204 results
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2. Optimal temporal decay rates for the compressible viscoelastic flows.
- Author
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Fu, Shengbin, Huang, Wenting, and Wang, Weiwei
- Subjects
CAUCHY problem ,COMPRESSIBLE flow - Abstract
For the Cauchy problem of the three-dimensional compressible viscoelastic flows, we establish the optimal temporal decay rates of the all-order spatial derivatives of the global strong solution in the weaker initial condition. The main novelty of this paper is that the optimal decay estimates of the highest-order derivatives of the solution is obtained by using spectral analysis and energy method, which can be considered as the further investigation to [X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal. 45 (2013) 2815–2833] with only the lower-order derivative estimates. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. Bifurcation Analysis of a Size-Structured Population Model.
- Author
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Cai, Yuting, Wang, Chuncheng, and Fan, Dejun
- Subjects
HOPF bifurcations ,CAUCHY problem ,COMPUTER simulation - Abstract
In this paper, a size-structured population model is studied. Choosing the coefficient of birth function as the varying parameter, we prove the existence of Hopf bifurcation and discuss the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The methods used in this paper include Hopf bifurcation theorem, center manifold theorem and normal form theory for the abstract Cauchy problems in a nondense domain. Numerical simulations are finally carried out to show the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. The regularity properties and blow-up of the solutions for nonlocal Schrödinger equations.
- Author
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Shakhmurov, Veli B. and Shahmurov, Rishad
- Subjects
- *
SCHRODINGER equation , *BLOWING up (Algebraic geometry) , *DIFFERENTIAL operators , *NONLINEAR Schrodinger equation , *NONLINEAR equations , *CAUCHY problem , *OPERATOR equations - Abstract
In this paper, the Cauchy problem for convolution-type nonlocal linear and nonlinear Schrödinger equations (NSEs) is studied. The equations include the general differential operators. The existence, uniqueness, Lp-regularity properties of linear problem and the existence, uniqueness, and blow-up at finite time of the nonlinear problem are obtained. By choosing differential operators including in equations, the regularity properties of a different type of nonlocal Schrödinger equation are studied. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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5. On a family of nonzero solutions to the heat equation ut=△u on ℝn+1 which vanish on an arbitrary n-dimensional hyperplane P⊂ℝn+1.
- Author
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Tsai, Dong-Ho and Nien, Chia-Hsing
- Subjects
- *
HEAT equation , *POWER series , *CAUCHY problem , *HYPERPLANES - Abstract
Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon.65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the 1 -dimensional heat equation u t = u x x on ℝ 2 with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the n -dimensional heat equation u t = △ u on ℝ n + 1 with zero initial data (i.e. vanishing on the hyperplane P : { t = 0 }) can be easily obtained as a consequence of the examples on ℝ 2. Finally, given an arbitrary hyperplane P ⊂ ℝ n + 1 , we can construct a family of nonzero solutions which vanish on P ⊂ ℝ n + 1. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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6. Stability and Bifurcation Analysis of a Spatially Size–Stage-Structured Model with Resting Phase.
- Author
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Li, Yajing and Liu, Zhihua
- Subjects
- *
BIFURCATION theory , *CAUCHY problem , *COMPUTER simulation , *HOPF bifurcations , *POPULATION dynamics - Abstract
In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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7. On the solutions for the Novikov equation.
- Author
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Coclite, Giuseppe Maria and di Ruvo, Lorenzo
- Subjects
- *
WATER waves , *WATER depth , *EQUATIONS - Abstract
The Novikov equation is a generalization of the Camassa–Holm one. It models the propagation of unidirectional shallow water waves on a flat bottom. In this paper, we prove the well-posedness of H 2 and H 3 solutions of the Cauchy problem, associated with this equation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
8. Existence and uniqueness of the mild solution of an abstract nonlocal semilinear fractional integrodifferential equation.
- Author
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Herzallah, Mohamed A. E. and Radwan, Ashraf H. A.
