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Journal articles on the topic 'Hamilton-Jacobi equations with unbounded data'

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Published: 11 March 2023

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1

Cosso, Andrea, Huyên Pham, and Hao Xing. "BSDEs with diffusion constraint and viscous Hamilton–Jacobi equations with unbounded data." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 53, no. 4 (November 2017): 1528–47. http://dx.doi.org/10.1214/16-aihp762.

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2

Nadolski, Adam. "Hamilton–Jacobi functional differential equations with unbounded delay." Annales Polonici Mathematici 82, no. 2 (2003): 105–26. http://dx.doi.org/10.4064/ap82-2-2.

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3

Zălinescu, Adrian. "Second order Hamilton–Jacobi–Bellman equations with an unbounded operator." Nonlinear Analysis: Theory, Methods & Applications 75, no. 13 (September 2012): 4784–97. http://dx.doi.org/10.1016/j.na.2012.03.028.

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4

Tataru, Daniel. "Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms." Journal of Mathematical Analysis and Applications 163, no. 2 (January 1992): 345–92. http://dx.doi.org/10.1016/0022-247x(92)90256-d.

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5

Barles, Guy, and Joao Meireles. "On unbounded solutions of ergodic problems in ℝmfor viscous Hamilton–Jacobi equations." Communications in Partial Differential Equations 41, no. 12 (November 14, 2016): 1985–2003. http://dx.doi.org/10.1080/03605302.2016.1244208.

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6

Armstrong, Scott N., and Panagiotis E. Souganidis. "Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments." Journal de Mathématiques Pures et Appliquées 97, no. 5 (May 2012): 460–504. http://dx.doi.org/10.1016/j.matpur.2011.09.009.

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7

Brändle, Cristina, and Emmanuel Chasseigne. "On unbounded solutions of ergodic problems for non-local Hamilton–Jacobi equations." Nonlinear Analysis 180 (March 2019): 94–128. http://dx.doi.org/10.1016/j.na.2018.09.015.

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8

Shimano, K. "A Class of HamiltonJacobi Equations with Unbounded Coefficients in Hilbert Spaces." Applied Mathematics and Optimization 45, no. 1 (January 1, 2002): 75–98. http://dx.doi.org/10.1007/s00245-001-0028-4.

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9

Qiu, Hong, and Jiongmin Yong. "Hamilton–Jacobi equations and two-person zero-sum differential games with unbounded controls." ESAIM: Control, Optimisation and Calculus of Variations 19, no. 2 (January 23, 2013): 404–37. http://dx.doi.org/10.1051/cocv/2012015.

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10

Stokols, Logan F., and Alexis F. Vasseur. "De Giorgi techniques applied to Hamilton–Jacobi equations with unbounded right-hand side." Communications in Mathematical Sciences 16, no. 6 (2018): 1465–87. http://dx.doi.org/10.4310/cms.2018.v16.n6.a1.

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11

Barles, Guy, and Emmanuel Chasseigne. "On the regularizing effect for unbounded solutions of first-order Hamilton–Jacobi equations." Journal of Differential Equations 260, no. 9 (May 2016): 7020–31. http://dx.doi.org/10.1016/j.jde.2016.01.021.

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12

Tataru, D. "Viscosity Solutions for Hamilton-Jacobi Equations with Unbounded Nonlinear Term: A Simplified Approach." Journal of Differential Equations 111, no. 1 (July 1994): 123–46. http://dx.doi.org/10.1006/jdeq.1994.1078.

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13

Engler, Hans. "On Hamilton–Jacobi equations in bounded domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 102, no. 3-4 (1986): 221–42. http://dx.doi.org/10.1017/s0308210500026317.

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SynopsisInitial-boundary value problems for nonlinear first order partial differential equations ∂tu + H(x, t, u, Dxu) = 0 and corresponding boundary value problems H(x, u, Dxu) = 0 are studied in bounded sets, using Crandal's and Lions' notion of viscosity solutions. We give pointwise conditions on the boundary data that guarantee the existence of such solutions and estimate their moduli of continuity in terms of continuity properties of the data. The results are applied to properties of the value function for certain differential games.

