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Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities
Michael Winkler
TL;DR
The paper addresses global solvability for chemotaxis systems with tensor-valued sensitivities, where a useful gradient-like structure is generally unavailable. It combines logarithmic a priori estimates and compactness for regularized solutions with a mild generalized solution concept, proving global generalized existence in every dimension without smallness restrictions on the initial data.
Problem
Tensor-valued sensitivities can destroy the gradient-like structure used in scalar-sensitivity chemotaxis, while existing global-existence results were restricted to two dimensions and small initial signal concentration.
Method
The paper derives compactness properties from logarithmic gradient estimates for regularized solutions and defines a generalized solution using weak and very weak formulations.
Results
For every n ≥1, suitably smooth nonnegative initial data yield at least one global generalized solution as an almost-everywhere limit of smooth classical solutions to regularized problems.
Takeaways & Limitations
Global generalized solvability extends to bounded smooth domains in arbitrary spatial dimension without smallness assumptions on the initial data.
Takeaways & Limitations
The logarithmic estimate provides only one-sided control for a key gradient term and does not directly support passage to a standard weak formulation.
Abstract
from arXiv · showhide
A chemotaxis system possibly containing rotational components of the cross-diffusive flux is studied under no-flux boundary conditions in a bounded domain $Ω\subset R^n$, $n\ge 1$, with smooth boundary, where the evolution of the signal is determined by consumption through cells. In contrast to related Keller-Segel-type problems with scalar sensitivities, in presence of such tensor-valued sensitivities this system in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal $L^2$ estimate for the gradient of the logarithm of the cell density as a starting point, we derive a series of compactness properties of solutions to suitably regularized versions of the system. Motivated by these, we develop a generalized solution concept which requires solutions to satisfy very mild regularity hypotheses only. On the basis of the above compactness properties, it is finally shown that within this framework, under a mild growth assumption on the sensitivity matrix and for all sufficiently regular nonnegative initial data, the corresponding initial-boundary value problem possesses at least one global generalized solution. This extends known results which in the case of such general matrix-valued sensitivities provide statements on global existence only in the two-dimensional setting and under the additional restriction that the initial signal concentration be suitably small.
1 Introduction
The paper studies globally solvable chemotaxis systems with tensor-valued sensitivities and signal consumption, where rotational fluxes remove the gradient-like structure available in scalar-sensitivity models. It establishes global generalized solutions in every spatial dimension without smallness restrictions on initial data, using regularization, compactness, and logarithmic estimates.
- Model and motivation: Tensor-valued sensitivities allow chemotactic motion not necessarily parallel to the signal gradient, including possible rotational flux components.The sensitivity is therefore modeled by a matrix-valued function rather than a scalar coefficient.
- Model and motivation: Signal consumption through cells motivates the central question of whether the resulting no-flux initial-boundary value problem can be solved globally in time.The question is linked to whether finite-time mass accumulation occurs.
- Existing gap: General matrix-valued sensitivities apparently lack the gradient-like structure that supports a priori regularity estimates in related systems.Before this work, global bounded solutions were available in bounded convex planar domains only under a sufficiently small initial signal concentration.
- Main framework: The paper assumes a bounded smooth domain in R^n with n ≥1 and studies suitably general cell-diffusion and sensitivity functions.The functions f and S are required to satisfy the paper’s stated structural assumptions.
- Main results: For nonnegative u0 ∈ C0(Ω̄) and v0 ∈ W 1,∞(Ω), the problem has at least one global generalized solution obtained as an almost-everywhere limit of smooth regularized solutions.The result applies under the assumptions on f and S and imposes no spatial-dimension restriction.
- Main results: The analysis uses a spatio-temporal estimate for ∇ln(uε+1), but its strong damping prevents the estimate from directly yielding the regularity needed for a standard weak formulation.The resulting one-sided control motivates a generalized solution concept based on ln(u+1) and an integral inequality.
2 A generalized solution concept
The generalized solution concept combines a weak formulation for the signal with a very weak supersolution formulation for the cell density. It is designed to retain the compactness available from logarithmic estimates while remaining compatible with classical solutions.
- Definition of generalized solutions: The formulation is built around the cross-diffusive equation, whose main difficulty is passing to the limit in terms involving the logarithmic transform of u.The relevant weak identity is formulated using test functions with the regularity specified for the generalized problem.
- Definition of generalized solutions: The cell-density formulation uses a strictly increasing φ and requires only mild regularity for u and v.The eventual existence proof chooses φ(s) = ln(s + 1).
- Definition of generalized solutions: The supersolution property is complemented by nonincrease of the total cell mass, which controls the time derivative from above in the generalized setting.This mass condition plays a subsolution-like role for the cell equation.
- Definition of generalized solutions: The signal component is treated as a global weak solution, while the cell equation is handled through a global very weak φ-supersolution.The pair is called a global generalized solution when these two componentwise requirements hold together with the prescribed nonnegativity and mass condition.
- Compatibility with classical solutions: For sufficiently regular nonnegative u and v, every global generalized solution is also a classical solution of the original system.This compatibility is established under continuity and C2,1 regularity in space-time.
3 Global solutions of regularized problems
The regularized chemotaxis problems use spatial and density cutoffs in the sensitivity, and each admits a global nonnegative classical solution. Basic mass and signal estimates follow from integration, testing, and the maximum principle.
- Regularization: The regularization defines Sε by multiplying the original sensitivity by spatial and density cutoff functions that converge monotonically to one.The cutoffs satisfy 0 ≤ ρε, χε ≤ 1 and approach one as ε decreases.