- Subjects
INTEGRO-differential equations ,SEMILINEAR elliptic equations ,CAUCHY problem - Abstract
The purpose of this paper is to study the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract nonlocal fractional integrodifferential equation, which has a distinctive nonlinear term. Two sufficient conditions on the existence of mild solutions will be displayed. Continuous dependence of solutions on initial conditions and local -approximate mild solutions will be discussed. An example will be given to elucidate the main results. The results obtained are based upon the method of semigroups, contraction mapping principle, and Krasnoselskii's fixed point theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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9. ON FRACTIONAL INTEGRALS AND DERIVATIVES OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION.
- Author
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NIETO, JUAN J., ALGHANMI, MADEAHA, AHMAD, BASHIR, ALSAEDI, AHMED, and ALHARBI, BOSHRA
- Subjects
FRACTIONAL calculus ,CAPUTO fractional derivatives ,INTEGRAL functions ,GENERALIZED integrals ,FRACTIONAL differential equations ,CAUCHY problem - Abstract
In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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10. On the global existence and time-decay rates for a parabolic–hyperbolic model arising from chemotaxis.
- Author
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Xu, Fuyi and Li, Xinliang
- Subjects
CAUCHY problem ,CHEMOTAXIS - Abstract
In this paper, we are concerned with the study of the Cauchy problem for a parabolic–hyperbolic model arising from chemotaxis in any dimension d ≥ 2. We first prove the global existence of the model in L p critical regularity framework with respect to the scaling of the associated equations. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we also establish the time-decay rates for the constructed global solutions. Our analyses mainly rely on Fourier frequency localization technology and on a refined time-weighted energy inequalities in different frequency regimes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
11. Rate of homogenization for fully-coupled McKean–Vlasov SDEs.
- Author
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Bezemek, Zachary William and Spiliopoulos, Konstantinos
- Subjects
STOCHASTIC differential equations ,CAUCHY problem ,STOCHASTIC systems ,ASYMPTOTIC homogenization - Abstract
In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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12. Long-time asymptotics for the complex nonlinear transverse oscillation equation.
- Author
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Geng, Xianguo, Liu, Wenhao, Wang, Kedong, and Chen, Mingming
- Subjects
NONLINEAR oscillations ,RIEMANN-Hilbert problems ,LAX pair ,NONLINEAR equations ,EQUATIONS ,CAUCHY problem ,SEPARATION of variables - Abstract
In this paper, we study the long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation. Based on the corresponding Lax pair, the original Riemann–Hilbert problem is constructed by introducing some spectral function transformations and variable transformations, and the solution of the complex nonlinear transverse oscillation equation is transformed into the solution of the resulted Riemann–Hilbert problem. Various Deift–Zhou contour deformations and the motivation behind them are given, from which the original Riemann–Hilbert problem is further transformed into a solvable model problem. The long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation is obtained by using the nonlinear steepest decent method. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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13. Well-posedness of the Cauchy problem for the kinetic DNLS on T.
- Author
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Kishimoto, Nobu and Tsutsumi, Yoshio
- Subjects
NONLINEAR Schrodinger equation ,CAUCHY problem ,HILBERT transform ,SOBOLEV spaces ,CUBIC equations - Abstract
We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus T = R / 2 π Z : ∂ t u − i ∂ x 2 u = α ∂ x (| u | 2 u) + β ∂ x [ H (| u | 2) u ] for (t , x) ∈ [ 0 , T ] × T , where the constants α , β are such that α ∈ R and β < 0 , and H denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces H s for s > 3 / 2. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when β = 0 , cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in H s (T) , s > 1 / 2. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term β ∂ x [ H (| u | 2) u ] , in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
14. Global existence results for semi-linear structurally damped wave equations with nonlinear convection.
- Author
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Dao, Tuan Anh and Takeda, Hiroshi
- Subjects
NONLINEAR wave equations ,CAUCHY problem ,WAVE equation ,NONLINEAR equations ,TRANSPORT equation - Abstract
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term ν (− Δ) 2 u t , where ν > 0 is a constant. As is now well known, the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain the global solutions with the sharp decay properties in time. Our main purpose in this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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15. Hopf Bifurcation of an Age-Structured Epidemic Model with Quarantine and Temporary Immunity Effects.