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14

Barles, Guy, and Jean-Michel Roquejoffre. "Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations." Communications in Partial Differential Equations 31, no. 8 (August 2006): 1209–25. http://dx.doi.org/10.1080/03605300500361461.

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15

Crandall, Michael G., and pierre-Louis Lions. "Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations." Illinois Journal of Mathematics 31, no. 4 (December 1987): 665–88. http://dx.doi.org/10.1215/ijm/1256063577.

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16

Barles, Guy, Olivier Ley, Thi-Tuyen Nguyen, and Thanh Viet Phan. "Large time behavior of unbounded solutions of first-order Hamilton–Jacobi equations in R N." Asymptotic Analysis 112, no. 1-2 (March 6, 2019): 1–22. http://dx.doi.org/10.3233/asy-181488.

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17

Crandall, Michael G., and P. L. Lions. "Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms." Journal of Functional Analysis 90, no. 2 (May 1990): 237–83. http://dx.doi.org/10.1016/0022-1236(90)90084-x.

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18

Bardi, Martino, and Yoshikazu Giga. "Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations." Communications on Pure & Applied Analysis 2, no. 4 (2003): 447–59. http://dx.doi.org/10.3934/cpaa.2003.2.447.

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19

Colombo, Rinaldo M., and Vincent Perrollaz. "Initial data identification in conservation laws and Hamilton–Jacobi equations." Journal de Mathématiques Pures et Appliquées 138 (June 2020): 1–27. http://dx.doi.org/10.1016/j.matpur.2020.03.005.

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20

Da Lio, Francesca, and Olivier Ley. "Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data." Applied Mathematics & Optimization 63, no. 3 (November 18, 2010): 309–39. http://dx.doi.org/10.1007/s00245-010-9122-9.

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21

JOLY, JEAN-LUC, GUY MÉTIVIER, and JEFFREY RAUCH. "HYPERBOLIC DOMAINS OF DETERMINACY AND HAMILTON–JACOBI EQUATIONS." Journal of Hyperbolic Differential Equations 02, no. 03 (September 2005): 713–44. http://dx.doi.org/10.1142/s0219891605000609.

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If L(t, x, ∂t, ∂x) is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets Ω0⊂ {t = 0}. The frozen constant coefficient operators [Formula: see text] determine local convex propagation cones, [Formula: see text]. Influence curves are curves whose tangent always lies in these cones. We prove that the set of points Ω which cannot be reached by influence curves beginning in the exterior of Ω0is a domain of determinacy in the sense that solutions of Lu = 0 whose Cauchy data vanish in Ω0must vanish in Ω. We prove that Ω is swept out by continuous spacelike deformations of Ω0and is also the set described by maximal solutions of a natural Hamilton–Jacobi equation (HJE). The HJE provides a method for computing approximate domains and is also the bridge from the raylike description using influence curves to that depending on spacelike deformations. The deformations are obtained from level surfaces of mollified solutions of HJEs.

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22

Díaz, G. "Blow-up time involved with perturbed Hamilton–Jacobi equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 1-2 (1992): 17–44. http://dx.doi.org/10.1017/s030821050002093x.

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SynopsisIn this paper we consider the evolution of positive bounded uniformly continuous data u0 by perturbed equations likeUnder general assumptions on u0, existence, uniqueness and regularity of the evolution u in the setare studied, where the blow-up function is given by .The exact blow-up rate of u is obtained. Uniqueness, regularity, decay at infinity of the function , as well a s a representation formula for the case m = 1, are also proved.

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23

Cardaliaguet, Pierre, and Luis Silvestre. "Hölder Continuity to Hamilton-Jacobi Equations with Superquadratic Growth in the Gradient and Unbounded Right-hand Side." Communications in Partial Differential Equations 37, no. 9 (March 29, 2012): 1668–88. http://dx.doi.org/10.1080/03605302.2012.660267.

