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We remove (α , λ) -discretely self-similar blow up for solutions to the Euler equations for α ≥ 3 2 , for which we allow sublinear growth for the profile. More precisely, we show that there are only spatial constant (α , λ) -discretely self-similar solutions v = c (t) having the sublinear growth at the infinity. For the proof, we establish a new a priori L loc 2 (R 3) estimate for the 3D Euler equations. [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics (MHD) system in R 3. Let (v , B , p) be a smooth solution to the stationary MHD equations in R 3. We show that if there exist smooth matrix valued potential functions Φ , Ψ such that ∇ ⋅ Φ = v and ∇ ⋅ Ψ = B , whose L 6 mean oscillations have certain growth condition near infinity, namely ⨍ B (r) | Φ − Φ B (r) | 6 d x + ⨍ B (r) | Ψ − Ψ B (r) | 6 d x ≤ C r ∀ 1 < r < + ∞ , then v = B = 0 and p =constant. With additional assumption of r − 8 ∫ B (r) | B − B B (r) | 6 d x → 0 as r → + ∞ , similar result holds also for the Hall-MHD system. [ABSTRACT FROM AUTHOR]

In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 (1) 1 (R n) , which corresponds to a critical case of the generalized Campanato spaces L q (N) s (R n). The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 (R n) ↪ L 1 (1) 1 (R n) ↪ C 0 , 1 (R n) , where B ∞ , 1 1 (R n) and C 0 , 1 (R n) are the Besov space and the Lipschitz space respectively. In particular L 1 (1) 1 (R n) contains non- C 1 (R n) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 (1) 1 (R n) , for which the solution to the Euler equations blows up in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in R 3 , having the diffusion term A p (u) = ∇ · (| D (u) | p - 2 D (u)) with D (u) = 1 2 (∇ u + (∇ u) ⊤) , 3 / 2 < p < 3 . In the case 3 / 2 < p ≤ 9 / 5 , we show that a suitable weak solution u ∈ W 1 , p (R 3) satisfying lim inf R → ∞ | u B (R) | = 0 is trivial, i.e., u ≡ 0 . On the other hand, for 9 / 5 < p < 3 we prove the following Liouville type theorem: if there exists a matrix valued function V = { V ij } such that ∂ j V ij = u i (summation convention), whose L 3 p 2 p - 3 mean oscillation has the following growth condition at infinity, ∫ - B (r) | V - V B (r) | 3 p 2 p - 3 d x ≤ C r 9 - 4 p 2 p - 3 ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the L 2 norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale. [ABSTRACT FROM AUTHOR]

In this paper we obtain new local blow-up criterion for smooth axisymmetric solutions to the three dimensional incompressible Euler equation. If the vorticity satisfies ∫ 0 t ∗ (t ∗ - t) ‖ ω (t) ‖ L ∞ (B (x ∗ , R 0)) d t < + ∞ for a ball B (x ∗ , R 0) away from the axis of symmetry, then there exists no singularity at t = t ∗ in the torus T (x ∗ , R) generated by rotation of the ball B (x ∗ , R 0) around the axis. This implies that possible singularity at t = t ∗ in the torus T (x ∗ , R) is excluded if the vorticity satisfies the blow-up rate ‖ ω (t) ‖ L ∞ (T (x ∗ , R)) = O 1 (t ∗ - t) γ as t → t ∗ , where γ < 2 , and the torus T (x ∗ , R) does not touch the axis. [ABSTRACT FROM AUTHOR]

We consider the Navier–Stokes equations in R 3 subject to the initial condition with initial velocity field in L loc 2 (R 3) such that lim sup R → + ∞ R − 1 ‖ u 0 ‖ L 2 (B (R)) < + ∞. Our aim is to show the local existence of a weak solution, global existence of weak solution if C = 0 and the partial regularity in the sense of Caffarelli–Kohn–Nirenberg. [ABSTRACT FROM AUTHOR]

We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations. [ABSTRACT FROM AUTHOR]

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution v∈L∞(-1,0;L2(B(x0,r)))∩Lloc∞(-1,0;W1,∞(B(x0,r))) of the 3D Euler equations, where B(x0,r) is the ball with radius r and the center at x0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as t→0 are small enough, then ∇v(·,t) does not blow-up at t = 0 in B(x0, r). [ABSTRACT FROM AUTHOR]

We provide a local compactness lemma, that can be used for various models of incompressible fluids in a general domain, such as generalized Newtonian fluids. In particular, it helps for the proof of the existence of weak solution to those systems, when it comes to carrying out the passage to limit for sequences of weak solution to a corresponding approximate system. The key argument of our approach is the use of the local pressure decomposition in order to avoid the construction of a global pressure. [ABSTRACT FROM AUTHOR]

We prove the existence of a forward discretely self-similar solutions to the Navier–Stokes equations in R 3 × ( 0 , + ∞ ) for a discretely self-similar initial velocity belonging to L l o c 2 ( R 3 ) . [ABSTRACT FROM AUTHOR]

We study the scenario of discretely self-similar blowup for Navier–Stokes equations. We prove that at the possible blow-up time such solutions have only one point singularity. In case of the scaling parameter λ near 1, we remove the singularity. [ABSTRACT FROM PUBLISHER]

We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $${\mathbb{R}^2}$$ . We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations. [ABSTRACT FROM AUTHOR]

We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ru̇žička-Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $${L^{p, \infty} (\mathbb{R}^3)}$$ with $${p > \frac{3}{2}}$$ . [ABSTRACT FROM AUTHOR]

In this paper we prove three different Liouville type theorems for the steady Navier–Stokes equations in R 3 . In the first theorem we improve logarithmically the well-known L 9 2 ( R 3 ) result. In the second theorem we present a sufficient condition for the trivially of the solution ( v = 0 ) in terms of the head pressure, Q = 1 2 | v | 2 + p . The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee v = 0 . [ABSTRACT FROM AUTHOR]