- Author
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Liu, Lili, Zhang, Jian, Zhang, Ran, and Sun, Hongquan
- Subjects
BASIC reproduction number ,HOPF bifurcations ,QUARANTINE ,CAUCHY problem ,IMMUNITY ,EPIDEMICS ,REPRODUCTION - Abstract
In this paper, we investigate an epidemic model with quarantine and recovery-age effects. Reformulating the model as an abstract nondensely defined Cauchy problem, we discuss the existence and uniqueness of solutions to the model and study the stability of the steady state based on the basic reproduction number. After analyzing the distribution of roots to a fourth degree exponential polynomial characteristic equation, we also derive the conditions of Hopf bifurcation. Numerical simulations are performed to illustrate the results. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
16. The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws.
- Author
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Francesco, Marco Di and Stivaletta, Graziano
- Subjects
CONSERVATION laws (Physics) ,CONSERVATION laws (Mathematics) ,CAUCHY problem ,ENTROPY - Abstract
We consider the follow-the-leader particle approximation scheme for a 1 d scalar conservation law with non-negative compactly supported L ∞ initial datum and with a C 1 concave flux, which is known to provide convergence towards the entropy solution ρ to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density ρ n is a "discrete version of an entropy condition"; more precisely, under fairly general assumptions on f (which imply concavity of f) we prove that the continuum version (f (ρ) / ρ) x ≤ 1 / t of said condition allows to select a unique weak solution, despite (f (ρ) / ρ) x ≤ 1 / t is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition f ′ (ρ) x ≤ 1 / t. Said result relies on an improved version of Hoff's uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case f (ρ) = ρ (A − ρ γ) the one-sided Lipschitz condition can be improved to a discrete version of the classical (and "sharp") Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases "alternative" ones) of all steps of the convergence of the particle scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
17. The Lp-regularity properties of nonlocal wave equations and applications.
- Author
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Shakhmurov, Veli B.
- Subjects
NONLINEAR wave equations ,FRACTIONAL powers ,BANACH spaces ,NONLINEAR equations ,CAUCHY problem ,WAVE equation ,DIFFERENTIAL equations - Abstract
In this paper, the Cauchy problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in a Banach space E and convolution terms. Here, assuming enough smoothness on the initial data and on coefficients, the existence, uniqueness and regularity properties of local and global solutions are established in terms of fractional powers of a given sectorial operator A. We obtain the regularity properties of a wide class of wave equations by choosing the space E and the operator A which appear in the field of physics. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
18. The Cauchy problem for properly hyperbolic equations in one space variable.
- Author
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Spagnolo, Sergio and Taglialatela, Giovanni
- Subjects
CAUCHY problem ,EQUATIONS ,HYPERBOLIC differential equations - Abstract
In this paper, we consider the Cauchy problem for higher-order weakly hyperbolic equations assuming that the principal symbol depends only on one space variable and the characteristic roots τ j verify an inequality like τ j 2 (x) + τ k 2 (x) ≤ M (τ j (x) − τ k (x)) 2. We prove that the Cauchy problem is well-posed in ∞ if the operators with frozen coefficients are uniformly hyperbolic in the sense of Gårding. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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19. Data dependence of approximate current-vortex sheets near the onset of instability.
- Author
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Morando, Alessandro, Secchi, Paolo, and Trebeschi, Paola
- Subjects
MAGNETOHYDRODYNAMICS ,TRANSITION temperature ,INTEGRO-differential equations ,CAUCHY problem ,HADAMARD matrices - Abstract
The paper is concerned with the free boundary problem for two-dimensional current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo [On the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equations 8(4) (2011) 691-726] have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation was shown in [Approximate current-vortex sheets near the onset of instability, J. Math. Pures Appl. 105(4) (2016) 490-536; Existence of approximate current-vortex sheets near the onset of instability, J. Hyperbolic Differ. Equations]. In the present paper, we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