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24

Crandall, Michael G., and P. L. Lions. "Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions." Journal of Functional Analysis 97, no. 2 (May 1991): 417–65. http://dx.doi.org/10.1016/0022-1236(91)90010-3.

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25

Bertsch, Michiel, Flavia Smarrazzo, Andrea Terracina, and Alberto Tesei. "Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations." Journal of Hyperbolic Differential Equations 18, no. 04 (December 2021): 857–98. http://dx.doi.org/10.1142/s0219891621500259.

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We study the Cauchy problem for the simplest first-order Hamilton–Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron’s method for Hamilton–Jacobi equations, Duke Math. J. 55 (1987) 368–384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron’s method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect.

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26

Li, Tian-Hong, Jinghua Wang, and Hairui Wen. "Regularity and global structure for Hamilton–Jacobi equations with convex Hamiltonian." Journal of Hyperbolic Differential Equations 18, no. 02 (June 2021): 435–51. http://dx.doi.org/10.1142/s0219891621500132.

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We consider the multidimensional Hamilton–Jacobi (HJ) equation [Formula: see text] with [Formula: see text] being a constant and for bounded [Formula: see text] initial data. When [Formula: see text], this is the typical case of interest with a uniformly convex Hamiltonian. When [Formula: see text], this is the famous Eikonal equation from geometric optics, the Hamiltonian being Lipschitz continuous with homogeneity [Formula: see text]. We intend to fill the gap in between these two cases. When [Formula: see text], the Hamiltonian [Formula: see text] is not uniformly convex and is only [Formula: see text] in any neighborhood of [Formula: see text], which causes new difficulties. In particular, points on characteristics emanating from points with vanishing gradient of the initial data could be “bad” points, so the singular set is more complicated than what is observed in the case [Formula: see text]. We establish here the regularity of solutions and the global structure of the singular set from a topological standpoint: the solution inherits the regularity of the initial data in the complement of the singular set and there is a one-to-one correspondence between the connected components of the singular set and the path-connected components of the set [Formula: see text].

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27

Ito, Yuji, Kenji Fujimoto, and Yukihiro Tadokoro. "Kernel-Based Hamilton–Jacobi Equations for Data-Driven Optimal and H-Infinity Control." IEEE Access 8 (2020): 131047–62. http://dx.doi.org/10.1109/access.2020.3009357.

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28

Claudel, Christian G., and Alexandre M. Bayen. "Convex Formulations of Data Assimilation Problems for a Class of Hamilton–Jacobi Equations." SIAM Journal on Control and Optimization 49, no. 2 (January 2011): 383–402. http://dx.doi.org/10.1137/090778754.

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29

Blanc, Alain-Philippe. "Comparison principle for the Cauchy problem for Hamilton–Jacobi equations with discontinuous data." Nonlinear Analysis: Theory, Methods & Applications 45, no. 8 (September 2001): 1015–37. http://dx.doi.org/10.1016/s0362-546x(99)00432-0.

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30

Mitake, Hiroyoshi. "Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data." Nonlinear Analysis: Theory, Methods & Applications 71, no. 11 (December 2009): 5392–405. http://dx.doi.org/10.1016/j.na.2009.04.028.

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31

Panov, E. "On the decay of viscosity solutions to Hamilton–Jacobi equations with almost periodic initial data." St. Petersburg Mathematical Journal 32, no. 4 (July 9, 2021): 767–79. http://dx.doi.org/10.1090/spmj/1669.

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The Cauchy problem is treated for a multidimensional Hamilton–Jacobi equation with a merely continuous nonstrictly convex Hamiltonian and a Bohr almost periodic initial function. Under the condition that the Hamiltonian is not degenerate in resonant directions (laying in the additive group generated by the spectrum of the initial function), the uniform decay of the viscosity solution to the constant equal to the infimum of the initial function is established.