The present paper deals with the problem of local regularity of weak solutions to the Navier–Stokes equation in Ω × ( 0 , T ) with forcing term f in L 2 . We prove that u is strong in a sub-cylinder Q r ⊂ Ω × ( 0 , T ) if two velocity components u 1 , u 2 satisfy a Serrin-type condition. [ABSTRACT FROM AUTHOR]

We study partial regularity of weak solutions of the three-dimensional (3D) valued nonstationary Hall magnetohydrodynamics equations on R². In particular, we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two. [ABSTRACT FROM AUTHOR]

We study partial regularity of suitable weak solutions of the steady Hall magnetohydrodynamics equations in a domain $${\Omega \subset \mathbb{R}^3}$$ . In particular, we prove that the set of possible singularities of the suitable weak solution has Hausdorff dimension at most one. Moreover, in the case $${\Omega=\mathbb{R}^3}$$ , we show that the set of possible singularities is compact. [ABSTRACT FROM AUTHOR]

In the present paper we prove that a weak Leray solution to the Navier-Stokes equations in R³ × (0, T] is regular provided the gradient of one component belongs to the Prodi-Serrin class ∇u³ Є L4(0, T; L²). [ABSTRACT FROM AUTHOR]

We work on the boundary regularity for weak solutions to the stationary Stokes type equations with shear dependent viscosity. Using a weighted estimate near the boundary, we obtain the Hölder continuity of the solution for the shear thickening fluid without the convection term. [ABSTRACT FROM AUTHOR]

We address the existence of strong solutions to a system of equations of motion of an incompressible non-Newtonian fluid. Our aim is to prove the short-time existence of strong solutions for the case of shear thickening viscosity, which corresponds to the power law ν ( D ) = ∣ D ∣ q − 2 ( 2 < q < + ∞ ) . In particular, we find that global strong solutions exist whenever q > 2.23 … . The results are obtained by flattening the boundary and by using the difference quotient method. Near the boundary, we use weighted estimates in the normal direction. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Liouville-type theorem for the stationary MHD and the stationary Hall-MHD systems. Assuming suitable growth condition at infinity for the mean oscillations for the potential functions, we show that the solutions are trivial. These results generalize the previous results obtained by two of the current authors in Chae and Wolf (J Differ Equ 295:233–248, 2021). To prove our main theorems, we use a refined iteration argument. [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in R 3 with the viscous part of the stress tensor A p (u) = div (| D (u) | p − 2 D (u)) , where D (u) = 1 2 (∇ u + (∇ u) ⊤) and 9 5 < p < 3. We consider a weak solution u ∈ W l o c 1 , p (R 3) and its potential function V = (V i j) ∈ W l o c 2 , p (R 3) , i.e. ∇ ⋅ V = u. We show that there exists a constant s 0 = s 0 (p) such that if the L s mean oscillation of V for s > s 0 satisfies a certain growth condition at infinity, then the velocity field vanishes. Our result includes the previous results [5,6] as particular cases. [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary Navier–Stokes equations in R 3 . Let V = (V ij) be a potential function of a smooth solution u, which means u = ∇ · V . We show that if there exists 3 < s < + ∞ such that the L s mean oscillation of the potential function has certain growth condition near infinity, namely 1 | B (r) | ∫ B (r) | V - V B (r) | s d x ≤ C r min { s - 3 3 , s 6 } ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]

Highlights • Carbon ions induced more severe DNA double-strand breaks than photons. • Carbon ion-induced double-strand breaks were repaired slower and less efficiently. • Carbon ions led to stronger cell cycle arrest and more apoptosis than photons. • PTEN−/− cells were more dependent on homologous recombination after carbon ions versus photons. Abstract Background and purpose Carbon ion radiotherapy is a promising therapeutic option for glioblastoma patients due to its high physical dose conformity and greater biological effectiveness than photons. However, the biological effects of carbon ion radiation are still incompletely understood. Here, we systematically compared the biological effects of clinically used carbon ion radiation to photon radiation with emphasis on DNA repair. Materials and methods Two human glioblastoma cell lines (U87 and LN229) were irradiated with carbon ions or photons and DNA damage response was systematically analyzed, including clonogenic survival, induction and repair of DNA double-strand breaks (DSBs), cell cycle arrest and apoptosis or autophagy. γH2AX foci were analyzed by flow cytometry, conventional light microscopy and 3D superresolution microscopy. Results DSBs were repaired delayed and with slower kinetics after carbon ions versus photons. Carbon ions caused stronger and longer-lasting cell cycle delays, predominantly in G2 phase, and a higher rate of apoptosis. Compared to photons, the effectiveness of carbon ions was less cell cycle-dependent. Homologous recombination (HR) appeared to be more important for DSB repair after carbon ions versus photons in phosphatase and tensin homolog (PTEN)-deficient U87 cells, as opposed to PTEN-proficient LN229 cells. Conclusion Carbon ions induced more severe DSB damage than photons, which was repaired less efficiently in both cell lines. Thus, carbon ion radiotherapy may help to overcome resistance mechanisms of glioblastoma associated with DNA repair for example in combination with repair pathway-specific drugs in the context of personalized radiotherapy. [ABSTRACT FROM AUTHOR]

Abstract: We study a system of partial differential equations describing the steady flow of a heat conducting incompressible fluid in a bounded three dimensional domain, where the right-hand side of the momentum equation includes the buoyancy force. In the present work we prove the existence of a weak solution under both the smallness and a sign condition on physical parameters and which appear on the right hand side. [Copyright &y& Elsevier]