20. REGULARIZATION OF CAUCHY PROBLEM FOR 2D TIME-FRACTIONAL DIFFUSION EVOLUTION EQUATIONS.
- Author
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BINH, TRAN THANH, BINH, NGUYEN PHUC, THANG, BUI DINH, and LONG, LE DINH
- Subjects
HEAT equation ,CAUCHY problem ,INITIAL value problems ,SEPARATION of variables ,NONLINEAR equations ,EVOLUTION equations - Abstract
In this work, we focus on an initial value problem for a class of 2D time-fractional diffusion evolution equations with Riemann–Liouville fractional derivative. There are three new results in this paper. First of all, the existence and ill-posedness result (in the sense of Hadamard) in three cases which are consisting of homogeneous, inhomogeneous, and nonlinear problems are considered. Next, by using the Fourier truncation method, we show that regularized problems are well-posed. Finally, some demonstrated examples are presented to test the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
21. QUALITATIVE ANALYSIS OF SEIRS ENDEMIC MODEL BOTH FROM PDEs AND ODEs PERSPECTIVE.
- Author
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ZHANG, LEI, SAEED, TAREQ, WANG, MIAO-KUN, AAMIR, NUDRAT, and IBRAHIM, MUHAMMAD
- Subjects
ORDINARY differential equations ,PARTIAL differential equations ,CAUCHY problem ,FINITE difference method - Abstract
An age-structured susceptible–exposed–infected–recovered–susceptible (SEIRS) endemic model is proposed in this analysis utilizing the tools of partial differential equations. Because of different outflows and inflows that are lopsided by migration and demographics factors, the population is supposed to be not constant. To demonstrate that the model is well-posed, an abstract Cauchy problem is developed from the proposed system. The simple reproduction number R 0 is used to analyze the local and global behavior of the disease-free equilibrium. The disease present equilibrium point is shown to exist and be stable locally under appropriate assumptions and conditions. We consider the age-free parameters and the problem is converted into an ordinary differential equations (ODEs) model. The ODEs model is investigated for disease-free and endemic equilibria and the global stability of each equilibrium is presented therein. A few simulations are carried out and discussed at the end of the paper to explain the central theorem of the study. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
22. Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum.
- Author
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Lai, Suhua, Xu, Hao, and Zhang, Jianwen
- Subjects
NAVIER-Stokes equations ,FLUIDS ,CAUCHY problem ,HEAT equation - Abstract
This paper is concerned with the Cauchy problem of Navier–Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in ℝ 3 . For less regular data and weaker compatibility condition than those proposed by Cho–Kim [Existence results for viscous polytropic fluids with vacuum, J. Differ. Equ. 228 (2006) 377–411], we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is more regular. We also establish the global well-posedness of classical solutions with large oscillations and vacuum in the case when the initial total energy is suitably small. The exponential decay estimates of the global solutions are obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation.
- Author
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He, Shangqin and Feng, Xiufang
- Subjects
CAUCHY problem ,HELMHOLTZ equation ,MATHEMATICAL regularization ,PROBLEM solving - Abstract
In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
24. Conformal scattering and the Goursat problem for Dirac fields in the interior of charged spherically symmetric black holes.
- Author
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Mokdad, Mokdad
- Subjects
BLACK holes ,DIRAC equation ,CAUCHY problem ,WAVE equation ,SCHWARZSCHILD black holes ,REISSNER-Nordstrom metric - Abstract
In this paper, we construct a conformal scattering theory for Dirac fields in the interior of a Reissner–Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
25. C-Semigroups, subordination principle and the Lévy α-stable distribution on discrete time.
- Author
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Alvarez, Edgardo, Díaz, Stiven, and Lizama, Carlos
- Subjects
DISTRIBUTION (Probability theory) ,DIFFERENCE operators ,CAUCHY problem ,EVOLUTION equations - Abstract
In this paper, we introduce the notion of Lévy α -stable distribution within the discrete setting. Using this notion, a subordination principle is proved, which relates a sequence of solution operators — given by a discrete C -semigroup — for the abstract Cauchy problem of first order in discrete-time, with a sequence of solution operators for the abstract Cauchy problem of fractional order 0 < α < 1 in discrete-time. As an application, we establish the explicit solution of the abstract Cauchy problem in discrete-time that involves the Hilfer fractional difference operator and prove that, in some cases, such solution converges to zero. Our findings give new insights on the theory, provide original concepts and extend as well as improve recent results of relevant references on the subject. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
26. Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations.