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32

ZHAO, YINCHUAN, TAO TANG, and JINGHUA WANG. "REGULARITY AND GLOBAL STRUCTURE OF SOLUTIONS TO HAMILTON–JACOBI EQUATIONS II: CONVEX INITIAL DATA." Journal of Hyperbolic Differential Equations 06, no. 04 (December 2009): 709–23. http://dx.doi.org/10.1142/s0219891609001976.

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The paper is concerned with the Hamilton–Jacobi (HJ) equations of multidimensional space variables with convex initial data and general Hamiltonians. Using Hopf's formula (II), we will study the differentiability of the HJ solutions. For any given point, we give a sufficient and necessary condition such that the solutions are Ck smooth in some neighborhood of this point. We also study the characteristics of the equations which play important roles in our analysis. It is shown that there are only two kinds of characteristics, one never touches the singularity point, but the other one touches the singularity point in a finite time. Based on these results, we study the global structure of the set of singularity points for the solutions. It is shown that there exists a one-to-one correspondence between the path connected components of the set of singularity points and path connected component of the set {(Dg(y),H(Dg(y)))| y ∈ ℝn}\ {(Dg(y), conv H(Dg(y)))| y ∈ ℝn}, where conv, H is the convex hull of H. A path connected component of the set of singularity points never terminates as t increases. Moreover, our results depend only on H and its domain of definition.

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33

Canepa, Edward S., and Christian G. Claudel. "Networked traffic state estimation involving mixed fixed-mobile sensor data using Hamilton-Jacobi equations." Transportation Research Part B: Methodological 104 (October 2017): 686–709. http://dx.doi.org/10.1016/j.trb.2017.05.016.

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34

Bokanowski, Olivier, Nadia Megdich, and Hasnaa Zidani. "Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data." Numerische Mathematik 115, no. 1 (October 17, 2009): 1–44. http://dx.doi.org/10.1007/s00211-009-0271-1.

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35

Tu, Son N. T. "Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension." Asymptotic Analysis 121, no. 2 (January 7, 2021): 171–94. http://dx.doi.org/10.3233/asy-201599.

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Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1.

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36

Liu, Youqiong, Li Cai, Yaping Chen, and Bin Wang. "Physics-informed neural networks based on adaptive weighted loss functions for Hamilton-Jacobi equations." Mathematical Biosciences and Engineering 19, no. 12 (2022): 12866–96. http://dx.doi.org/10.3934/mbe.2022601.

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<abstract><p>Physics-informed neural networks (PINN) have lately become a research hotspot in the interdisciplinary field of machine learning and computational mathematics thanks to the flexibility in tackling forward and inverse problems. In this work, we explore the generality of the PINN training algorithm for solving Hamilton-Jacobi equations, and propose physics-informed neural networks based on adaptive weighted loss functions (AW-PINN) that is trained to solve unsupervised learning tasks with fewer training data while physical information constraints are imposed during the training process. To balance the contributions from different constrains automatically, the AW-PINN training algorithm adaptively update the weight coefficients of different loss terms by using the logarithmic mean to avoid additional hyperparameter. Moreover, the proposed AW-PINN algorithm imposes the periodicity requirement on the boundary condition and its gradient. The fully connected feedforward neural networks are considered and the optimizing procedure is taken as the Adam optimizer for some steps followed by the L-BFGS-B optimizer. The series of numerical experiments illustrate that the proposed algorithm effectively achieves noticeable improvements in predictive accuracy and the convergence rate of the total training error, and can approximate the solution even when the Hamiltonian is nonconvex. A comparison between the proposed algorithm and the original PINN algorithm for Hamilton-Jacobi equations indicates that the proposed AW-PINN algorithm can train the solutions more accurately with fewer iterations.</p></abstract>

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37

NOCHETTO, RICARDO H., and GIUSEPPE SAVARÉ. "NONLINEAR EVOLUTION GOVERNED BY ACCRETIVE OPERATORS IN BANACH SPACES: ERROR CONTROL AND APPLICATIONS." Mathematical Models and Methods in Applied Sciences 16, no. 03 (March 2006): 439–77. http://dx.doi.org/10.1142/s0218202506001224.

