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In this paper, we investigate singularity formation for the high-order generalized Fokas–Olver–Rosenau–Qiao equation which include the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation), and we are devoted to understanding how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we show the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, we investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces B p , r s (R) with 1 ≤ p , r ≤ ∞ and s > m a x { 1 + 1 p , 3 2 } , which improve the previous work in Sobolev spaces H s (R) = B 2 , 2 s (R) with s > 3 2 (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces B p , r s (R) with 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and s > m a x { 1 + 1 p , 3 2 } . [ABSTRACT FROM AUTHOR]

This paper investigates a noteworthy phenomenon within the framework of Majda-Biello systems, wherein the inclusion of lower-order terms can enhance the well-posedness of the system. Specifically, we investigate the initial value problem (IVP) of the following system: u t + u xxx = - v v x , v t + α v xxx + β v x = - (u v) x , (u , v) | t = 0 = (u 0 , v 0) ∈ H s (R) × H s (R) , x ∈ R , t ∈ R , where α ∈ R \ { 0 } and β ∈ R . Let s ∗ (α , β) be the smallest value for which the IVP is locally analytically well-posed in H s (R) × H s (R) when s > s ∗ (α , β) . Two interesting facts have already been known in literature: s ∗ (α , 0) = 0 for α ∈ (0 , 4) \ { 1 } and s ∗ (4 , 0) = 3 4 . Our key findings include the following: For s ∗ (4 , β) , a significant reduction is observed, reaching 1 2 for β > 0 and 1 4 for β < 0 . Conversely, when α ≠ 4 , we demonstrate that the value of β exerts no influence on s ∗ (α , β) . These results shed light on the intriguing behavior of Majda-Biello systems when lower-order terms are introduced and provide valuable insights into the role of α and β in the well-posedness of the system. [ABSTRACT FROM AUTHOR]

This paper investigates a class of planar Schrödinger–Poisson systems with critical exponential growth. The conditions for the local well‐posedness of the Cauchy problem in the energy space are defined. By introducing innovative ideas and relaxing some of the classical growth assumptions on nonlinearity, this study shows that such a system has at least two standing waves with a prescribed mass. One wave is a ground‐state standing wave with positive energy, and another one is a high‐energy standing wave with positive energy. In addition, with the help of the local well‐posedness, it is shown that the set of ground‐state standing waves is orbitally stable. [ABSTRACT FROM AUTHOR]

In the present paper, a new model equation with the Coriolis effect is derived from the incompressible and irrotational two-dimensional shallow water in the equatorial region according to the formal asymptotic procedures. This new model equation is called the rotational b -family of equations, which generalize the rotational Camassa-Holm equation. Subsequently, the local well-posedness and the blow-up criteria for the rotational b -family of equations on the Sobolev space H s , for s > 3 / 2 , are presented. Additionally, we study the effects of the Coriolis force and nonlocal higher nonlinearities on wave-breaking phenomena for both the rotational b -family of equations and the rotational Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem for the fractional nonlinear Schrödinger (fNLS) equation. First, we establish the local well-posedness in the fractional Sobolev spaces Hγ with γ> 0 for the fNLS equation under less regularity assumptions for the nonlinear term than previous work. Based on the local well-posedness result, we then prove that the fNLS equation is globally wellposed in Hγ> with γ> > 0 if 4s/d < n > nc(g) and the initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the well‐posedness, blow‐up phenomena, and continuity of the data‐to‐solution map of the Cauchy problem for a two‐component higher order Camassa–Holm (CH) system. The local well‐posedness is established in Besov spaces Bp,11p×Bp,12+1p$B_{p,1}^{\frac{1}{p}} \times B_{p,1}^{2+\frac{1}{p}}$ with 1≤p<∞$1 \le p < \infty$, which improves the local well‐posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution‐to‐data map, that is, the ill‐posedness is derived in Besov space Bp,∞s−2×Bp,∞s$B_{p,\infty }^{s - 2} \times B_{p,\infty }^s$ with 1≤p≤∞$1 \le p \le \infty$ and s>max{2+1p,52}$s>\max \lbrace 2+\frac{1}{p},\frac{5}{2}\rbrace$. Then, the nonuniform continuous and Hölder continuous dependence on initial data for this system are also presented in Besov spaces Bp,rs−2×Bp,rs$B_{p,r}^{s - 2} \times B_{p,r}^s$ with 1≤p,r<∞$1 \le p,r < \infty$ and s>max{2+1p,52}$s > \max \lbrace {2+\frac{1}{p},\frac{5}{2}}\rbrace$. Finally, the precise blow‐up criteria for the strong solutions of the two‐component higher order CH system is determined in the lowest Sobolev space Hs−2×Hs$H^{s-2}\times H^s$ with s>52$s>\frac{5}{2}$, which improves the blow‐up criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509–1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a new weakly dissipative quasilinear shallow-water waves equation, which can be formally derived from a model with the effect of underlying shear flow from the incompressible rotational two-dimensional shallow water in the moderately nonlinear regime by Wang, Kang and Liu (Appl Math Lett 124:107607, 2022). Considering the dissipative effect, the local well-posedness of the solution to this equation is first obtained by using Kato's semigroup theory. We then establish the precise blow-up criterion by using the transport equation theory and Moser-type estimates. Moreover, some sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied according to the different real-valued intervals in which the dispersive parameter θ being located. It is noteworthy that we need to overcome the difficulty induced by complicated nonlocal nonlinear structure and different dispersive parameter ranges to get corresponding convolution estimates. [ABSTRACT FROM AUTHOR]