- Author
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Bertsch, Michiel, Smarrazzo, Flavia, Terracina, Andrea, and Tesei, Alberto
- Subjects
VISCOSITY solutions ,HAMILTON-Jacobi equations ,NEUMANN problem ,CAUCHY problem ,MATHEMATICS - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
27. On uniqueness of solutions to conservation laws verifying a single entropy condition.
- Author
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Krupa, Sam G. and Vasseur, Alexis F.
- Subjects
CONSERVATION laws (Mathematics) ,CONSERVATION laws (Physics) ,HAMILTON-Jacobi equations ,ENTROPY ,BURGERS' equation ,CAUCHY problem - Abstract
For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z.55(5) (1994) 116–129 (in Russian), Math. Notes55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math.62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
28. More on Bifurcations and Dynamics of Traveling Wave Solutions for a Higher-Order Shallow Water Wave Equation.
- Author
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Li, Jibin and Chen, Guanrong
- Subjects
DYNAMICAL systems ,BIFURCATION diagrams ,CAUCHY problem ,NONLINEAR wave equations ,ORDINARY differential equations - Abstract
This paper studies the dynamics of traveling wave solutions to a shallow water wave model with a large-amplitude regime in phase space. The corresponding traveling wave system is a singular planar dynamical system with two singular straight lines. By using the method of dynamical systems, bifurcation diagrams are obtained. The existence of solitary wave solutions, periodic wave solutions, peakon, pseudo-peakon solution, periodic peakon solutions and compacton solutions are determined under different parameter conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
29. Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations.
- Author
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Bahri, Yakine, Ibrahim, Slim, and Kikuchi, Hiroaki
- Subjects
PROBLEM solving ,EQUATIONS ,SCHRODINGER equation ,CAUCHY problem ,STANDING waves - Abstract
In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed v , traveling wave solutions exist and converge to the zero wave as the velocity tends to 1. Finally, we solve the Cauchy problem for initial data in L x 2 H y s (ℝ 2) , with s > 1 2 . The critical case s = 1 2 still stands as an interesting open problem. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
30. The -Cauchy problem on pseudoconvex domains and applications.
- Author
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Saber, Sayed
- Subjects
PSEUDOCONVEX domains ,NEUMANN problem ,LAPLACIAN operator ,KAHLERIAN manifolds ,CAUCHY problem - Abstract
Let be a complex manifold of dimension and be a weakly pseudoconvex domain with smooth boundary in . Let be a holomorphic line bundle over which is positive on a neighborhood of . Let be the -times tensor product of for positive integer . The purpose of this paper is to study the -problem with support conditions in for forms of type , with values in . Applications to the -problem for smooth forms on boundaries of are given. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
31. Orbital stability vs. scattering in the cubic-quintic Schrödinger equation.
- Author
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Carles, Rémi and Sparber, Christof
- Subjects
NONLINEAR Schrodinger equation ,CAUCHY problem ,SCHRODINGER equation ,SOLITONS ,CONFORMAL field theory - Abstract
We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is already known that all solitons are orbitally stable. In dimension two, we show that if the initial data belong to the conformal space, and have at most the mass of the ground state of the cubic two-dimensional Schrödinger equation, then the solution is asymptotically linear. For larger mass, solitary wave solutions exist, and we review several results on their stability. Finally, in dimension three, relying on previous results from other authors, we show that solitons may or may not be orbitally stable. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
32. Stability of traveling wave fronts for nonlocal diffusive systems.
- Author
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Meng, Yanling, Yu, Zhixian, and Zhang, Shengqiang
- Subjects
SOBOLEV spaces ,EXPONENTIAL stability ,CAUCHY problem - Abstract
This paper is concerned with stability of traveling wave fronts for nonlocal diffusive system. We adopt L 1 -weighted, L 1 - and L 2 -energy estimates for the perturbation systems, and show that all solutions of the Cauchy problem for the considered systems converge exponentially to traveling wave fronts provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