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Nonlinear evolution equations governed by m-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order [Formula: see text]. Applications to scalar conservation laws and degenerate parabolic equations (with or without hysteresis) in L1, as well as to Hamilton–Jacobi equations in C0are given. The error analysis relies on a comparison principle, for the novel notion of relaxed solutions, which combines and simplifies techniques of Benilan and Kružkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall–Liggett error estimate.

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38

Galaktionov, V. A., and A. E. Shishkov. "Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 5 (October 2003): 1075–119. http://dx.doi.org/10.1017/s0308210500002821.

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We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as t → T−. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(T−t)−γ → ∞ as t → T−, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.

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39

Fomin, Igor, and Sergey Chervon. "Exact and Slow-Roll Solutions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints." Universe 6, no. 11 (October 29, 2020): 199. http://dx.doi.org/10.3390/universe6110199.

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We investigate the ability of the exponential power-law inflation to be a phenomenologically correct model of the early universe. We study General Relativity (GR) scalar cosmology equations in Ivanov–Salopek–Bond (or Hamilton–Jacobi like) representation where the Hubble parameter H is the function of a scalar field ϕ. Such approach admits calculation of the potential for given H(ϕ) and consequently reconstruction of f(R) gravity in parametric form. By this manner the Starobinsky potential and non-minimal Higgs potential (and consequently the corresponding f(R) gravity) were reconstructed using constraints on the model’s parameters. We also consider methods for generalising the obtained solutions to the case of chiral cosmological models and scalar-tensor gravity. Models based on the quadratic relationship between the Hubble parameter and the function of the non-minimal interaction of the scalar field and curvature are also considered. Comparison to observation (PLANCK 2018) data shows that all models under consideration give correct values for the scalar spectral index and tensor-to-scalar ratio under a wide range of exponential-power-law model’s parameters.

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40

Torres-Ulloa, C., S. Berres, and P. Grassia. "Foam–liquid front motion in Eulerian coordinates." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2220 (December 2018): 20180290. http://dx.doi.org/10.1098/rspa.2018.0290.

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A mathematical model formulated as a system of Hamilton–Jacobi equations describes implicitly the propagation of a foam–liquid front in an oil reservoir, as the zero-level set of the solution variable. The conceptual model is based on the ‘pressure-driven growth’ model in Lagrangian coordinates. The Eulerian mathematical model is solved numerically, where the marching is done via a finite volume scheme with an upwind flux. Periodic reinitialization ensures a more accurate implicit representation of the front. The numerical level set contour values are initially formed to coincide with an early time asymptotic analytical solution of the pressure-driven growth model. Via the simulation of the Eulerian numerical model, numerical data are obtained from which graphical representations are generated for the location of the propagating front, the angle that the front normal makes with respect to the horizontal and the front curvature, all of which are compared with the Lagrangian model predictions. By making this comparison, it is possible to confirm the existence of a concavity in the front shape at small times, which physically corresponds to an abrupt reorientation of the front over a limited length scale.

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41

Hao, Yun, Yiyi Zhu, and Jong-Ping Hsu. "Experiments on Frequency Dependence of the Deflection of Light in Yang-Mills Gravity." EPJ Web of Conferences 168 (2018): 02004. http://dx.doi.org/10.1051/epjconf/201816802004.

