions with memory, Israel J. Math., 39 (1981) , 83-100.  doi: 10.1007/BF02762855.
      G. Haddad , Monotone viable trajectories for functional-differential inclusions, J. Differential Equations, 42 (1981) , 1-24.  doi: 10.1016/0022-0396(81)90031-0.
      G. Haddad, Functional viability theorems for differential inclusions with memory, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 179–204, URL http://www.numdam.org/item?id=AIHPC_1984__1_3_179_0. doi: 10.1016/S0294-1449(16)30427-9.
      C. J. Himmelberg  and  F. S. Van Vleck , An extension of Brunovský's Scorza Dragoni type theorem for unbounded set-valued functions, Math. Slovaca, 26 (1976) , 47-52. 
      S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, vol. 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997, Theory. doi: 10.1007/978-1-4615-6359-4.
      S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II, vol. 500 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000, Applications. doi: 10.1007/978-1-4615-4665-8_17.
      A. Hunt  and  C. Surulescu , A multiscale modeling approach to glioma invasion with therapy, Vietnam J. Math., 45 (2017) , 221-240.  doi: 10.1007/s10013-016-0223-x.
      J. Jarník  and  J. Kurzweil , On conditions on right hand sides of differential relations, Časopis Pěst. Mat., 102 (1977) , 334-349, 426. 
      P. O. Kasyanov , V. S. Mel'nik  and  S. Toscano , Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued $w_{λ_0}$-pseudomonotone maps, J. Differential Equations, 249 (2010) , 1258-1287.  doi: 10.1016/j.jde.2010.05.008.
      J. Kelkel  and  C. Surulescu , On some models for cancer cell migration through tissue networks, Math. Biosci. Eng., 8 (2011) , 575-589.  doi: 10.3934/mbe.2011.8.575.
      J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Math. Models Methods Appl. Sci., 22 (2012), 1150017, 25pp. doi: 10.1142/S0218202511500175.
      P. E. Kloeden  and  T. Lorenz , Nonlocal multi-scale traffic flow models: Analysis beyond vector spaces, Bull. Math. Sci., 6 (2016) , 453-514.  doi: 10.1007/s13373-016-0090-5.
      K. Kuratowski  and  C. Ryll-Nardzewski , A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 13 (1965) , 397-403. 
      C. Le Bris  and  P.-L. Lions , Renormalized solutions of some transport equations with partially $W^{1, 1}$ velocities and applications, Ann. Mat. Pura Appl., 183 (2004) , 97-130.  doi: 10.1007/s10231-003-0082-4.
      Y. S. Ledyaev , Criteria for viability of trajectories of nonautonomous differential inclusions and their applications, J. Math. Anal. Appl., 182 (1994) , 165-188.  doi: 10.1006/jmaa.1994.1074.
      T. Lorenz , Shape evolutions under state constraints: A viability theorem, J. Math. Anal. Appl., 340 (2008) , 1204-1225.  doi: 10.1016/j.jmaa.2007.08.030.
      T. Lorenz, Mutational Analysis, vol. 1996 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, A joint framework for Cauchy problems in and beyond vector spaces. doi: 10.1007/978-3-642-12471-6.
      T. Lorenz , A viability theorem for set-valued states in a Hilbert space, J. Math. Anal. Appl., 457 (2018) , 1502-1567.  doi: 10.1016/j.jmaa.2017.08.011.
      T. Lorenz and C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Preprint, TU Kaiserslautern, Department of Mathematics, 2013, URL https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3440.
      T. Lorenz  and  C. Surulescu , On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, Math. Models Methods Appl. Sci., 24 (2014) , 2383-2436.  doi: 10.1142/S0218202514500249.
      A. Marigonda  and  M. Quincampoix , Mayer control problem with probabilistic uncertainty on initial positions, J. Differential Equations, 264 (2018) , 3212-3252.  doi: 10.1016/j.jde.2017.11.014.
      V. S. Melnik  and  J. Valero , On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998) , 83-111.  doi: 10.1023/A:1008608431399.
      D. Motreanu and N. Pavel, Tangency, Flow Invariance for Differential Equations, and Optimization Problems, vol. 219 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1999.
      M. Nagumo , Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan, 24 (1942) , 551-559. 
      K. J. Painter , J. M. Bloomfield , J. A. Sherratt  and  A. Gerisch , A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bull. Math. Biol., 77 (2015) , 1132-1165.  doi: 10.1007/s11538-015-0080-x.
      T. Rzeżuchowski, Scorza-Dragoni type theorem for upper semicontinuous multivalued functions, Bull. Acad. Polon. Sci. Sér. Sci. Math., 28 (1980), 61–66 (1981).
      G. Scorza Dragoni, Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile, Rend. Semin. Mat. Univ. Padova, 17 (1948), 102–106, URL http://www.numdam.org/item/RSMUP_1948__17__102_0.
      S. Z. Shi , Nagumo type condition for partial differential inclusions, Nonlinear Anal., 12 (1988) , 951-967.  doi: 10.1016/0362-546X(88)90077-6.
      S. Z. Shi , Optimal control of strongly monotone variational inequalities, SIAM J. Control Optim., 26 (1988) , 274-290.  doi: 10.1137/0326016.
      S. Z. Shi , Viability theorems for a class of differential-operator inclusions, J. Differential Equations, 79 (1989) , 232-257.  doi: 10.1016/0022-0396(89)90101-0.
      G. V. Smirnov, Introduction to the Theory of Differential Inclusions, vol. 41 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
      A. A. Tolstonogov , On the Scorza-Dragoni theorem for multivalued mappings with a variable domain, Mat. Zametki, 48 (1990) , 109-120,160.  doi: 10.1007/BF01236303.
      A. Ülger , Weak compactness in $L^1(μ, X)$, Proc. Amer. Math. Soc., 113 (1991) , 143-149.  doi: 10.2307/2048450.
      K. Yosida, Functional Analysis, vol. 123 of Grundlehren der mathematischen Wissenschaften, Sixth edition, Springer-Verlag, Berlin-New York, 1980.
      M. Z. Zgurovsky and P. O. Kasyanov, Qualitative and Quantitative Analysis of Nonlinear Systems, vol. 111 of Studies in Systems, Decision and Control, Springer, Cham, 2018, Theory and applications.
      M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing. III, vol. 27 of Advances in Mechanics and Mathematics, Springer, Heidelberg, 2012, Long-time behavior of evolution inclusions solutions in earth data analysis. doi: 10.1007/978-3-642-28512-7.
      M. Z. Zgurovsky and V. S. Mel'nik, Nonlinear Analysis and Control of Physical Processes and Fields, Data and Knowledge in a Changing World, Springer-Verlag, Berlin, 2004, With a preface by Roman Voronka. doi: 10.1007/978-3-642-18770-4.
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