33. Global existence and blow-up solutions of the radial Schrödinger maps.
- Author
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Zhai, Jian and Zheng, Bo-Wen
- Subjects
INTEGRO-differential equations ,CAUCHY problem ,SOBOLEV spaces ,SCHRODINGER equation ,INTEGRALS - Abstract
This paper studies the Cauchy problem of radial inhomogeneous Schrödinger maps (ISM) which arises from the integrable model of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system. Through a complex transformation the radial ISM is equivalent to an integro-differential Schrödinger equation. A new weighted Sobolev space 𝒲 β 1 , l (ℝ +) is introduced and the well-posedness of integro-differential Schrödinger equations, including the integral radial IMS, with small spherically symmetric initial data in one-dimensional energy space 𝒲 1 1 , 2 (ℝ +) is established. Furthermore, for n ≤ 2 , we prove the existence of blow-up solutions for the integral radial ISM. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
34. The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings.
- Author
-
Blanc, Xavier, Danchin, Raphaël, Ducomet, Bernard, and Nečasová, Šárka
- Subjects
CAUCHY problem ,IDEAL gases ,VECTOR fields ,EULER equations ,HELMHOLTZ equation ,COUPLES - Abstract
We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field u 0 such that the spectrum of D u 0 is positive and bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends Grassin and Serre's papers [Existence de solutions globales et régulières aux équations d'Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d'Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
35. Modeling and numerical simulations of multilane vehicular traffic by active particles methods.
- Author
-
Zagour, M.
- Subjects
- *
COMPUTER simulation , *CAUCHY problem , *SIMULATION methods & models , *GAME theory , *MATHEMATICAL models - Abstract
This paper deals with the modeling and numerical simulations of multilane vehicular traffic according to the kinetic theory of active particles methods. The main idea of this theory is to consider each driver–vehicle system as a micro-system, where the microscopic state of particles is described by position, velocity, and activity which is an appropriate variable for modeling the quality of the driver-vehicle. The interactions between micro-systems are modeled by stochastic game theory. This leads to the derivation of a mathematical model within the framework of the approach of kinetic theory. The well-posedness of the related Cauchy problem for the spatially homogeneous case is established. Numerical simulations are carried out to show the ability of the proposed model to reproduce the empirical data such as the asymptotic property in time and the emerging behavior of clusters with a particular focus on the road environment conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. L2-blowup estimates of the wave equation and its application to local energy decay.
- Author
-
Ikehata, Ryo
- Subjects
RAYLEIGH waves ,BLOWING up (Algebraic geometry) ,WAVE equation ,CAUCHY problem - Abstract
We consider the Cauchy problems in R n for the wave equation with a weighted L 1 -initial data. We derive sharp infinite time blowup estimates of the L 2 -norm of solutions in the case of n = 1 and n = 2. Then, we apply it to the local energy decay estimates for n = 2 , which is not studied so completely when the 0 th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the L 2 -norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520]. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. Stability and Bifurcation Analysis in a Predator–Prey Model with Age Structure and Two Delays.
- Author
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Wang, Yujia, Fan, Dejun, and Wei, Junjie
- Subjects
HOPF bifurcations ,CAUCHY problem ,PLANE curves ,BIFURCATION theory ,AGE ,COMPUTER simulation - Abstract
In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
38. Bifurcation Analysis of a Predator–Prey Model with Age Structure.
- Author
-
Cai, Yuting, Wang, Chuncheng, and Fan, Dejun
- Subjects
HOPF bifurcations ,CAUCHY problem ,MANIFOLDS (Mathematics) ,COMPUTER simulation ,AGE - Abstract
In this paper, a predator–prey model with age structure in predator is studied. Using maturation period as the varying parameter, we prove the existence of Hopf bifurcation for the model and calculate the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The method we employed includes Hopf bifurcation theorem, center manifolds and normal form theory for the abstract Cauchy problems with nondense domain. Under a certain set of parameter values, it turns out that subcritical Hopf bifurcation may occur, indicating that the increment of maturation period could stabilize the steady state, which is initially unstable and enclosed by a stable periodic solution. In addition, stability switches will also take place. Numerical simulations are finally carried out to show the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