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In Yang-Mills gravity based on flat space-time, the eikonal equation for a light ray is derived from the modified Maxwell’s wave equations in the geometric-optics limit. One obtains a Hamilton-Jacobi type equation, GLµv∂µΨ∂vΨ = 0 with an effective Riemannian metric tensor GLµv. According to Yang-Mills gravity, light rays (and macroscopic objects) move as if they were in an effective curved space-time with a metric tensor. The deflection angle of a light ray by the sun is about 1.53″ for experiments with optical frequencies ≈ 1014Hz. It is roughly 12% smaller than the usual value 1.75″. However, the experimental data in the past 100 years for the deflection of light by the sun in optical frequencies have uncertainties of (10-20)% due to large systematic errors. If one does not take the geometric-optics limit, one has the equation, GLµv[∂µΨ∂vΨcosΨ+ (∂µ∂vΨ)sinΨ] = 0, which suggests that the deflection angle could be frequency-dependent, according to Yang-Mills gravity. Nowadays, one has very accurate data in the radio frequencies ≈ 109Hz with uncertainties less than 0.1%. Thus, one can test this suggestion by using frequencies ≈ 1012 Hz, which could have a small uncertainty 0.1% due to the absence of systematic errors in the very long baseline interferometry.

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42

Blanes, Sergio, Fernando Casas, Cesáreo González, and Mechthild Thalhammer. "Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations." IMA Journal of Numerical Analysis, March 20, 2020. http://dx.doi.org/10.1093/imanum/drz058.

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Abstract This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge–Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker–Campbell–Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.

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43

Goffi, Alessandro. "Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations." Journal of Evolution Equations, June 8, 2021. http://dx.doi.org/10.1007/s00028-021-00720-3.

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AbstractWe investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $$s\in (1/2,1)$$ s ∈ ( 1 / 2 , 1 ) . As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

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44

Cirant, Marco, and Alessandro Goffi. "Maximal $$L^q$$-Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games." Annals of PDE 7, no. 2 (August 22, 2021). http://dx.doi.org/10.1007/s40818-021-00109-y.

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AbstractIn this paper we investigate maximal $$L^q$$ L q -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.

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Barles, Guy, and Alessio Porretta. "Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations." ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, December 2, 2009, 107–36. http://dx.doi.org/10.2422/2036-2145.2006.1.07.

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Kraaij, Richard C., and Mikola C. Schlottke. "Comparison Principle for Hamilton-Jacobi-Bellman Equations via a Bootstrapping Procedure." Nonlinear Differential Equations and Applications NoDEA 28, no. 2 (March 2021). http://dx.doi.org/10.1007/s00030-021-00680-0.

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AbstractWe study the well-posedness of Hamilton–Jacobi–Bellman equations on subsets of $${\mathbb {R}}^d$$ R d in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker–Varadhan theory of large deviations for occupation-time measures. To allow for this flexibility, we assume that the internal Hamiltonian and cost functional have controlled growth, and that they satisfy an equi-continuity estimate uniformly over compact sets in the space of controls. In addition to establishing the comparison principle for the Hamilton–Jacobi–Bellman equation, we also prove existence, the viscosity solution being the value function with exponentially discounted running costs. As an application, we verify the conditions on the internal Hamiltonian and cost functional in two examples.

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Tien, Phan Trong, and Tran Van Bang. "Hamilton–Jacobi equations for optimal control on junctions with unbounded running cost functions." Applicable Analysis, August 28, 2019, 1–17. http://dx.doi.org/10.1080/00036811.2019.1643012.

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Barles, Guy, Alexander Quaas, and Andrei Rodríguez-Paredes. "Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in R N." Communications in Partial Differential Equations, November 19, 2020, 1–26. http://dx.doi.org/10.1080/03605302.2020.1846561.

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Basco, V., and H. Frankowska. "Hamilton–Jacobi–Bellman Equations with Time-Measurable Data and Infinite Horizon." Nonlinear Differential Equations and Applications NoDEA 26, no. 1 (February 2019). http://dx.doi.org/10.1007/s00030-019-0553-y.

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Soravia, Pierpaolo. "Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness." Advances in Differential Equations 4, no. 2 (January 1, 1999). http://dx.doi.org/10.57262/ade/1366291416.

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