This paper aims to understand a blow-up mechanism on a family of shallow-water models with linear dispersion, which are linked with the modified Camassa-Holm equation and the Novikov equation. We first demonstrate the local well-posedness of the model equation in Besov spaces. Our blow-up analysis begins with two cases where the first case is 2 k 1 + 3 k 2 ≠ 0 and then we deduce the results on the curvature blow-up in finite time. To overcome the lack of conservation in the functional due to weak linear dispersion, we can determine a suitable alternative via a slight modification to conserved quantity H 2 [ u ] (see Lemma 4.1). Furthermore, we explore the formation of singularities in another case when nonlocal terms are absent. Lastly, we investigate the Gevrey regularity and analyticity of solutions for Cauchy problem within a specified range of Gevrey-Sobolev spaces by employing the generalized Ovsyannikov theorem and study the continuity of the data-to-solution mapping. [ABSTRACT FROM AUTHOR]

A shallow water wave equation including the famous Degasperis-Procesi model is considered. Firstly, the L2 conservation law for the equation is derived. Secondly, using the methods of transport equation, we establish the boundedness of solutions for the shallow water wave equation. Finally, utilizing the approaches to construct Lyapunov functions, we find local-in-space wave breaking criteria of its solutions on the line $ \mathbb {R} $ R and circle $ \mathbb {S} $ S under certain restrictions on the coefficients of the equation. [ABSTRACT FROM AUTHOR]

We study the initial-boundary vaule problem of the long and short wave equations posed on the half line with initial datas (u 0 , n 0) ∈ H s 0 (R +) × H s 1 (R +) and boundary datas (h , f) ∈ H 2 s 0 + 1 4 (R +) × H s 1 (R +) . We show the local well-posedness by giving the bilinear estimates of the coupling terms for the equations in suitable spaces of Bourgain type. Moreover, we consider results concerning ill-posedness for the system. Finally, the system is proved to be globally well-posed in Sobolev spaces H 1 (R +) × L 2 (R +) . [ABSTRACT FROM AUTHOR]

This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we considered a wave-structure interaction problem modeling surface water waves (governed by a Boussinesq system) interacting with a fixed partially immersed object and vertical lateral walls, which can be reduced to a transmission problem for a Boussinesq system. The local well-posedness of the strong solutions for the Cauchy problem in Sobolev space $ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ with $ s>\frac{3}{2} $ was obtained. Under some assumptions, the uniqueness and existence for the weak solutions in lower Sobolev space $ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ with $ 0

We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation$ iu_{t} +\Delta^{2} u = \lambda |x|^{-b}|u|^{\sigma}u,\;u(0) = u_{0} \in H^{s} (\mathbb R^{d}), $where $ \lambda\in \mathbb R $, $ d\in \mathbb N $, $ 0\le s<\min\left\{2+\frac{d}{2},d\right\} $, $ 00 $. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in $ H^{s} $ for the IBNLS equation in both of subcritical case $ \sigma<\sigma_{c}(s) $ and critical case $ \sigma = \sigma_{c}(s) $. We also prove that the IBNLS equation is globally well-posed in $ H^{s} $, if the initial data is sufficiently small and $ \frac{8-2b}{d}\le \sigma\le \sigma_{c}(s) $ with $ \sigma<\infty $. [ABSTRACT FROM AUTHOR]

In this paper, we mainly revisit a weakly dissipative periodic two‐component b$$ b $$‐family system. Considering the dissipative effect, the local well‐posedness is first obtained for the system by applying the Kato's semigroup theory. We then utilize the characteristics line method to get one blow‐up criterion with the dispersive parameter k1∈(1,3]$$ {k}_1\in \left(1,3\right] $$. Finally, the other blow‐up criterion is derived with regard as considering the odevity of initial data by using energy method. When the parameters λ,ki(i=1,2,3)$$ \lambda, {k}_i\left(i&#x0003D;1,2,3\right) $$ belong to suitable range, our obtained results supplement the corresponding blow‐up criteria, which was derived by Liu and Yin (Nonlinear Analysis: Real World Applications 12 (2011) 3608–3620). [ABSTRACT FROM AUTHOR]