39. The Cauchy problem for a two-dimensional generalized Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces.
- Author
-
Yan, Wei, Li, Yongsheng, Huang, Jianhua, and Duan, Jinqiao
- Subjects
CAUCHY problem ,SOBOLEV spaces ,KADOMTSEV-Petviashvili equation ,EQUATIONS ,MATHEMATICS - Abstract
The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation u t + | D x | α ∂ x u + ∂ x − 1 ∂ y 2 u + 1 2 ∂ x (u 2) = 0 , α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces H s 1 , s 2 ( R 2) with s 1 > − α − 1 4 and s 2 ≥ 0. Second, we prove that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − (α − 1) (3 α − 4) 4 (5 α + 3) if 4 ≤ α ≤ 5. Finally, we show that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − α (3 α − 4) 4 (5 α + 4) if α > 5. Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl.79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl.90 (2008) 338–352]. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
40. Global Dynamics of a Lotka–Volterra Competition Diffusion System with Nonlocal Effects.
- Author
-
Han, Bang-Sheng, Yang, Yinghui, Bo, Wei-Jian, and Tang, Huiling
- Subjects
DIFFUSION ,CAUCHY problem - Abstract
This paper is concerned with the global dynamics of a Lotka–Volterra competition diffusion system having nonlocal intraspecies terms. Based on the reconstructed comparison principle and monotone iteration, the existence and uniqueness of the solution for the corresponding Cauchy problem are established. In addition, the spreading speed of the system with compactly supported initial data is considered, which admits uniform upper and lower bounds. Finally, some sufficient conditions for guaranteeing the existence and nonexistence of Turing bifurcation are given, which depend on the intensity of nonlocality. Comparing with the classical Lotka–Volterra competition diffusion system, our results indicate that a nonconstant periodic solution may exist if the nonlocality is strong enough, which are also illustrated numerically. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
41. Existence of strong solutions to the rotating shallow water equations with degenerate viscosities.
- Author
-
Duan, Ben, Luo, Zhen, and Zhou, Yan
- Subjects
SHALLOW-water equations ,WATER depth ,CAUCHY problem ,CORIOLIS force ,VISCOSITY - Abstract
In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
42. Development of singularities of the Skyrme model.
- Author
-
McNulty, Michael
- Subjects
SKYRME model ,STATISTICAL smoothing ,NONLINEAR wave equations ,CAUCHY problem ,BLOWING up (Algebraic geometry) - Abstract
The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper, we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what's known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the 5 + 1 -dimensional Skyrme Model in the strong-field limit, and use that to conclude that the solution to the corresponding Cauchy problem blows-up in finite time, starting from a particular class of everywhere smooth initial data. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
43. Stability of the composite wave for compressible Navier–Stokes/Allen–Cahn system.
- Author
-
Luo, Ting, Yin, Haiyan, and Zhu, Changjiang
- Subjects
CAUCHY problem ,EULER equations ,NONLINEAR analysis ,CLASSICAL solutions (Mathematics) - Abstract
This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier–Stokes/Allen–Cahn system which is a combination of the classical Navier–Stokes system with an Allen–Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of χ (x , t) ≡ 1 for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier–Stokes/Allen–Cahn system. The proof is mainly based on a basic energy method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
44. Periodic Solutions of a Delayed Eco-Epidemiological Model with Infection-Age Structure and Holling Type II Functional Response.