- Global classical solvability: For every ε ∈ (0, 1), the regularized problem has a global nonnegative classical solution.Global solvability follows after local existence because the regularized sensitivity vanishes for sufficiently large cell density.
- Basic estimates: The regularized system preserves the initial total cell mass for all positive times.This identity is obtained by integrating the first equation over the domain.
- Basic estimates: The signal remains bounded through the maximum principle, using nonnegativity of the cell density and consumption rate.The estimate is derived from the second equation and the assumed nonnegativity of f.
- Basic estimates: Testing the signal equation with vε and with a constant yields additional dissipation and time-integrated bounds.Both estimates rely on the nonpositive contribution of the consumption term.
4 Estimates for ln(uε + 1)
The analysis obtains an integral estimate for the gradient of ln(uε + 1), then combines it with time-derivative control to establish compactness. This compactness is strong in space-time and weak in the Sobolev topology.
- A priori estimates: The first estimate controls the space-time gradient of ln(uε + 1), providing the main a priori bound for the cell density.The estimate is derived by testing the first regularized equation with 1/(uε+1).
- A priori estimates: Integrating the logarithmic estimate in time yields a bound with growth at most linear in the time horizon T.The resulting estimate is recorded as (4.3).
- Estimate mechanism: The proof uses integration by parts, Cauchy–Schwarz, Young’s inequality, and the embedding W_0^{m,2}(Ω) into L∞(Ω) when m > n/2.These steps control the terms involving the sensitivity and signal gradient.
- Compactness: For every T > 0, ln(uε + 1) is relatively compact in L2((0,T); W1,2(Ω)) weakly and in L2(Ω×(0,T)) strongly.The strong compactness follows from an Aubin–Lions argument using time-derivative bounds.
5 Compactness properties of (vε)ε∈(0,1)
The signal approximations form a strongly precompact family in space-time. This follows by combining spatial Sobolev bounds with time-derivative control and an Aubin–Lions argument.
- Signal compactness: For every T > 0, (vε)ε∈(0,1) is relatively compact in L2(Ω×(0,T)) with respect to the strong topology.The proof uses boundedness in L2((0,T); W1,2(Ω)) together with time-derivative control.
- Signal compactness: The compactness argument applies a variant of the Aubin–Lions lemma to the regularized signal sequence.Testing the signal equation against Sobolev functions provides the required temporal estimates.
6 Precompactness of (uεf(vε))ε∈(0,1)
Strong signal compactness and uniform estimates yield weak compactness of the consumption term uεf(vε), which is needed when passing to the limit in the signal equation and taxis system.
- Motivation: Strong precompactness of ∇vε is required to pass to the limit in the taxis term.The analysis therefore supplements weak convergence of the signal gradient with stronger compactness properties.
- Consumption term: The inhomogeneity hε := uεf(vε) is treated as the source term in the semilinear heat equation for vε.The subsequent estimates seek superlinear integrability and compactness for this source.
- Weak compactness: The required equi-integrability is obtained from the available estimates for uε and vεf(vε), together with bounds on f(vε).The argument uses the uniform signal bound and estimates established earlier.
- Weak compactness: For each T > 0, the family (uεf(vε))ε∈(0,1) is relatively compact in L1(Ω×(0,T)) with respect to the weak topology.This conclusion follows from equi-integrability and Pettis’ theorem.
7 Passing to the limit. Solution properties of v
A subsequence of regularized solutions converges to a limit pair whose signal component v is shown to satisfy the weak formulation of its equation.
- A subsequence εj→0 is extracted so that regularized solutions converge to nonnegative limit functions u and v.
- The limit signal v is a weak solution of the corresponding signal equation.
- The verification uses convergence of nonlinear terms and passage to the limit in the tested regularized signal equation.
8 Strong precompactness of (∇vε)ε∈(0,1)
The analysis establishes strong compactness of the signal gradients by combining temporal averaging, energy inequalities, and entropy-based lower bounds.
- The argument combines an upper estimate with the entropy identity and a lower bound obtained through lower semicontinuity.
- A limit inequality for v is established first, outside a null set of times, and then used to prove gradient convergence.
- The temporal averaging lemmas provide pointwise and weak convergence properties needed in the limiting procedure.
- Temporal averages and carefully chosen test functions are used to derive estimates for the regularized signal gradients.
- The target compactness result is strong convergence ∇vε→∇v in L2(Ω×(0,T)) along a subsequence.
9 Solution properties of u. Proof of Theorem 1.1
The limiting cell density u satisfies the generalized solution requirements, and combining these properties with those of v proves global generalized existence.
- The limit u belongs to L∞((0,∞);L1(Ω)) and satisfies the mass inequality required by the generalized solution framework.
- The limit u is a global very weak φ-supersolution of the cell equation.
- Passing to the limit uses the regularized equation, uniform bounds, strong convergence, and lower semicontinuity.
- The proof of global existence follows by combining the properties of u and v with the subsequence construction.
10 Appendix
The appendix records approximation and convergence lemmas used to justify temporal averaging and strong convergence arguments.
- Steklov averages converge almost everywhere to the underlying function as the averaging interval tends to zero.
- For p>1, Steklov averages converge weakly in Lp, while bounded averages converge weak-star in L∞.
- A nonnegative sequence with weak L1 convergence and almost-everywhere convergence satisfies a strong-convergence criterion.
- A dominated-convergence lemma supplies an additional limit passage for bounded and square-integrable sequences.