We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e., the improved regularity of the integral term in Duhamel’s formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases of the modified Zakharov–Kuznetsov equation, the cubic and quintic nonlinear Schrödinger equation, and the quartic Korteweg–de Vries equation. [ABSTRACT FROM AUTHOR]

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed "odd viscosity", becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of a priori estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected. In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of good unknowns , which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $ \alpha\rightarrow1^- $. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.

In this paper, we delve into an analysis of the fifth-order Fokas-Olver-Rosenau-Qiao (FORQ) equation. We demonstrate local well-posedness for this equation under the condition that an initial data u 0 ∈ H s (R) for some s > 7 / 2 . We examine global existence and blow-up scenario. Additionally, we study the persistence properties of the equation in weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the "(sound) wave-part" of the data in Sobolev space and "transport-part" in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a "wave-part", that is, geometric wave equations for the velocity components, density and enthalpy, and a "transport-part", that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm H 2 + of the variables in the "wave-part" and the Hölder norm C 0 , 0 + of the variables in the "transport-part" can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption H 2 + is the optimal result for the variables in the "wave-part". Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory. [ABSTRACT FROM AUTHOR]

We study in this paper the local well‐posedness of classical solutions to the Cauchy problem for a kinetic self‐organized model of Cucker–Smale type for collective motions, which includes two cases of velocity alignment mechanisms: normalized and nonnormalized reorientation cases. The main concern is the a priori estimates for both cases. Our treatments rely on two key ingredients: One point is to deal with the possible singularity and high nonlinearity arising from the normalized/nonnormalized mean velocity, and the other is to control the growth with respect to the microscopic velocity variable in the whole space by employing weighted estimates. [ABSTRACT FROM AUTHOR]

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m , n) ∈ C k (R) ∩ W k , 1 (R) with k ∈ N ∪ { 0 } . This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂ t m (t , x) = ∂ x [ m (t , x) (u (t , x) - ∂ x u (t , x)) (u (- t , - x) + ∂ x (u (- t , - x))) ] , where m (t , x) = 1 - ∂ x 2 u (t , x) . Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class C k ∩ W k , 1 . Moreover, we derive criteria for blow-up of the local solution in this class. [ABSTRACT FROM AUTHOR]

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm. [ABSTRACT FROM AUTHOR]

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More precisely, we investigate some $L^2$ -estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot)\|_{L^2(G)}$ by further assuming the $L^1(G)$ -regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$ [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when α → 1 − . The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the well-posedness of a diffuse interface model of non-Newtonian two-phase flows in ℝ 3 . First, we prove the local existence of a unique strong solution provided that the initial velocity and initial concentration of two phases are sufficiently regular. Then, by using the energy estimates and standard continuity argument, we show that there exists a unique global strong solution provided that the initial velocity and initial concentration are sufficiently small. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problem for the ab-family of equations with cubic nonlinearities, which contains the integrable modified Camassa–Holm equation ( a = 1 3 , b = 2 ) and the Novikov equation ( a = 0 , b = 3 ) as two special cases. When 3 a + b ≠ 3 , the ab-family of equations does not possess the H 1 -norm conservation law. We give the local well-posedness results of this Cauchy problem in Besov spaces and Sobolev spaces. Furthermore, we provide a blow-up criterion, the precise blow-up scenario and a sufficient condition on the initial data for the blow-up of strong solutions to the ab-family of equations. Our blow-up analysis does not rely on the use of the conservation laws. [ABSTRACT FROM AUTHOR]

This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.

The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.

In this paper, the local well‐posedness (which means that the initial‐to‐solution map φ↦u$$ \varphi \mapsto u $$ is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ with 1≤p≤+∞,1≤r<+∞$$ 1\le p\le &#x0002B;\infty, 1\le r&lt;&#x0002B;\infty $$ and s>1+dp$$ s&gt;1&#x0002B;\frac{d}{p} $$ was established, and the solution of PEKS (Keller‐Segel system with β>0$$ \beta &gt;0 $$) converges to the solution of HEKS (Keller‐Segel system with β=0$$ \beta &#x0003D;0 $$) as the diffusion coefficient β→0$$ \beta \to 0 $$ in these Besov spaces was proved. In addition, we show that the solution of this parabolic‐elliptic chemotaxis‐growth system is local well‐posedness in the critical spaces Bp,11+dp(ℝd)$$ {B}_{p,1}&#x0005E;{1&#x0002B;\frac{d}{p}}\left({\mathbb{R}}&#x0005E;d\right) $$ but ill‐posed (not continuous) in Besov spaces B2,∞s(ℝd)$$ {B}_{2,\infty}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$. Moreover, we show a further continuity result that the initial‐to‐solution map φ↦u$$ \varphi \mapsto u $$ is Hölder continuous in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ was also obtained. [ABSTRACT FROM AUTHOR]