- Author
-
Yang, Peng and Wang, Yuanshi
- Subjects
HOPF bifurcations ,CAUCHY problem ,INFECTIOUS disease transmission ,INCUBATION period (Communicable diseases) ,COMPUTER simulation ,EQUILIBRIUM - Abstract
This paper is devoted to the study of a new delayed eco-epidemiological model with infection-age structure and Holling type II functional response. Firstly, the disease transmission rate function among the predator population is treated as the piecewise function concerning the incubation period τ 2 of the epidemic disease and the model is rewritten as an abstract nondensely defined Cauchy problem. Besides, the prerequisite which guarantees the presence of the coexistence equilibrium is achieved. Secondly, via utilizing the theory of integrated semigroup and the Hopf bifurcation theorem for semilinear equations with nondense domain, it is found that the model exhibits a Hopf bifurcation near the coexistence equilibrium, which suggests that this model has a nontrivial periodic solution that bifurcates from the coexistence equilibrium as the bifurcation parameter τ crosses the bifurcation critical value τ 0 . That is, there is a continuous periodic oscillation phenomenon. Finally, some numerical simulations are shown to support and extend the analytical results and visualize the interesting phenomenon. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
45. The Cauchy problem for an Oldroyd-B model in three dimensions.
- Author
-
Wang, Wenjun and Wen, Huanyao
- Subjects
CAUCHY problem ,SOBOLEV spaces ,NAVIER-Stokes equations ,DIMENSIONS ,MATHEMATICS ,EQUATIONS - Abstract
We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci.15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to η , in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing L t 2 L x 2 -norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
46. Lp time asymptotic decay for general hyperbolic–parabolic balance laws with applications.
- Author
-
Zeng, Yanni
- Subjects
JACOBIAN matrices ,CAUCHY problem ,GAS flow ,CONSERVATION laws (Physics) ,CHEMICAL reactions - Abstract
We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term for relaxation, damping or chemical reaction. The viscosity matrix and the Jacobian matrix of the lower-order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper, we obtain optimal L p decay rates for p ≥ 2. Our result is general and applies to models such as Keller–Segel equations with logarithmic chemotactic sensitivity and logistic growth, and gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic–parabolic conservation laws and hyperbolic balance laws, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
47. A local theory for a fractional reaction-diffusion equation.
- Author
-
Viana, Arlúcio
- Subjects
REACTION-diffusion equations ,CAUCHY problem ,FRACTIONAL differential equations ,PARTIAL differential equations ,HEAT equation ,CHARACTERISTIC functions - Abstract
In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like | u | ρ 2 − 1 u and ∂ χ B ∂ ν | u | ρ 1 − 1 u , and χ B is the characteristic function of a ball B. Here, we are interested in the solvability of the problem when singular initial data u 0 are taken in L p . Eventually, we give sufficient conditions to the nonexistence of positive global solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
48. On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in.
- Author
-
Li, Jinkai, Titi, Edriss S., and Xin, Zhouping
- Subjects
LIQUID crystals ,CAUCHY problem ,ESTIMATION theory ,COEFFICIENTS (Statistics) ,PERFORMANCE evaluation - Abstract
This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen-Leslie system of liquid crystal models in , with both general Leslie stress tensors and general Oseen-Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen-Leslie system. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
49. Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption.
- Author
-
Iagar, Razvan Gabriel, Laurençot, Philippe, and Sánchez, Ariel
- Subjects
BURGERS' equation ,CAUCHY problem ,LOTKA-Volterra equations ,ABSORPTION ,POROUS materials - Abstract
We study the dynamics of the following porous medium equation with strong absorption ∂ t u = Δ u m − | x | σ u q , posed for (t , x) ∈ (0 , ∞) × ℝ N , with m > 1 , q ∈ (0 , 1) and σ > 2 (1 − q) / (m − 1). Considering the Cauchy problem with non-negative initial condition u 0 ∈ L ∞ (ℝ N) , instantaneous shrinking and localization of supports for the solution u (t) at any t > 0 are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Stochastic Navier–Stokes equations perturbed by Lévy noise with hereditary viscosity.
- Author
-
Mohan, Manil T. and Sritharan, Sivaguru S.
- Subjects
NAVIER-Stokes equations ,VISCOSITY ,CAUCHY problem ,NON-Newtonian fluids ,NOISE - Abstract
In this paper, we study the stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise in three dimensions with a hereditary viscous term which depends on the past history. We establish the local solvability of the Cauchy problem for such systems. The local monotonicity property of the nonlinear term of the cutoff problem and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. Finally, we show that the global solvability results hold under smallness condition on the initial data and suitable assumptions on the noise coefficients. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
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