Our main aim in the current study is to examine the mathematical properties of a generalized 2-component Camassa-Holm system with a weakly dissipative term. Firstly, we acquire the theorem of well-posedness in locally for the generalized system with weak dissipation. Then, we demonstrate that this system can reveal the blow-up phenomenon. Finally, we acquire the theorem of global existence utilizing a method of the Lyapunov function. [ABSTRACT FROM AUTHOR]

In this paper, the sixth-order Boussinesq equation is studied. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the “bad” fourth term Δu in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$. This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with "good" linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in ℝ3$$ {\mathrm{\mathbb{R}}}&#x0005E;3 $$. This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with "good" linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

We investigate the initial value problem of a very general class of 3 + 1 non-Newtonian compressible fluids in which the viscous stress tensor with shear and bulk viscosity relaxes to its Navier–Stokes values. These fluids correspond to the non-relativistic limit of well-known Israel–Stewart-like theories used in the relativistic fluid dynamic simulations of high-energy nuclear and astrophysical systems. After establishing the local well-posedness of the Cauchy problem, we show for the first time in the literature that there exists a large class of initial data for which the corresponding evolution breaks down in finite time due to the formation of singularities. This implies that a large class of non-Newtonian fluids do not have finite solutions defined at all times. [ABSTRACT FROM AUTHOR]

In this paper, we consider the local well‐posedness of the 1D Green–Naghdi system. This system describes the evolution of water waves over an uneven bottom in the shallow water regime in terms of the water depth h and the horizontal velocity u. Using a Lagrangian formulation of this system on a Sobolev‐type diffeomorphism group, we prove local well‐posedness for (h,u)$(h,u)$ in the Sobolev space ([1−ξ]+Hs(R))×Hs+1(R),s>1/2$([1-\xi ]+H^s(\mathbb {R})) \times H^{s+1}(\mathbb {R}),\; s {>} 1/2$, where ξ:R→R$\xi :\mathbb {R}\rightarrow \mathbb {R}$ is the parameterization of the bottom and where we assume that the water surface has an equilibrium at height 1. This improves the present local well‐posedness range by one degree. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô's lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term. [ABSTRACT FROM AUTHOR]

This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates. [ABSTRACT FROM AUTHOR]

We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device are reformulated as a transmission problem related to the interaction between waves and a fixed partially immersed structure. By imposing the conservation of the total fluid-OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in Bocchi et al. (ESAIM Proc Surv 70:68–83, 2021). We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored. [ABSTRACT FROM AUTHOR]

This paper is concerned with the local well-posedness, wave breaking, blow-up rate and global existence of solutions for a weakly dissipative coupled Camassa–Holm system. First, by using Kato's theory, we obtain the local well-posedness of solutions in the Sobolev space. We then derive the wave breaking mechanism of solutions, and obtain the blow-up rate of blow-up solutions. Finally, we establish the global existence of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain T d , for space dimensions d = 2 , 3 . We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case k ≥ 0 ; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces H s , for any s > 1 + d / 2 . We expect this regularity to be optimal, due to the degeneracy of the system when k ≈ 0 . We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations. [ABSTRACT FROM AUTHOR]

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ∂ t u − Δ u = | u | α u , α > 0 with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of life-span are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for life-span using dilation methods and we give new life-span estimates for Hardy-Hénon parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces B p , r s with s > 1 + 1 p and s = 1 + 1 p , r = 1 , p ∈ [ 1 , ∞). Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces. [ABSTRACT FROM AUTHOR]

The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed in the linearization and we do not assume the Friedrich's symmetrizability of the system. This coupling results in linear energy estimates that prevent the application of Banach's contraction principle. A metric fixed point theorem is developed in order to conclude the local existence and uniqueness of solutions. We show that the boundedness in the high norm and contraction in the low norm can be incorporated into the formulation of the fixed point by introducing the notion of a closed extension of the solution map. We apply our results to the Cattaneo-Christov system for viscous compressible fluid flow, a system of equations whose inviscid part is not hyperbolic. [ABSTRACT FROM AUTHOR]